electromagnetic waves in stratified media. including supplemented material

OTHER TITLES IN THE SERIES IN

ELECTROMAGNETIC WAVES

Electromagnetic Diffraction and Propagation Problems — FOCK Ionospheric Sporadic-E — SMITH and MATSUSHITA (Editors) The Scattering of Electromagnetic Waves from Rough Surfaces — BECKMANN and SPIZZICHINO

Electromagnetic Scattering — KERKER Electromagnetic Theory and Antennas — JORDAN (Editor) The Propagation of Electromagnetic Waves in Plasmas — GINZBURG

8 Tropospherìc Radiowave Propagation beyond the Horizon — Du CASTEL Dipole Radiation in the presence of conducting Half-space — BANOS

Vol. 10 Electrical Methods in Geophysical Prospecting — KELLER and FRISCHKNECHT

Vol. 11 Electromagnetic Wave Theory — BROWN (Editor) Vol. 12 The Plane Wave Spectrum Representation of Electromagnetic Fields —

CLEMMOW

Vol. 13 Basic Theory of Waveguide Functions and introductory Microwave -Network Analysis — KERNS and BEATTY

Vol. 14 V.L.F. Radio Engineering — WATT Vol. 15 Antennas in Inhomogeneous Media — GALEJS

Vol. Vol. Vol.

Vol. Vol. Vol. Vol. Vol.

1 2 4

5 6 7 8 9

ELECTROMAGNETIC WAVES IN

STRATIFIED MEDIA REVISED EDITION INCLUDING

SUPPLEMENTED MATERIAL

by

JAMES R. WAIT, Fellow, I.E.E.E. Professor of Electrical Engineering, University of Colorado, Boulder, Colorado, USA

P E R G A M O N PRESS OXFORD · NEW YORK · TORONTO

SYDNEY · BRAUNSCHWEIG

Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523

Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street,

Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig

Copyright © 1962 and 1970 Pergamon Press Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of

Pergamon Press Ltd.

First edition 1962 Second edition 1970

Library of Congress Catalog Card No. 79-100362

Printed in Great Britain by age Bros. (Norwich) Ltd. 08 0066364

TO GERTRUDE, LAURA AND GEORGE

PREFACE TO SECOND EDITION IN this second edition, I have made a number of changes in order to call attention to recent investigations. I am particularly indebted to J. Heading and D. B. Large who brought a number of corrections and typographical errors to my attention.

In order to bring the reader up to date, I have added a number of my recent papers (co-authored in some cases). These describe recent work on electromagnetic waves in stratified media. Numerous references to other related investigations are included in the bibliographies of these appended papers.

I wish to thank Mrs. Eileen Brackett for her continued assistance and A. J. Steel for his sustained interest in the present series of monographs.

Boulder J. R. WAIT

PREFACE TO FIRST EDITION THIS book was written primarily to be used as a reference; however, the material was also presented in graduate courses in Wave Propagation at the University of Colorado and the Technical University of Denmark. Although the book is basically of a theoretical nature, numerous numerical examples and references to experimental data are included. Comprehension of the material requires knowledge of electromagnetism and mathematical analysis at the undergraduate level.

Much of the subject matter is based on the author's own investigations. Some of these have been published previously in Technical Notes and in the Journal of Research of the National Bureau of Standards over the period 1956-1962. These portions of the work were carried out at the Boulder Laboratories of the National Bureau of Standards with support extended by the Cambridge Research Laboratories of the U.S. Air Force and the Advanced Research Projects Agency.

It is a pleasure to thank the following individuals: H. Bremmer, K. G. Budden, D. D. Crombie, A. G. Jean, S. Maley, K. A. Norton, W. L. Taylor, A. D. Watt and F. J. Zucker for numerous and helpful discussions; K. P. Spies for extensive assistance in the calculations and critical readings of the manuscript; Mrs. Eileen Brackett for her painstaking care in the typing and preparation of the manuscript, and Mrs. L. C. Walters for helpful editorial comments.

I am particularly indebted to my wife, Gertrude Wait who prepared the subject index and helped with proof reading. Finally, I wish to thank the publisher, I. R. Maxwell and his able assistants, J. P. H. Connell and Miss Felicity Slatter for the careful attention they gave to the book during production and printing.

Boulder J. R. WAIT

xii

Chapter T

GENERAL INTRODUCTION

1. SCOPE OF THE SUBJECT

This book is concerned with electromagnetic waves in media whose properties vary in one particular direction. The variation may consist of abrupt or continuous changes. In this sense the media may be generally classified as stratified. This is an idealization of many situations which occur in nature. For example, there is a tendency for our terrestrial atmosphere to occur in horizontal layers. Similarly, the properties of the earth's crust do not vary significantly in the horizontal direction. Consequently, a proper understanding of wave phenomena in stratified media is of great practical importance. The major theoretical task is to find solutions of Maxwell's field equations which satisfy the appropriate conditions imposed by the various boundary conditions.

Rather than present a sterile and formal treatment couched in the language of the mathematician, a physical approach is adopted. Furthermore, applications to the real world are used as illustrations of the theoretical principles. Special emphasis is given to radio waves in the frequency range from 3 to 30 kc/s which is described as v.l.f. (very low frequency). These frequencies correspond to wavelengths ranging from 100 to 10 km. In such cases the ionized layers in the upper atmosphere do indeed behave as stratified media since the scale of the irregularities is usually small compared with the wavelength. In addition, many comparisons are made with experimental data on v.l.f. radio transmissions since only in this way can we ascertain if the adopted theoretical model has any real significance. An important feature of these waves is their low attenuation which enables communications at global distances. In addition, they exhibit remarkably stable phase characteristics.

To avoid any misconception that the stratified models are only applicable to v.l.f. radio propagation, the results are also applied to e.l.f. (extremely low frequency) which covers the range from 3 kc/s down to about 1.0 c/s. A discussion of the mode theory of tropospheric radio propagation at v.h.f. (very high frequency) is also included. A more detailed summary of the book is given in Section 3 of this chapter.

The book contains extensive references to published papers. For convenience of the reader each chapter is followed by its own list of references. Also included is a list of "additional" references which are germane to the subject of the chapter, but are not specifically cited.

1

2 Electromagnetic Waves in Stratified Media

2. NOTATION AND SOME BASIC IDEAS

Through the book the rationalized MKS system of units is used. Since attention is only confined to linear phenomena, the electrical properties of the medium (or media) can also be defined in terms of the constants

e, the dielectric constant (Fm)

σ, the conductivity (mho/m)

μ, the permeability (H/m)

In some cases these "constants" will be functions of the coordinate. Locally, however, they are always considered constant. In later chapters e and σ are regarded as tensors to account for the anisotropie characteristics of magneto-plasmic media.

Nearly always the time factor in this book is exp(+*W) when ω is the angular frequency and t is the time. Consequently, the actual electric field e{i) is related to the complex phasor E by

e(t) = Real part of (Eei(ut).

To explain some of the basic notation a very short exposition of plane electromagnetic waves in a homogeneous medium is presented.

Ohm's law in the complex form is

J = (σ + /€ω)Ε (1)

where J is the current density vector and E is the electric field vector. The dimensions of J are A/m2 and those of E are V/m. The analogous relation for magnetic quantities is

Β = μ Η (2)

where B is the magnetic vector density and H is the magnetic vector intensity. The dimensions of B are wb/m2 and those of H are A/m.

In source-free media the above vector quantities are related by

curl E = - ψωΆ (3)

and

curl H = (σ + /€ω)Ε. (4)

These are Maxwell's equations.

For a homogeneous medium

curl curl E = grad div E — div grad E = — ψω{σ + /βω)Ε. (5)

General Introduction 3

Since div E = 0, this can be reduced to

(V2 _ y 2 ) E = 0 (6)

where V2 = div grad is the Laplacian operator (which operates on the rectangular components of E) and y2 = ίμω(σ + ieœ). The quantity y is defined as the propagation constant.

As a simple preliminary problem, the fields are assumed not to vary in either the x or y directions with reference to a conventional coordinate system (*, y, z). Furthermore, the electric field is taken to have only an x component Ex. Therefore, Eq. (6) reduces to

and the solutions are e+?z and e-?2. Therefore, the general solution is

Ex = Ae?* + B Q-y* (8)

where A and B are constants. The magnetic field component then has only a y component given by

1 dEx H = - - -? = - η-ΐΑ &ζ + η~ιΒ e-yz (9) y Ιμω dz

where η = [ΐμω/(σ + ΐ€ω)]* is by definition the characteristic impedance of the medium for plane wave propagation. Remembering that the time factor is eiwi, it can be seen that the term Bt~yz is a wave travelling in the positive z direction with a diminishing amplitude, and the term A&z is a wave travelling in the negative z direction with a diminishing amplitude. The quantity η is thus equal to the complex ratio of the electric and magnetic field components in the x and y directions, respectively, for plane waves in an unbounded homogeneous medium.

The quantities defined by

y = [ϊμω(σ + 7€ω)]* and η = [ / ρ / ( σ + lew)]4

are sometimes called the secondary constants. In the case of free space

e = €0 = 8.854 X IO"12 F/m

μ = μο = 4π X 10~7 H/m

σ = 0


and then y = ik where k = (ε0μο)έω = 2π/λ and λ is the wavelength. Furthermore

η = ηο = (μο/*ο)έ = 120τ7 Ω.

3. SUMMARY OF SUBJECT MATTER IN FOLLOWING CHAPTERS

In Chapter II, a general analysis for the electromagnetic response of a plane stratified medium consisting of any number of parallel homogeneous layers is presented. The solution is first developed for plane-wave incidence and then generalized to both cylindrical and spherical-wave incidence. Numerical results for interesting special cases are presented and discussed. The application of the results to surface-wave propagation over a stratified ground is considered in some detail.

In Chapter III, the reflection of electromagnetic waves from planar stratified media is discussed in a relatively concise manner. Attention is confined to special forms of conductivity (or dielectric constant) profiles which lead to solutions in terms of Bessel functions. Most of the results, in equivalent forms, have already appeared in the literature. The chapter is essentially a consolidation of known solutions and their (sometimes novel) applications to the determination of reflection coefficients.

In Chapter IV, the oblique reflection of plane electromagnetic waves from a continuously stratified medium is considered. Various approximate procedures are employed. For the slowly varying profiles, the WKB method and its extension are most suitable. However, certain modifications must be made when the ray has a turning point. It is shown that under this situation, the phase integral method is applicable. Finally, when the medium is rapidly varying, an alternative approach is adopted which is particularly suitable at low frequencies.

In Chapter V, the basic theory of wave propagation around a sphere is given. By utilizing the concept of surface impedance, the derivations are greatly simplified. The formal solution in the form of a slowly convergent series is transformed to a more useful form by following the method of Watson. A further transformation is made in order to obtain a formula which is suitable for very small curvature of the surface. The influence of a concentric in-homogeneous atmosphere, with a smooth and monotonically varying profile, is also considered.

In Chapter VI, a self-contained treatment of the waveguide-mode theory of propagation is presented. The model of a flat earth with a sharply bounded homogeneous isotropie ionosphere is treated for both vertical and horizontal dipole excitation. The properties of the modes are discussed in considerable detail. The influence of earth curvature is also considered by reformulating the problem using spherical wave functions of complex order. The modes in such a curved guide are investigated and despite the initial complexity of the


general solution, many interesting and limiting cases are treated in simple fashion to yield useful and convenient formulas for v.l.f. propagation.

In Chapter VII, the mode theory of v.l.f. propagation is considered from a somewhat different viewpoint. Taking note of the fact that the important modes for long distance propagation are near grazing, suitable approximate forms are introduced at the outset rather than at the end of the analysis. The derived formulas are used to obtain numerical results for the attenuation, phase velocity and excitation of the dominant modes in v.l.f. radio propagation. The physical and practical significance of these results are described.

In Chapter VIII, the influence of a steady or d.c. magnetic field on reflection from ionized media is considered in some detail. Initially, the geometry is chosen so that the wave propagation is essentially transverse to the d.c. magnetic field. Under this condition, the analogy with non-uniform transmission line theory is exploited to obtain specific results in a relatively simple manner. For an arbitrarily oriented magnetic field, the formalism is a great deal more complicated. However, meaningful results can be obtained when certain approximate procedures are adopted. The theory in this chapter is used to obtain reflection coefficients for a sharply bounded ionosphere with the inclusion of the terrestrial magnetic field.

In Chapter IX, approximate techniques for solving the v.l.f. modal equation are described. Essentially, the idea is to expand the logarithm of the reflection coefficient in a power series of C, the cosine of the angle of incidence. Using this approach, curves of attenuation rates for a wide variety of conditions are obtained. Both the terrestrial magnetic field and the earth's curvature are included in the analysis. The results are then compared with experimental data obtained from many sources beginning with the early field strength data of Round et al [1925] to the most recent 'sferics" data of Taylor [1961].

In Chapter X, the mode theory of propagation of electromagnetic waves at extremely low frequencies (1.0-3000 c/s) is considered. Starting with the representation of the field as a sum of modes, approximate formulas are presented for the attenuation and phase constants. Certain alternate representations of the individual modes are mentioned. These are used as a basis for describing the physical behavior of the field at large distances from the source, particularly near the antipode of the source. At the shorter distances, where the range is comparable to the wavelength, the spherical-earth mode series is best transformed to a series involving cylindrical wave functions. This latter form is used to evaluate the near field behavior of the various field components.

The effect of the earth's magnetic field is also evaluated using both quasi-longitudinal and transverse-type approximations.

In Chapter XI, the physical connections between mode and ray theory are developed. The starting point is the exact representation of the field in terms of an integral. It is shown, when the integrand is expanded, that the individual


terms can be identified with ray contributions. The formulation permits a straightforward discussion of focussing in horizontally stratified media.

Finally, in Chapter XII, the theory of propagation in a spherically stratified medium is considered. The profile of modified refractive index M(h) is allowed to have a minimum with height. Particular attention is paid to the case when M(h) may be approximated by a parabolic form. Here the analysis closely follows the recent work of Fock, Weinstein, and Belkina in the U.S.S.R. Other approaches such as the Eckersley phase integral method and Furry's mode theory for the bilinear profile are also considered.

GENERAL REFERENCES

The following are selected texts or review papers which deal with subject matter related to the present book. They are recommended as supplementary reading. AL'PERT, IA. L., GINZBURG, V. L. and FEINBURG, E. L. (1953) Radio wave propagation,

State Printing House for Technical-Theoretical Literature, Moscow. AL'PERT, IA. L. (1960) Ionospheric Propagation of Radio Waves, Acad. of Sciences of USSR,

Moscow. (Translation of chapter on Long Waves available as Translation T5-60 from the National Bureau of Standards, Boulder, Colorado, USA.)

BANOS, A., (1965). Electromagnetic Fields of a Dipole in a Conducting Half-space, Monograph on Electromagnetic Waves, Pergamon Press.

BREKHOVSKIKH, L. M. (1960) Waves in Layered Media, Academic Press, New York. BREMMER, H. (1958) Propagation of electromagnetic waves, Handbuch der Physik, 16, 423-639, Springer-Verlag, Berlin. BUDDEN, K. G. (1961) Radio Waves in the Ionosphere, Cambridge University Press. COLLIN, R. E. (1960) Field Theory of Guided Waves, McGraw-Hill, New York. EWING, M., JARDETZKY, W. and PRESS, F. (1958) Elastic Waves in Layered Media, McGraw-

Hill, New York. FOCK, V. A. (1946) The Diffraction of Radio Waves Around the Earth, Acad. of Sciences of

USSR, Moscow. FRANZ, W. (1957) Theorie der Beugung elektromagnetischer Wellen, Ergebnisse der

angewandten Mathematik, Pt. 4, Springer-Verlag, Berlin. HARRINGTON, R. F . (1961) Time Harmonie Electromagnetic Fields, McGraw-Hill, New York. HÖNL, H., MAUE, A. W. and WESTPFAHL, K. (1961) Theorie der Beugung, Handbuch der

Physik, 25, 218-583, Springer-Verlag, Berlin. LOGAN, N. A., et al. (1959) General research in diffraction theory—Vols. I and II, Reports

LMSD 288087 and 288088, Lockheed Aircraft Corp., Missiles and Space Division, Sunnyvale, California.

LOGAN, N. A. and YEE, K. S. (1962) A mathematical model for diffraction by convex surfaces, Proc. Symposium of Electromagnetic Waves, pp. 139-180, University of Wisconsin Press.

MARCUVITZ, N. (1958) General electronic waveguides, Research Report R-692-58, Polytechnic Institute of Brooklyn, New York.

MENTZER, J. R. (1955) Scattering and Diffraction of Radio Waves, Pergamon Press, London and New York.

MOORE, R. K. and BLAIR, W. E. (1961) Dipole radiation in a conducting half-space, / . Res. Nat. Bur. Stand., 65D, (Radio Prop.) 547-563.

OFFICER, C. B. (1958) Introduction to the Theory of Sound Transmission, McGraw-Hill, New York.

OLINER, A. A. (1961) Investigations on guiding and radiating microwave structures, Final Report No. PIBMRI-934-61, Polytechnic Institute of Brooklyn, New York.

SCHELKUNOFF, S. A. (1943) Electromagnetic Waves, Van Nostrand, New York. VAN DE HÜLST, H. C. (1957) Light Scattering by Small Particles, Wiley, New York.


WAIT, J. R. (1959) Electromagnetic Radiation From Cylindrical Structures, Pergamon Press, New York and London.

WAIT, J. R. (1961) The electromagnetic fields of a horizontal dipole in the presence of a conducting half-space, Canad. / . Phys., 39, 1017-1028.

WAIT, J. R. (1962) The propagation of electromagnetic waves along the earth's surface, Proc. Symposium of Electromagnetic Waves, 243-290, University of Wisconsin Press (ed. by R.E. Langer).

ZUCKER, F. J. (1961) Surface and leaky wave antennas, Chapter 16 in Antenna Engineering Handbook (ed. by H. Jasik), McGraw-Hill, New York.

Added in Proof: BUDDEN, K. G. (1962) The waveguide mode theory of wave propogation, Prentice-Hall,

New York.

B

Chapter II

REFLECTION OF ELECTROMAGNETIC WAVES FROM HORIZONTALLY STRATIFIED MEDIA

Abstract—A general analysis is presented for the electromagnetic response of a plane stratified medium consisting of any number of parallel homogeneous layers. The solution is first developed for plane-wave incidence and then generalized to both cylindrical and spherical-wave incidence. Numerical results for interesting special cases are presented and discussed. The application of the results to surface-wave propagation over a stratified ground is considered in some detail.

1. INTRODUCTION

The propagation of radio waves along the surface of the ground has been discussed from a theoretical standpoint for many years. As long ago as 1907 Zenneck showed that a wave, which was a solution of Maxwell's equations, travelled without change of pattern over a flat surface bounding 2 homogeneous media of different conductivity and dielectric constants. When the upper medium is air and the lower medium is a homogeneous dissipative ground, the wave was characterized by a phase velocity greater than that of light and a small attenuation in the direction along the interface. Furthermore, this Zenneck surface wave, as it has been called, is highly attenuated with height above the surface.

In 1909 Sommerfeld solved the problem of a vertical dipole over a homogeneous ground (half-space). In an attempt to explain the physical nature of his solution, he divided the expression for the field into a "space wave" and a "surface wave". Both parts, according to Sommerfeld, are necessary to satisfy Maxwell's equations and the appropriate boundary conditions. The surface-wave part varied inversely as the square root of the range, and it was identified as the radial counterpart of the plane Zenneck surface wave. For many years it was believed that the Sommerfeld surface wave was the predominant component of the field radiated from a vertical antenna over a finitely conducting ground. Much later an error in sign in Sommerfeld's 1909 paper was pointed out by Norton [1935], which also partly accounted for the unusual calculated field-strength curves of Rolf [1930]. At about this time there was a series of papers by Weyl, van der Pol, Niessen, Wise, and Norton deriving more accurate representations for the field of the dipole. A discussion of this later work has been given by Bouwkamp [1950]. Norton

8

Reflection of Electromagnetic Waves 9

[1936, 1937, and 1941], in particular, has developed his formula for the field components to a stage where numerical results can readily be obtained. It is now generally accepted that the Sommerfeld surface wave (or the radial Zenneck wave) does not bear any similarity to the total field of a vertical di-pole over a homogeneous conducting earth. In fact, the field excited by a dipole varies as 1/rf where d is the distance for low frequencies and varies as l/d2 for high frequencies. Norton has suggested that the field in air of a dipole over a homogeneous ground be expressed as a sum of 3 components : a direct ray (or primary influence), a reflected ray which is to be modified by an appropriate Fresnel reflection coefficient and a correction term. Norton has described the first and second components as the space wave; the third or correction term, the surface wave. This seems to be a logical step although, taken separately, the space and surface waves of Norton are not solutions of Maxwell's equations. On the other hand, his "space wave" is the contribution which would be derived on the basis of geometrical optics, and his "surface wave" is the correction from wave theory. This latter term will be called the "Norton surface wave" as distinct from the Zenneck and Sommerfeld surface waves, and the trapped surface waves discussed below.

It was pointed out by Sommerfeld [1899] many years ago that a straight cylindrical conductor of finite conductivity can act as a guide for electromagnetic waves. Some time later, Goubau [1950, 1952] demonstrated that such a cylindrical surface wave can be launched with reasonable efficiency from a coaxial line whose outer surface is terminated in a conical horn. The improvement in the transmission properties by coating the wire with a thin dielectric film has also been discussed in detail by Goubau [1952]. The plane counterpart of the Goubau-Sommerfeld cylindrical surface wave is obtained when a flat metallic surface is coated with a dielectric film. Atwood [1951] has discussed the nature of the surface waves which can exist in a structure of this type. When the film thickness is small compared to the wave-length, the single propagating mode has a phase velocity slower than that of light and is attenuated rapidly above the surface in the air. Such surfaces have been called inductive since the normal surface impedance, looking into the surface, is almost purely imaginary for a low-loss dielectric on a well conducting base. A similar type of surface wave can exist over a corrugated surface [Rotman, 1951 ; Elliott, 1954; Zucker, 1954] which is inductive if the periodicity and depth of the corrugation are small compared to the wavelength.

The excitation of surface waves on dielectric-clad plane conducting surfaces has been discussed by Tai [1951] for a line current source and Brick [1954] for a dipole source. Corresponding treatments for corrugated surfaces have been given by Cullen [1954] for a line magnetic or infinite slot source and Barlow and Fernando [1956] for a vertical electric dipole source. In the case of dipole excitation, the field varies predominantly as the inverse square root of the distance along both the dielectric-clad and the corrugated surfaces.

Surface waves of this type which have a phase velocity less than that of light can be called "trapped surface waves" since they carry most of their energy within a small distance from the interface [Barlow and Cullen, 1953; Marcu-vitz, 1952J.

In this chapter, 3 aspects of the problem are discussed. The first is a general analysis of reflection of plane waves from a parallel stratified medium consisting of M homogeneous slabs, the second is the extension to line source over the stratified medium, and the third is the generalization to dipoles or current elements over the stratified medium. In addition, the evaluation of an important integral by the modified saddle-point method is outlined in an appendix.

2. PLANE WAVE INCIDENCE

A plane wave with a time factor exp(/atf) is incident at an angle 0 on a stratified medium composed of M homogeneous layers. The electric vector is in the plane of incidence (xz plane). The situation is illustrated in Fig. 1

( < ( ( ( <

\ \ \ \ \ \ \

\ \ \ \ \ \ \ °M€MMM

K ( ( ( ( ({( ( ( ( ( (Z,

^^

i / / / / y i / / Ì z m - i hm

\ " \ \ \ \ V \ \ \ \ , Z M - 1

FIG. 1. A stratified medium consisting of M homogeneous layers.

where the y axis is out of the paper. The electrical constants of the layers are am, em and μΜ where the subscripts m indicate the wth layer below the surface.

From symmetry it can be seen that the magnetic field has only a y component and for the mth layer, it is a solution of the equation

(V2 - yl)Hmy = 0 (1) where

y« = iflWV> - ε/πμ„,ω2 with real part of ym > 0.

The general solution is of the form

tfw, = [ a m e - ^ + fcme^]e — iXx (2) where u^ = λ2 + γ„ and λ can take any value. However, real part of um > 0· The incident field Hinc can be written

Hine Oy

Therefore, in equation (2) a0 e uoz e ίλχ can be identified with H™yc if a0 = H0

■ = jj g-yocostf .z y^ g-yosin^jc

a-M0Z Q-iXx .

and ik = yo sino. Consequently, fc0eW0Ze~iA* is a reflected wave and the angle of reflection is 0.

The boundary conditions at the interface z = 0, z = zx,. are that the tangential fields should be continuous. Now since

z = z, m - l

£*,*= -(<rro + toßmy .t8Hm

dz

this means that the boundary conditions can be written

(3)

(4)

where m = 0, 1, 2, .. , M - 2, M - 1. Imposing the condition that only outgoing waves are permissible in the

lowest layer (which is semi-infinite) it follows that bM = 0. The boundary conditions then lead to 2(M — 1) equations which are linear in am and bm to solve for 2(M — 1) unknowns in terms of the known coefficient a0. The solution* is

where

Z±=Kt

Z T — JY-J

KO + Z I

Z2 + Ki tanh ujii

(5)

Xi + Z2 tanhWi/ii

Z3 + K2 tanh u2h2

K2 + Z3 tanh u2h2 (6)

Zm — K, + Xm tanh umh

m lKm + Zm+l tanh umhm

%M-l ~ ^ M -^ M + ^ M - 1 tanhwM-i/ijv,-! ^ Μ - ι + ^Mtanhw M _ 1 ft M _ 1

' The quantity λ should not be confused with the wavelength.


where

Km — u„ σ + ΐωε,

·, and um = (A2 + y£),/2

(Note that hm is the thickness of the mth slab). The quantity b0/a0 is the ratio of the amplitude of the reflected wave to the

amplitude of the incident wave. It is denoted by R^ to indicate that the electric field of the incident wave (and also that of the reflected wave) is in the plane of incidence.

The present problem has a well defined analogy in transmission line theory.

*2 *M-2 * M - I

FIG. 2. Transmission line analogy for the stratified medium of M layers.

In this analogy each section of the multiple tandemmed line is to correspond to a slab. The voltage across the line is Emx and the current is Hmy for the mth section. The propagation constant is um and the characteristic or surge impedance is Km of the mth slab. The incident wave which comes from the left in Fig. 2 is given by

a0 e"Mor

and the wave reflected at the junction, z = 0, is

b0 eM0Z

The input impedance of the line which is the ratio of the voltage to the current at z = 0 is Zx. Furthermore, the impedance at the junction z = zm is Z m + 1 . With this analogy and a knowledge of the behavior of one dimensional transmission lines, one could write down the solution of the 2 dimensional reflection problem.

Some features of the wave problem will now be discussed. The quantity Zi will play an important role in the following. Because

Zi = E0xIH0y]z:=0 = EixlHly]z=0

it is called the surface impedance, being the ratio of the tangential fields at the air-ground interface. In the case of normal incidence, 0 = 0 and λ = 0 so that um = ym and Km = ηηί where

and

For example, in the case of a homogeneous ground (ht -► oo),

%i = *h> Ko = *7o and the reflection coefficient becomes simply

R bo = n2-rb_ " «o i/o + fi

Another special case of considerable interest is when Θ approaches 90 degrees corresponding to glancing incidence, then

Um = (Vm - ylf\ Km = (Vm ~ ϊΙΥ^η, + Ì<DSm)

which yields for the homogeneous ground

Z i - K i - i h C l - y S / t f ) * (8) This special value of Zl9 which relates the tangential fields in the limiting case of glancing incidence, turns out to be an important quantity in further work. For this reason, it is denoted by Zv where the superscript v indicates that the electric field in the air is nearly vertical for \γχ\ > |y0|. This fact can be shown by evaluating the wave tilt which is defined by

W-iH (9) and is the complex ratio of the horizontal to the vertical electric field in the air just above the ground. It readily follows that for the general case [Norton, 1935]

w = EJHly] = ιωε0Ζί

E0zIH0y\z=0 yosin0 In the case of a homogeneous ground

. ( l - 4 sin2 of w _ ιωε0Κ1 _ηι\ 7ί )

y0 sin θ η0 sin 0 which for grazing incidence becomes

W.llU-*)* =(*)*>(!-*)« (12) no \ 111 W 7ι \ yJ

For radio frequencies (ω £ 106) and moderate conducting ground (σ ^ 10" 2), |yo/Vi| is of the order of 10"2 and thence Wis small and hence \E0z\ is much greater than \E0x\.


To indicate in a simple way as is possible, the influence of stratification on the reflection of waves from the ground surface, a 2 layer case will be considered. This is effected by letting h2 -> oo. Furthermore, it will be assumed that |)Ί/7Ο| a nd ^/Vol > 1· Then for any angle of incidence

um = ym(l-y4sm2e)2czym (m = l,2)

and K»> = n Ar^l» ("» = 1,2)

This leads to the simple relation

Zi = QKt (13) where

(yily2) + tanhy1h1 ß = l + ( y i /y2 ) tanhr i / l

f ° r '« = ^ ,

and where Q is the correction to the characteristic impedance Kt of the upper layer to account for the presence of the lower layer. Note then, if ly^il > 1, ß = 1. It can be said that the lower layer is not detectable when \Q\ is within 5 per cent of unity. Such a condition is met when {σφ0ώγ/2Η > 3.

It should also be noted that Q relates the wave tilts for a stratified (2 layer) ground and that of a homogeneous ground by

W9ÉW0Q where

An example is here quoted to illustrate the order of magnitude of the quantities involved:

Frequency,/= ω/2π =125 kc/s Upper layer conductivity, σ1 = 10"3 mho/m Dielectric constant of air, ε0 = 8.854 x 10"12 F/m Dielectric constant of ground, 6i = 10·ε0 Magnetic permeability, μ = 4π x 10~7 H/m.

For these values W0 = 0.082 Ζ.41.Γ

for a homogeneous ground. In the case of a 2 layer ground where hx is finite

W=W0Q

= 0.0S2\Q\L(41.r + q)

where q is the argument of Q expressed in degrees. For frequencies of this order είω/σί and ε2ω/σ2 are small (in the above example ε1ω/σ1 = 0.0069). Therefore, yt = (ΐσίμω)ί/2 and y2 computation of Q is then given by

(ίσ2μω)/ζ. A formula suitable for

e = frifo)** + tanh V(QK where V = (σ1μω)ί/2Η1 (14) 1 + (σίΙσ2)1/2 tanh y/(i)V

= tanh[V(i)K + tanh"1(σ1/σ2),/2]

If σ2 > σι corresponding to a highly conducting substratum,

β s tanh yJ(x)V (15)

or if σ2 < σ! corresponding to an insulating substratum,

ßseothVOP (16> In the above example, the parameter F can be replaced by hJlQ where hi is

the thickness of the upper stratum in meters. The function Q and its argument q is plotted in Fig. 3a and 3b as a function of Kfor various values of the ratio σ2/σι.

3. EXTENSION TO PERPENDICULAR INCIDENCE

In the preceding problem, the incident plane wave has the electric vector contained in the plane of incidence (and the magnetic vector parallel to the interfaces). For this reason, it is called parallel incidence. The other important case is when the electric vector is perpendicular to the plane of incidence. This is termed perpendicular incidence.

Again choosing the plane of incidence to be the (xz) plane, the incident wave now has only a y component of the electric field. By analogy to Eq. (2), the general solution is of the form

^ = [^e - M - z + 5meM-z]e -ίλχ (17) where

u^ = λ2 + y„, a0 is the amplitude of the incident wave, B0 is the amplitude of the reflected wave, and EM = 0. In this case

rr dEmy

so that the boundary conditions now become

-Ήι- i,y

Ο'/Ί«-I«)" = (</<»,ω> c£„,

dz dz

(18)

(19)


o.o .05 0.1 02 03 0.4 0.5 1.0 2 0 304.0 50

FIG. 3a. Amplitude of correction factor for a two-layer ground.

80c

60e

40e

20e

q 0«

-20 e

-40 -4π°μ-

-60«

h

f-h

h

^ ^ ^ ^

L

L. ..,.- i .

0

0 . 2 ^ __

" Ξ ΐ Ξ ^ — ' 0 . 5

C=I.O

" — - _ _ 2 . 0 ■— Γ -—-—_3.0 ^-

- - - ^ 5 J D ^

°2———*^^

! ! 1

b

C=(CT-, /σ2)Τ

,. I ! . ! I . J .05 0.1 0.2 0.4 0.5 1.0 2.0 30 4.0 5.0

V:(oi/v^)Th,

FIG. 3b. Phase of correction factor for a two-layer ground.


These are transformable to the boundary conditions for parallel incidence by making the substitutions : Emy by Hmy9 /μ„,ω by am + ismco. Then using the previous results, the solution for perpendicular incidence can be written down:

Bo N0-Yi

where «o Wo + Yl

(20)

Y = Α Γ Ym+i + Nmtenhumhm m mNm+Ym+ltanhumhm

K }

for m = 1, 2, 3, . . . , M — 1 and

In the preceding

Ν~ = Ϊ1Γ7\ (22)

where as before t/m = (λ2 + yffi. The quantity B0/ä0 which is the ratio of the amplitude of the reflected wave

to the incident wave is denoted by R±. There is a similar transmission line analogy for this problem which need not be pointed out.

In analogy to the surface impedance function Zl9 the quantity Y± is a surface admittance and is given by

^ι = —H0xIE0y]z=0 = -HlxIEiy]z=0 (23)

In the case of a homogeneous ground at glancing incidence (0 -> π/2), it follows that

y1 = N1 = (1/?1χΐ - yg/rî)% (24)

which is denoted ΓΛ where the superscript h indicates that the electric field is horizontal in contrast to the near vertical electric field associated with Zv. It is interesting to note that

YhZ» = {\-ylh\) (25) and for \y\\ > \y2

Q\ YhZv s 1

4. IMPEDANCE MATCHING AND NATURAL OSCILLATIONS IN STRATIFIED MEDIA

In this section, some remarks will be made concerning the nature of extreme conditions where the reflection coefficient on a plane stratified media becomes zero or infinite. The discussion will be confined primarily to parallel incidence, although the results are easily carried over to perpendicular incidence.


As indicated, the reflection coefficient R^ is given by

«■-èri (26) where

K0 = Μ0/ίωε0 = (λ2 — kl)i/2liœe0 and Zi is the normal impedance at the interface z = 0. In the case of a homogeneous half space (ht -> oo)

Zi = K1 = ηίΙ(σι + iœej = (A2 + γ2ί)ί/2Ι(σί + ΐωεθ (27)

The condition for matching is that R^ = 0 corresponding to the absence of a reflected wave. This requires that

X o - Z i (28) For a homogeneous half space, the condition is simply

K0 = Kt (29)

which when solved for λ yields

Λ ,p«îM-j4/râf (30)

In the case of no magnetic permeability contrast (ßt = μ0), the above simplifies* to

(yl + yl)* and if both media are perfect dielectrics γχ = iki and y0

== ^ ο »

' Λ = + , 2 " , ! ■ / , (3D

^Ä (32) But, since λ — k0 sin 0, the preceding equations for a condition of matching can be employed to determine the angle of incidence when there shall be no reflection. In the case of the 2 dielectric half spaces, the condition is

s i n 0 = ± ( f c f T ^ (33)

or tanO= ±(kjk0)

This latter equation is well known from optics and there the angle Θ is known as the Brewster angle and the ratio ki/k0 is called the relative refractive index.

* In this case the wave tilt W = yo/n as can be deduced from Eq. 11.

It is indeed very interesting to observe that if kl is not real, (or γ1 not imaginary), the quantity λ or 0 required for matching is complex. That is, in the language of optics, the Brewster angle for an absorptive homogeneous medium is complex. The existence of plane waves with a complex angle of incidence will not be discussed at this stage, but it is interesting to note that the so-called Zenneck surface wave is very similar to the disturbance propagating along the interface when there is a condition of wave matching. His solution is briefly described below.

Zenneck [1907] postulated that a surface wave could propagate along the interface z = 0 between 2 homogeneous media. He assumed that in the upper medium (z < 0)

H0y = b0e+UQZe-iXx (34) and in the lower medium

^ i y = « i e " a i Z e " U x (35)

for propagation in the x direction. The signs prefixing u0 and ux were chosen to insure that the field decayed to zero for \z\ -> oo. The corresponding electric fields are then obtained from Maxwell's equations;

£ 0 y = - X 0 b 0 e M o z e - ^ (36) and

Ei X ^ e - ^ e " ^ (37) The boundary conditions then require that

b0 = al and

Ko + K^O or ^ + ^ ^ = 0 ε0 st — ισίιω

The latter condition cannot be met for any value of λ if the imaginary parts of w0 and ut are to be greater than zero. It now appears that Zenneck solved the matching equation K0 = Kx in place of K0 = — Kv

Returning to the general case, it is seen that

has a pole (i.e. becomes infinite) when K0 + Zi = 0 (39)

This is called the "resonance condition". For the homogeneous half-space, this resonance condition is simply

K0 + Kt = 0

which is identical to Zenneck's equation for a surface wave.


While it is not possible to set up a surface wave in the sense envisaged by Zenneck, it is possible to retain his solutions if the disturbance in the upper medium is a wave containing a factor e"M°z in place of e+M°z, and the boundary condition becomes K0 = Kx which corresponds to the "matching" condition.

In the case of stratified media it may well happen that the "resonance" condition K0 + Zi = 0 is satisfied. Rewriting this as

Z1 = i - ^ (40)

and since for a surface wave u0 is to be mainly real (i.e. small imaginary component), then Ζγ is to be mainly imaginary (in the positive sense). Such a surface characteristic is described as inductive. The quantity λ which describes the transverse propagation is mainly real and somewhat greater than k0 [i.e. λ2 = u\ + k2].

A simple example of an inductive surface is a highly conducting plane coated with a thin uniform film of dielectric. From Eq. (6) with Z 2 = 0 (or y2 = oo)

Zt = Kx tanh uji^ (41)

where h1 is the thickness of the film. For thin films, and μ0 = μ±

Zi s Kiujii = ίμοω/ΐ! 1 —p\ (42)

This can be further approximated to

Zi S ιμοωΗ,ΙΙ - (fco/feO2] (43)

since λ is near k0. Such a surface is therefore almost purely inductive. In this case

uQ = ε0μ0ω2Ε = kffi and

λ = (u2 + k2f> = fe0(l + (/c0R)2)1/2 S fc0(l + (k0h)2l2) where

h = Ä^l - kl/kl)

The wave has a phase velocity in the transverse (x) direction which is 1 — (k0h)2/2 times that of free space and it decays rapidly in the z direction. This approximate solution is valid if k0H (or &0^i) is small compared to unity.

In general the determination of the poles in the reflection coefficient i?H leads to transcendental equations which must be solved by numerical or graphical means. A discussion of surface waves on dielectric coated conductors without the restriction that hi be small has been given by Atwood [1951].

Further remarks regarding surface waves will be made in a later section following the analysis of the excitation problem.

5. LINE SOURCE EXCITATION

In problems of reflection from stratified media, it is more meaningful from a physical standpoint to consider the source a localized distribution of current. No physical source in existence produces pure plane waves, although in certain limiting cases the fields have characteristics closely akin to plane waves. An example is a line source of electric current. It was shown that at large distances from the source the electric and magnetic field were mutually orthogonal to one another and to the direction of propagation [Wait, 1953]. (See for example, Eq. 99). It might then be conjectured that the reflected wave for a line source located high over a plane interface could be computed from the plane wave reflection coefficient. Such is the case if certain limitations are imposed.

It is the purpose of this section to extend the plane wave solution to the case of an (infinite) line source of (constant) electric current located at height h above the stratified medium considered in the previous section. The coordinate of the line source is taken as (o, y, —h) as indicated in Fig. 4.

/ / / 7 / /

/ / / / / /

σΜ€Μ^Μ

/ / / / / Y / / / / z m - i hm

t i t < ( * { t t t Z m

\ \ \ \ \ \"Λ \ \ \ ZM-I

FIG. 4. Line source over a layered half-space.

In view of the symmetry, the electric field has only a y component Em for any of the layers (the subscript y on Em is dropped for convenience). The

direct, or primary, field Ep0 in the region z < 0 (above the interface) is

expressible as

£g = ^ f ^ K0[y0lx> + (z + hfn (44)

where K0 is the modified Bessel function. To convert the expression for E% to a more useful form, it is written as a

Fourier integral as follows

Ep0=\ A(k)Q-ikxàk (45) -I

By making use of the inverse property of the Fourier integral, it follows that

Α(λ) = ±Γ E>0ea*dx (46)

— ϊμ0ωΙ Π°°ΚοΙ>οΙ>2 + (z + ft)2]54] eiA*dx (47) J —οο 4π2

This integral is evaluated to give*

= zlt^L e±Mz+*> w h e r e „0 = (A2 + y2)1/' and 2πΜ0

where the + sign is to be employed for (z + A) < 0 and the — sign for (z + A) > 0.

The resultant field in the space 0 > z > —A is then written in the form

E0 = ~ιμοωΙ | ll-i[e-«D(«+*) + R±(X) e"»i*-*)]exp(-iAx) dl (48) 4 π J-oo

where the term containing R±(k) accounts for the presence of the stratified medium at z = 0 and is as yet unknown. Equation (48) satisfies the equation (V2 — yl)E0 = 0 and behaves in the proper manner as z -► —A and x -► 0. The corresponding expression for E0 in the region —h>z> — oo is identical to Eq. (48) except that the (—) sign on the first exponential should be changed to a + sign. E0 then decays properly to zero as z -> — oo.

The integral representation for E0 has a clear physical meaning. When the symbol λ is identified with k0 sin Θ or — iy0 sin Θ, the field Εξ is a spectrum of plane waves of (complex) angle Θ of incidence. The complete field E0 contains a spectrum of both incident and reflected waves [Wait, 1953]. The

* Campbell and Foster [1948], pair No. 917.

structure of the integrand in the integral representation for E0 is identical to the corresponding factor in the plane wave solution in the previous section. In fact, the correspondence carries over to each of the sublayers. For example, in the present problem, for m = 0 to M,

Λοο

J —oo „(λ) e-·""2 + 5„(Â) e"mZ] e_u* άλ (49)

J —oo and

• u dE>» · n dE>» Wm(oHmx = -fo-> - Ψη^Η^ = —

The solution of the present problem is thus immediate;

RAA) - α0(λ) - Ν0(λ) + yx(A) ( 5 0 )

^-^^Sîr^ (51) Nm(X) + ym+1(/)tanh umhm

where

with m = 1, 2, ... , M - 1 and ΥΜ(λ) = ΝΜ(λ). Equation (48) along with (50) and (51) constitutes the formal solution of the

problem. The remaining task is the evaluation of the integral in (48). Except in certain limiting cases, certain approximations must be made in order to obtain a useful result.

The simplest limiting case is when the interface becomes highly conducting. Then, since yt £ oo, R±(X) £ — 1 and

= Zlt^L f + 0°w-i[e-Mo(z+/o _ Qu0(Z-h)i Q-ixX(iX ( 5 2 )

Both these integrals are of the same type and may be readily evaluated in view of Eqs. (46) and (47). Consequently, for z < 0,

E° = - ^ {KoD-oiy + (z + Λ)2]'/2] - KOIJOC*2 + (z - Λ)2]'/2]} (53)

It can be readily verified that (V2 — yl)E0 = 0 for — z > 0 and moreover, E0 = 0 for z = 0 which is the required boundary condition for a perfect conductor. It should be mentioned that an equally simple result is obtained for cases where μγ ^ oo corresponding to a rather hypothetical situation where the interface as z = 0 is behaving as a perfectly permeable medium. Then i?j_(/l) s +1 and the solution for E0 is identical to Eq. (53) with the second term inside the curly brackets changing sign.

c

When \y0h\ > 1 and h > -z, Eq. (53) for E0 can be written in the form*

-ΐμ0ωΙ/π\ί/2 e"yoRo

E«*< 2π ' $ (yoRof2 [-e-«o(z), _ e+«o(*)] e~i. -ίλχ (54)

where û0 = ((^)2 + yl)v\ 1=~ iy0 sin 0

and where g is the geometrical reflection point such that the ray IQ and the ray QP make equal angle B with the normal at g . The quantity JR0 is the length of the ray from the line source to the point of reflection Q. In fact, g is taken as the new origin of the cartesian system x9 y, z as indicated in Fig. 5.

p ( x , y , * î

FIG. 5. Geometrical interpretation for the secondary field.

Equation (54) can be simplified even further by noting that the quantity prefixing the square bracket is simply the amplitude of the incident wave. When this is normalized to unity,

E0 = [e-"°<z) - e+öo(z)] e_i-— iXx (55)

which is readily identified as an incident and reflected plane wave with an angle of incidence and reflection of 0. To generalize equation (55) for a stratified medium, the reflected wave would be simply modified by a reflection coefficient for an angle of incidence 9. The modified form would be

E0 = [e"öoz + R±(X) e+öoz] e -iXx (56)

A suggested modified version of Eq. (53) to account for the stratified nature of the medium would then be

E0 £ ^ ^ {Kobolx2 + (z + h)2!*] + RA9)Koboix2 + (* - Λ)2]*]}

(57) * The first term only of the asymptotic expansion of Ko(<x) is retained, i.e. Ko(oì) ^

β"α(π/2α)*.

when RJSP) is the reflection coefficient for a plane wave incident at an angle 9 on the interface z = 0. The angle 9 is determined by simple geometrical considerations as indicated above.* This result which has been derived by an elementary approach considers only the specular component. For smaller values of \z\ and h the situation becomes more complicated since surface waves within the layers may be excited. This question will not be pursued any further at the moment.

6. LINE SOURCE ON A HOMOGENEOUS MEDIUM

When the line source is situated at the interface (A = 0) between 2 media of semi-infinite extent with electrical properties ε0, μο an(* 0Ί, ε!, μ^ respectively, the electric field E0 is given by

- i ^ / p _ c o s A x e + M o Z d / l π Jo "i + "o

for z < 0. This is a special case of Eq. (48) with h = 0 and hi = oo. Noting that

1 Mi — Mn Mi - Un

ux + u0 u\- u\ y{ - 7o

it follows that, for z = 0

Since

— iu col if00 f °° ) E0= . ~ ° .1 Mi cos Ax dA— MoCosAxdyq (59)

rc(yi-yo)lJo Jo i

lim I cos λχ άλ = K0(y0x) (60) z->o Jo uo

which is equivalent to Eq. (45), it is seen thatf

(cos λχ)ιι0 άλ = ΙγΙ - ^-ήΚ0(γ0χ)γ0χ = -Κί(γ0χ)γ0χ-ί (61)

Making use of this result, the expression for the electric field becomes — ιμ0ωΙ

<y\ - 7oV Eo = ΖΠ 27"2 CVoX iCVo*) - y1xK1(y1x)] (62)

This formula is exact.

* Note then tan ö = x/(h — z). t In Eq. (61) use is made of dKi/dx — —KIJQL — Ko where a is the argument γοχ.

At large values of the argument

K^x)*{ùTc~nx (63)

^. = .,rT^W (64)

and if the real part of y^ is > 1,

-ΐμ0ω!γ0

<yì - yl)x Furthermore, if y0 = ik0 where k0x is a real number > 1, it follows that

-ΐμ0ωΙγ0 / π \ , / î __ynJ£ ^ -ίμ0ωΙγ0/ π V 0~n(yì-yl)x\2y0x)

~ J * o « M _ / _ « \ 'A - i ( t0X+n/4) i 6 5 ,

It is convenient to express this in terms of the primary field,

— ΐμ0ωΙ Epo =

In

in the following manner:

E0 2γ0 2ik0

™°»~^(uT'- ™

Epo~{y\-yl)x (yi + kl)x (67)

The vertical electric field in the interface, for the current line source in the interface, is given by

H0l = Hlz=-—ψ (68) ιμ0ω ox

= 7-2—ΓΚ-3 Pyo*Ki(yo*) + ylx2K0(y0x)

- ly.xKfy.x) - yWKoiyyx)-] (69) and, if the real part of ytx > 1,

ypi n(yl - yl)x2

which simplifies further, for the case \y0x\ > 1, to

"-»(a*1"·5* (71) where

η0 = ίμ0ωΙγ0 = μ0ω^0 = (μ0/ε0)1/2

Η0ζ s _, / ° 2. 2 ρκΛνοχ) + y0x^o(ro^)] (70)

The field at very low frequencies near the line source is also of interest. For example, if \y0x\ <ξ 1,

Ηα^ l0z = 2 3 ny\x

[2 - 2y1*X1(y1x) - (yi*)3X0(yi*)] (72)

which approaches —Ιβπχ as ^Λ; approaches zero. The magnetic field H0z expressed as a ratio to the primary field Ηξζ

(= -Ιβπχ) is plotted in Fig. 6 as a function of \yLx\ under the assumption that είωΙσί s 0.

ι.υ

LU CO

^ 0.8 o Q. C/) ÜJ £T ω 0 . 6 > H

< _J ω 0.4 Cu

N 0.2 o

X

V \

-

-

/

S 1

-0

\^ y>

1

X^v

0 1 2 3 4 5 6 ΙΤίχ^ίο- ,^ω^χ

FIG. 6. Response of a line source on a homogeneous half-space.

7. LINE SOURCE OVER A THIN LAYER

The general solution for M layer can also be specialized conveniently to the case of a line source at a height h over a thin sheet of thickness hx and conductivity σ1. The following substitutions in Eq. (48) are made

μη = μ0, M = 2, y2 = y0

where it is supposed that the medium above and below the sheet is homogeneous. The electric field for z < 0 is given by

i λχ άλ) -iuncoll , , x f^e^-^cosAxdA) / w E0 = —p— Ko(yori) - Χ0(7οΓ2) + — — (73)

In \ Jo ue + Mo / where i/0 + i/i tanh «i/ i i

t/e = « i — - — — Mi + M0 tanh «!«!

rî = (z + Λ)2 + * 2 and r\ = (z - ft)2 + x2

(74)

For small values of the sheet thickness ht and large values of the conductivity au

tanh uihi £ u^h^ and ε1ω/σ1 4.1 and therefore

«ι*ι=(λ2 + ??)*ι£2ΐί (75) where

q = a^ocohjl With these approximations

ue s M0 4- 2/g and then

„ ~ίμ0ω/ Γ^ / Λ ^ / χ f °° β"0(ζ"Λ) cos Ax dAl £0 = - ^ Γ — ^ofto'i) - K0(y0r2) + — — (76) In l Jo «o + *<? J

Using the result appropriate for large q9

_i^_4.£(s)-(_ir (,, and noting

s) — <78)

it follows that, asymptotically,*

— iu ml { °° f — lY" /d\m+i 1 £ ° S ~ ^ ~ |*o(Vo»-i) - Xo(y0r2)+ ^ ^ p r ( ^ ) KoiVo^J (79)

where use has been made of the relation •oo 6 « ο ( ζ - Λ ) ί. 0 M 0

cos λχ άλ = K0(Vor2) (80)

valid for (z - h) < 0. The field E0 can thus be developed into an (asymptotic) series of inverse

powers of qr2 where the coefficients involve derivatives with respect to the argument of the modified Bessel functions. Using the relations

KQ = — Kx

and

* A more rigorous derivation of this series expansion is given by Wait [1953].

the fl'th derivative of K0 can be expressed in terms of K0 and Kx. The first few terms of the resulting series are given below:

E ° = ~ ^ Γ iKo(y°ri) - Ko(y°r2)+T(y°r2>q)i (81)

where

s — y0(z - Κ)Κ^0ν2) qr2

+ ^ 2 [v2(* - h)2K0(y0r2) - y0r2(l - 2 (-^l)JKi(y0r2)^

(83) 1 1

+ terms containing -—-^, -—-τ , etc. (qr2)3 (qr2)4

where qr2 = ϋγμ^ω\ιχγ2\2 and z < 0. It can be readily verified that the field below the thin conducting sheet

(z > h) is given by

E2 = f^LT(y0ruq)l for z > 0 (84)

As a check it can be seen that EQ = E2 at z = A. Expressions for the magnetic field components can be found from Maxwell's equations :

1 dEm 1 d£L Hmx = z - ^ and Hmx=—, - = for m = 0, 2 m* ίμ0ω dz mz ιμ0ω dx The above expansions developed in inverse powers of (qr2) are particularly

suitable for highly conductive sheets when the source or the observer are not near the sheet. Alternative expansions can be developed for smaller values of qr2. These are developed as follows. At low frequencies where y0rx and y0r2 are small compared to unity, y0 can be set equal to zero and it readily follows that

Eo = Z ^ [ l o g ( r 2 / r 1 ) + r0(/?)] (85)

where ,<0 e- A i cos/be

άλ and ß = h-z (86) io λ + iq

Γο(/0 = ίο Inserting the power series expansion for cos λχ, Ε0 becomes

mf0 (2m!) Jo λ+iq

Since

where

it follows that

ί e-λβ -άλ= -e+w<,Ei(-ij?g) (88)

= - — ι J+ißq *

o λ + iq

Ei(-ißq)=-\ '—ài (89)

00 C-l)mxZm /d\2m

τ·<»--Σ.4ίτ' y ^ - ^ (9o) where

g = ßq = (h — ζ)/ι1σ/ιω/2

This is an ascending series expansion in g. When x = 0, it reduces to

T0(ß)=-QV«Ei(-ißq) (91) Using the latter result it follows that the fields E0 and E2 are in the plane x = 0 are given by

L \z + -^z-h^Ei(-i\z-h\q) (92)

2π L \z + Λ| for z < 0, and

£ 2 = ± | ^ ί β·Ίζ+Λ^Εί(-ί|ζ + fc|9) (93)

for z > 0. The exponential integral Ei(—ig) with imaginary argument can be expressed

in terms of the cosine and sine integrals as follows

Ei(-^) = Ci( ) + i^-Si(^)J (94) where

i °° cos t C9 sin t

— di and Site) = I —at (95) are tabulated [Janke and Emde, 1945] for real values of g.

It might be mentioned that T0(ß) for x > 0 can also be expressed in terms of exponential integrals, but then the argument is complex. For example

1 foo ~-λ(β-ίχ) i [co -λ(β + 1χ)

2J 0 λ + iq 2 Jo λ + iq

= - ± eVi[e",fXEi(-i(i8 - ix)«) 4- e'^Ei(-i()? + ix)«)]

Unfortunately, the exponential integral of general complex argument is not well tabulated.


8. THE RADIATION FIELD OF THE LINE SOURCE FOR ANY NUMBER OF LAYERS

The evaluation of the integrals for the general case of any number of layers is not readily carried out unless certain approximating conditions are introduced. The field E0, for the case of the line source and the observer in the interface (z = A = 0), is conveniently written

2π Jo «o + w» where

"•-"»Li + C Y ^ ^ M i J (98)

where Y2 is the wave admittance associated with the lower layers and is given by Eq. (48) et al. When hi becomes sufficiently large, it is seen that ue s ui and the fields correspond to a homogeneous half space.

Now, when the sub-surface layers are highly conducting, such that

WHvol it follows that um = (A2 + y2)1/2 = ym because the important values of λ are of the order of |y0|. To this approximation

ue s (A2 + yj)* s ye

where ye can be regarded as an effective propagation constant. It is approximately given by

y. Ä [ ( l y ^ O + t a n h ^ M y i ^ U + i ^ / ^ t a n h u ^ J ^ o

For a 2 layer medium (A2 -> oo),

y e = yi/ß where

(Yl/y2) + tanhyA ^ " l + WtìtanhrA v ;

which is identical to Eq. (13). In analogy to Eq. (64) for the homogeneous half space, the field E0 for the

stratified medium can be written

E^^^Kti^Q* ïor\yex\>\ (101)

The factor Q2 can be interpreted as a correction factor to account for the presence of stratification in the half space. A numerical discussion of Q has already been given.

9. MAGNETIC LINE SOURCE OVER A STRATIFIED MEDIUM

In the previous section, a line source of electric current / over a stratified medium was discussed. The corresponding formal solution for a line source of magnetic current K is obtained by making the transformations: I -+ K, am + i8mco -» z>mû), / / νυ -» am + iemœ9 Ez -> Hz, Hx = -Ex and i/y -►-£,,. The nature of the resulting integrals in the solution is, however, more complicated than in the electric-current counterpart.

The integral form of the solution can be written, for z < 0 ' f + 00

J —oo H0 ■■ ΐ ϊ ^ | [e*"·^*) + Ä„(A) e"*'-»] e""* dA (102)

where the + sign is to be used for (z + h) < 0 and the — sign for (z + h) > 0. The reflection coefficient is now given by

Ä | i ( A ) = §r l i (103)

where K0 and Zx have their usual meaning. For a homogeneous half space, for example, Z1 = K^ and if μχ «= μ0,

Α | ( Λ ) _ φ · ^ 11 rîwo + ?ο"ι

The coefficient Α||(λ) can have a pole when yjwo + Vowi = 0 which is the "resonance" condition discussed previously. This complicates the evaluation of the integral and eliminates the possibility of using the formulae developed for the line current source in the previous section. [There was no pole in the reflection coefficient RL(X) for a homogeneous half space, since there is no solution of u0 + ut = 0].

If attention is restricted, however, to the field a large distance from the surface, or if the height h is large, it can be expected that contributions from the pole (which are surface waves) are small compared to the specular component. Then with analogy to Eq. (57), the field is approximately given by

Ho = = ^ {Koboix2 + (z + ft)2]*] + R^Koboi*2 + (z - ft)2]*]}

(104)

where R\\(B) is the reflection coefficient for a plane wave (parallel polarization) incident at an angle Θ on the interface (z = 0) where tan Θ = x/(h — z). This result is only valid in the asymptotic sense and hence to the same accuracy the formula reads

where only the first term of the asymptotic expansion of K0 has been retained. As can be verified by the saddle point method (see next section), this result for H0 quoted above is the first term of an asymptotic expansion.

10. MAGNETIC LINE SOURCE OVER A DIELECTRIC COATED CONDUCTOR

As mentioned above, the integrals appearing in the solution for parallel polarization are complicated by the existence of poles. This occurs even in the case of a homogeneous half space. Another relatively simple situation occurs when the upper layer is a pure dielectric and the region below is a metallic conductor [Cullen, 1954]. The situation is illustrated in Fig. 7.

K i

y^s^y

(o, o, -h)

€o

• . . . €i

σ2

. / * · . . . . μ9 · ·

/ / V ' / V / = CD

• * Ϊ . • ·.·> · ; . ; . · ;.

Fig. 7. Magnetic line source over a dielectric coated conductor.

Noting that Z2 = 0, the field for z < 0 can be expressed by

— ΐε0ωΚ tfo =

where

4π ; r+°° Γ 0 ±«ο(ζ + Λ ) +

u o i'fcoA ett0(x_„/i ^ax άλ J-oo L U0 + ik0A J

AaaJgttanhMif a n d K i = «i rio σχ + ΐωελ

The integral has a pole where u0 + ik0A = 0, and this is denoted by λχ. Furthermore, there is a branch point where u0 = 0 which occurs when λ = k0.

Now for small values of Ai (i.e. k^h^ <ζ 1) Δ is also small (compared to one), and therefore

A*'-lo no K'-D

Therefore, to a first approximation, Δ does not depend on λ the integration variable and the surface impedance Ζί(^η0Α) is almost purely inductive. The pole is then given by

where λ, s fc0[l + (fe0/i)2],/2 = koll + (fc0/i)2/2]

Ä-Ml-fco/fc?)

(107)

The integral for H0 is now in a form where it can be evaluated directly by the modified saddle point method. A similar integral appears in the case of a dipolar excitation. For the present purpose, however, it is possible to develop the first few terms of the asymptotic expansion in a direct manner.

+ R

Branch Line

Fig. 8. Integration contour in the λ plane.

The integration contour is shown in Fig. 8. The integration in the original integral is along C0 from — oo to + oo. The pole λ^ and the branch point k0 can have a small but vanishing negative imaginary part if the conductivity of the air and the dielectric is finite. The contour C0 then passes above λί and k0 but passes below — λ1 and — k0 as indicated.

Following standard practice in contour evaluation of integrals, the contour C0 is closed by an infinite semi-circle (Ce + Cb) in the lower plane with an indentation along Cb, Cc and Cd to the branch point at k0. The integral along C0 is then equal to the integral along the complete closed contour C minus the


individual contributions of Ca, Cbi Cc, Cd and Ce. The contributions from Ca and Cb vanish as R -* oo since the real part of u0 > 0. Furthermore, the integral along the branch lines can be obtained as an inverse series in x (which is asymptotic in nature) by expanding the integrand in a Taylor series and integrating term by term. The integral around the complete contour C is simply — Ini times the residue at the pole Xt. Finally, it follows that the integral along C0 is — Ini x the residue at λί minus the contribution along the branch lines.

The final result reads for x > (h — z)

H0 = - ε 0 α > κ ί ^ Q-Uph Q+UpZ e"Ul*

+ .ermsco„Uini,1g^,iJ-L?i,elc.]

where "p = V ( ^ - f e o ) = - i f c 0 A (108)

The first term in the square brackets is the residue at the pole λγ and the succeeding terms are from the branch line integration. The latter series is asymptotic and is useable if

which requires that the pole λ^ is not too near k0. Such a condition occurs when the thickness ht of the dielectric coating approaches zero. It is then necessary to apply the modified saddle point method [Wait, 1957].

11. THE FIELDS OF A VERTICAL ELECTRIC DIPOLE OVER A STRATIFIED HALF-SPACE

In the previous section, the excitation of the fields over a stratified half-space was by a fine source. In this case the problem is a two-dimensional one. It is the purpose of this section to mention the extension to the case where the source is a dipole. This problem was formally solved by Sommerfeld in 1909 for the homogeneous half-space.

A vertical electric dipole is to be located at z = — A over the stratified region of M layers. The dipole is to be regarded as a current element of length ds carrying a current /.

The situation is illustrated in Fig. 9 where it is noted that cylindrical coordinates (p, φ, z) are employed with the z axis pointing downwards as usual. The primary influence in the upper region (z < 0) can be expressed in

1

\

1

1

1

' I

i!

9 ^

<=o / X o

*ι σ ι Μ ι

e2 σ2 μζ

«M °M MM

Fig. 9. Vertical electric dipole over an M-layered half-space.

terms of a Hertz vector which has only a z component πJ, which hereafter is denoted^'. Furthermore

id«? e" i f c o R o

*'-r^-V- (109) 4πιωε0 Κ0

where Ä 0 - [ p 2 + (z + A)a]*

It is now assumed that the total fields can be expressed everywhere in terms of a Hertz vector which also has only a z component, denoted \j/m. (The justification for doing this is attained by verifying that the final solution satisfies all the boundary conditions and behaves properly at the source and at infinity.) For the wth layer, \j/m satisfies

( A 2 - y ^ m = 0 (110) except at the source where

for R0 -* 0. This suggests writing

Ψο = Ψρ + ΦΌ (111)

where ψΌ is the "secondary influence". Since

( V 2 - r ô W = 0 (112) then

(V2-rô)<As = 0 (113) In cylindrical coordinates and for azimuthal symmetry, the equation for

\l/m becomes

6έ'έ+£-4>-° (114) It can be readily seen that solutions of this equation are made up of linear combinations of

p±«mz Jottp) Υ0(λρ)

where, as usual,

and

for m = 0, 1, 2, ... , M. Again by convention, the real part of um is to be taken positively. Since the resulting solution is to be finite on the axis p = 0 when z Φ —A, the Υ0(λρ) Bessel function can be rejected since it behaves as log p for small values of p. The JQ(Xp) Bessel function on the other hand, is finite at p = 0.

As in the previous solution for the line source, the parameter λ can take any value (with a few exceptions). The general solution for \l/m is then written in the form

Ψ» = j™ iaJX) e""»* + bjji) cP«]J0(^) ^ (115)

for m = 1, 2, ... , M — 1. In region (0), for z < 0

Ψο = Ψρ+ Γ VQ{X) ewoV0(Ap) άλ (116)

and in region (M), for z > zM_x

^M = JJ«M(A)e"—dA (117)

To express φρ as an integral of the appropriate type, use is made of the Fourier-Bessel transforms which read

Κρ) = ^9{λ)30{λρ)λάλ (118) and

gß) = jy(p)J0(.Xp)pdp (119)

subject to the conditions that the integrals exist. Letting /(p) = e~ikop/p

9(λ) = j\-ikopJ0(Xp)dp (120)

This integral is a standard type and is given by

^'wh&'i <12» and therefore

e""0' f" Jp(Ap) e-'*°*° - J : (122)

To generalize the preceding to any other value of z, the integrand must contain the factor β

±Μο(ζ+Λ) since e~ikoRo/R is a solution of the wave equation and necessarily becomes e~ifcop/p as z -> —A. Consequently

e ^ f - J ^ e ^ ^ ( 1 2 3 )

^o Jo uo

where the sign of the exponent is chosen so that the integral converges. (i.e. + for (z + h) < 0 and - for (z + h) > 0).

The z component of the Hertz vector for the upper region (z < 0) can then be written

where

Ψο=\ «o + b°W Jo(Ap)A ^ - 0 , Emz={-yi + ^ m (125)

and

H»P = 0, //,„, = ^ -£=, J/m2 = 0 (126)

and since £mp and Zsm</> are continuous at the interfaces zm(m = 0 to M — 1), it follows that

dij/Jdz and (am + iœem)\l/m

Reflection of Electromagnetic Waves 39 are also continuous at the same interfaces. This leads to 2(M — 1) equations to solve for the 2(M - 1) unknown coefficients. It readily follows that

where -^Γ-*(Α)-Κ0(Λ) + Ζι(Λ) (127)

iOl) + Km(A)tanh «mftm

iCm(/l) + Zm+1(/)tanh umhm

The algebraic form of R(X) is identical to Eq. (5) for the plane wave reflection coefficient for parallel incidence. It is thus clear that the field of a dipole over a stratified half space can be regarded as a spectrum of plane waves whose angle Θ of incidence and reflection is related to the variable λ by λ = k0 sin Θ. In this case, the wave normals generate a family of cones co-axial with the z axis.

Equations (127) and (128) in conjunction with Eq. (124) constitute the complete formal solution. Approximate evaluations [Wait, 1957] can be carried out using the same approach in the previous sections or by a modified saddle point method [Wait, 1954]. In view of the similarity with the line source excitation, it does not seem justified to discuss this in detail and the essential results will be simply quoted.

As has been demonstrated, Z^X) for a highly conducting half-space or even for a dielectric coated conductor, is mainly determined by the electrical properties of the layers. In other words, R(X) can be replaced by

J K i ) - ^ (129) u0 + ik0A

where Δ = ZJf/o = ZJllOn is assumed to be constant. The form of φ0 is then

ψ0=J-iL p r e±M.+*>+"o-^oA e t t 0 ( 2_j (λ άλ

4πίωε0]0 [ u0 + ik0A J This can be rewritten as follows

:

Ids [e~ik<>Ro Q~ikoR

where

and

ψ0 4πΐωε0 [ ί 0 ϋ , J v '

*ο = [Ρ2 + (ζ + Λ)2]*, R, = [ρ2 + (ζ - Ä)2]*

p=pOMA)c-^-^) Jo ("o + ^οΔ)"ο

D

This integral has been previously evaluated [Wait, 1954, 1957]. Subject to |Δ| <ζ 1, it is given by

Q-ikoRi P s i(npeyA e"w erfc(iw1/2) — — (133)

« i where

w =4+^ί Pe = zi^RlA2 = lPeleü

and 2 f ° _ v 2 erfc(îw/2) = — e * άχ

- (?)»-- F ( w ) ] — — (134)

The expression for P can also be written

where F(w) = 1 - ί(πνν)1/2 e"w erfc(iw1/2) (135)

It is noted that when z = h = 0 / d s e"ifcop , x

^ = 2niœe0 p W

and the vertical electric field is given by

E ψΛ,,-*^ {orkp>l 2np

The function F(pe) can be regarded as the correction to the field of a dipole on the surface of a perfectly conducting plane. For \p\ <£ 1 it approaches unity. F(pe) has the same functional form as the ground wave attenuation of Norton [1935-1941] who presented numerical values for the case where (in the present notation), b is in the quadrant 0 to —180 degrees. In the case of a homogeneous ground Δ has a phase angle in the range 0 degrees (for a perfect dielectric) to 45 degrees (for a good conductor). The corresponding values of b are — 90 degrees and 0 degrees respectively. In the case of a stratified ground, however, the phase angle of Δ may be outside this range. In the particular case of a two-layer conducting half-space the parameter pe can be written

Pe = PiQ2 where p1 = —L2£fy,1fo0)2

where

0-\0\ci«~ ( w J + t a n h y ^ 1 1 " l + f r i ^ t a n h y ^

in terms of the propagation constants γχ and y2 of the upper and lower layers and the thickness Λ1 of the upper layer. This function Q was discussed in Section 2. From Fig. 3b it is seen that for a highly conducting substratum q can become positive. Consequently, the phase angle b can exceed 0 and would approach 90 degrees for

|y2/yi|-*oo and

Vi^i < 1

In analogy to the terminology introduced by Sommerfeld for the homogeneous half-space, pe is here described as an effective numerical distance. The propagation factor

is plotted in Fig. 10a and 10b as a function of \pe\ for a range of b values. The curves for b < 0 correspond to those computed by Norton and it is interesting to note that for this region \Fe\ never exceeds unity. On the other hand when b > 0 \Fe\ may exceed unity and when b = 90 degrees this effect is most pronounced. Apparently, in the case where b is positive, the energy is being guided to some extent along the surface. This effect can be seen in the asymptotic development of F(pe) which for (pe) > 1 reads

(136)

when

and

WI1CI1

1 1 x 3 1 x 3 x 5 1 x 3 x 5 x 7 ( ^ = ~ ^ ^ ~ ~ ( W (2Per "'

- 2 π < 6 < 0

In > b > 0

(137)

The term —2iyJ(npe) c~Pe has all the characteristics of a surface wave. It is not present in the asymptotic development when b is negative. At b = 0, 2π or — 2π, pe is real and this term vanishes asymptotically. The trapped surface wave is most predominant when b = 90 degrees which corresponds to Δ or Zi being positive imaginary. For example, if the surface is coated by a thin dielectric film of thickness hY with a dielectric constant εΐ9

Z1 s ιμωΗ^ί - eQje{\ s ιχ (138)

which is purely imaginary. Then since

Pe = ^ 2 £ ( Z 1 / » / o ) 2 (139)

l . O

r> m LU Γ o σΐ Q .

() 1_

2.8 2.6 2.4 2.2 2.0

1.8 1.6

1.4 1.2

10 0.8 0.6 0.4 0.2

^

/ ^

/^ j

" ^ ^ -50*0-- 6 0 · ^

#

^

^ /

/ /

<$

/ / , /

■ -■ ^50

——^0°

s——^

â^

\ . \ V

\ \

\ \ \ \

>\\ \ |p^ ^Ιϋ

\ \ \ λ v\ V Λ

\ \ ^

a

^ t ^ , 0.001 0.002 0.005 001 0.02 0.05 0.1 0.2 0.5 I 2 5 10 20 S

Effective Numerical Distance |pel

FIG. 10a. Amplitude of propagation factor for an impedance boundary.

350

300

250

200

150

100

A. è 1 ^

/ / II

a, jl 1 11/ i

V /

• - \ \ \

/ • - NN

b

b '75* 1

40" 30° 20e

10° I 0e

-20*

- 40 e

-60e

- 9 0 ·

-no· 1

0.02 0.05 0.1 0.2 0.5 I 2 5 10 20

Effective Numerical Distance |p e | 50 100

FIG. 10b. Phase of the propagation factor for an impedance boundary.

Pe = \Pe\ QÌn/2 or b = π/2 rad = 90 degrees. The attenuation factor now has the asymptotic development

FM « -2 e"«V(.W> =-'"·' + 2H + il^P - 8H5 - «*» where

k0pX \Pe\ =

and the vertical electric field is

Pe]=^m Εο.*ψ±ο->Ρ(ρ1 (141)

2πρ which shows that the (trapped) surface wave component varies as

which has the characteristics of a cylindrical wave travelling in the positive p with a phase velocity

(-ΙΓ times that of light.

The (trapped) surface wave is not excited when b is negative and then the distant electric field varies asymptotically as

1 (koP)'

. Q-ikop

which has a phase velocity equal to that of light. When the source dipole and the observer are both raised above the surface

(i.e. A > 0 and — z > 0), the field in the upper half space can be expressed conveniently in the asymptotic sense as the sum of three partial fields in the manner

* « - » - + *♦+ « « 4 ^ ( 1 4 2 )

where ε = 0 for arg w < 0

and ε = 1 for arg w > 0

An asymptotic development shows that -ikRo ,C _ Δ \ Q-ikRt

*-^T+(cTî)— <1 4 3>

f 1 1 x 3 1 x 3 x 5 \ e~ikR> ψ" " \p(l + C/Αγ + 2p2(l + CIA)5 + 4 / ( 1 + C/Δ)7 + " · ) Rt

( 1 4 4 )

where

and

2Δ , e"ifcKl

* ■ = " Ä T C [ 2 i ^ ( 7 C W ) e " w ] Ί * Γ ( 1 4 5 )

Λ k - z k + lzl . „ . C = = —, k = 27r/wavelength = k0

A = Zji/o In terms of I/O, the vertical electric field components for k0p > 1 are given by

£0 2^feâ( l-C2) iAo In the above ψα can be identified as a geometrical optical term being just

the primary field e~ikRoIR0 and a specularly reflected component e"ikRiIRi modified by a reflection coefficient (C — A)/(C + Δ). i/^ is an asymptotic expansion containing terms varying as R^2, R^3

9 etc. \\/s which is only present for positive b values, represents a (trapped) surface wave and has a phase velocity less than that of light. When b is negative there is no (trapped) surface wave present.

Sometimes \j/a itself is called a space wave and φ0 a surface wave. This usage would correspond to that of Norton [1935-1941]. This would be an obvious designation when the half space is homogeneous since ψ3 is zero and there is no trapped surface wave excited.

Equations (143), (144) and (145) enable one to compute the field in the upper half space (z < 0) for an electric dipole located at z = —A in terms of the ratio Δ of the surface impedance Z t to that of free space η0. As a first approximation ZY could be replaced by Ζ^λ) for λ = 0 which would be the normal surface impedance. Clearly, a better value would be Zt(X^) where ks is the saddle point of the integral in Eqs. (130) or (132). This would mean that

Δ = - ^ (146) rjo

where

where S-P/K!=»V(I-C2)

In the case of a homogeneous half space, with μί = μ0

Δ = — ι±Ζ = u s \ (147) »7ο (σι + ιωει)//0

^-lU-risA* (148) »ίο\ *Ιο /


The coefficient (C - A)/(C + Δ) in Eq. (143) now can be identified as the Fresnel reflection coefficient for a plane wave with parallel polarization incident at angle 0. The equations are now in complete agreement with the results of Norton [1935-1941] for a vertical electric dipole over a homogeneous conducting ground.

The corresponding treatment for a vertical magnetic dipole located at z = — h over the M layered half space is almost identical to the above. The quantity ψ0 is then to be identified with the z component of the magnetic Hertz vector and then Δ is to be defined by

where Υχ is the surface admittance at the interface z = 0.

12. SOME EXPERIMENTAL MEASUREMENTS

A number of investigations on the influence of stratification on the ground wave of radio transmission have been published. Many years ago Hack [1908] demonstrated that the tilt of the Zenneck wave was affected by sub-surface stratifications. During World War II, Grosskopf and Vogt [1940, 1941] extended this work. They indicated that variable frequency measurements of the electrical constants of the ground could be satisfactorily accounted for in terms of a two-layer model. Unfortunately, their interpretations can lead to certain errors since the Zenneck wave was assumed to be the major component of the ground wave. As shown in Sections 4 and 11, the wave tilt of the Zenneck wave is not the same as the total ground wave of a dipole transmitter. The wave tilt in the latter case is almost identical to that for a vertically polarized plane wave at grazing incidence.

Quite recently, Eliassen [1957] made a very systematic study of ground wave propagation over actual stratified media. In the first series of measurements he chose a thick ice-covered lake near Odnes, Norway. The transmitter was located on the shore and the measuring equipment was situated in the middle of the lake at 2 different sites spaced about 1 kilometer apart. The wave tilt WfoT frequencies in the range 1-10 Mc/s were then found from the characteristics of the "polarization ellipse." The results were repeated at a lower frequency using transmissions from the Klofta broadcasting station at 218 kc/s. For both sites, the water depth was over 30 m and ice thicknesses were approximately 46 and 80 cm.

Eliassen's results, of the amplitude and phase of W9 are shown in Figs. 11a and lib, respectively. The dots (for 46 cm ice depth) and the circles (for 80 cm ice depth) represent the "best of 3 measurements." The "O" curve, in each case, is calculated for a homogeneous half-space of relative dielectric constant ε2/ε0 of 90 and a conductivity σ2 of 1.8 mmho/m. These values are

appropriate for the lake water. At the lower frequencies, it appears that this curve fits the data reasonably well. However, at the higher frequencies, the experimental points depart considerably from the ·'homogeneous".

0.30

0.20

UJ

0.10 h

h~

L

—

, , , , , - „ ! , , , o ON 80 cm ICE • ON 46 cm ICE

O - ^

o ■"■ ^ 1 ^ ·

• —

, . ......1 . . .

,..„, J

h = 8or

oy J / · / H

o 1

I I

I II

70 w _ ΰ«0 a: CD ΓΛ uj 50 s 5s40 u. °30 S 20 CL

10

1 1—ι ; I M I 1 1—r-ρτττπ <

-h = 8 θ / !

/ ~Λ

% 9 M Q J

/ · / - ^ / · / J \ ^ / / 20,

>ν ν >/# / y\ ^^>——^^^ >/ >/ r ^ : : = ί τ Τ ^ Χ

1 ^ ι 1 . 1 Oi E 1 ι ι 1 > ι n i ι—ι ι I ι > n i 3

0.1 1 10 0.1 I 10 FREQUENCY (Mc/S)

FIG. 1 la and b. Measurements of wave tilt on an ice-covered lake (Eliassen, 1957) and calculated curves for a two-layer model.

It is evident that the ice introduces appreciable changes in the wave tilt at frequencies above 2 Mc/s. For further comparison with theory, 3 additional curves are drawn in accordance with Eq. (10) for a two-layer earth and at grazing incidence (i.e. φ = 90 degrees). Here the relative dielectric constant ε^εο of the ice was taken to be 4 and the ice thicknesses h are chosen to be 20, 40 and 80 cm. It is evident that the experimental results are in good accord with these calculations.

Eliassen [1957] also indicates that the field strength vs. distance curves are in accord with theory [Wait, 1953] if the numerical distance/? in the Sommerfeld formula has the more general meaning (see Section 11).

It is also of interest to note that seasonal changes in the effective ground constants can be interpreted in terms of the variable stratification of the soil. Again, it was Eliassen [1957] who made three series of measurements near the Kjeller airport in 1955:

(a) Feb. 22-23, dry air, 1 ft of snow, ground frozen, air temperature, - 5 to -10°C;

(b) May 13-16, no snow or ice, very wet ground after heavy rain, temperature + 10°C.

(c) July 26-28, dry air, very dry earth surface, temperature + 30°C. The terrain was ideal for wave tilt measurements being a flat area covered

with short grass only and ordinarily quite wet. The experimental points are shown in Figs. 12a and 12b. For the calculations, a simplified model of the


situation is introduced. An infinitely deep substratum of high conductivity is located below a poorly conducting upper layer formed by frost in winter and drought in summer. In the spring it is assumed homogeneous throughout with σχ = σ2 = 13 mmho/m and ε2/ε0 = ει/εο = 50. The latter condition yields the OA curves, in Figs. 12a and 12b, calculated from Eq. (12) with μι = μ0. For the spring data this fits the \W\ curve very closely; however,

- 0.20

t i 0.10 Ld

â 0.05

0.02

i

—

o

" , , , , , , „ 1

I 1 1 | M M . 1

Jpt*^ — à WINTER :

• SPRING 1 o SUMMER

1 ,! I 10

FREQUENCY (Mc/S)

FIG. 12a and b. Measured seasonal variations of wave tilt over a flat clay area (Eliassen) and calculated curves for a two-layer model.

there is some departure in the phase. The best curve for the experimental phase data is shown by OA'. Assuming that σχ = σ2/10 and εχ = ε2/10, the theoretical modification resulting from the presence of the upper layer is also shown in Figs. 12a and 12b for h = 20 and 40 cm. It is again evident that the stratified two-layer model explains the experimental data. Of course, every detail of the variation is not accounted for. For example, the departure of OA and OA' in the phase curves at the lower frequencies is due, no doubt, to deep-lying layers which do not exhibit seasonal variations.

Another series of measurements of wave propagation over a stratified earth was reported recently by Stanley [I960]. The relative field strength of a vertically polarized wave at 236 kc/s was measured as a function of distance over several radial paths in the vicinity of Point Barrow, Alaska. Within the first 100 miles, the attenuation of the field was actually less than one would expect for a perfectly conducting ground plane. It was then noted that highly conducting saline layers or lenses exist at depths from 10 to 100 ft in the area around Point Barrow. Thus, it was considered reasonable to approximate the ground by a two-layer model. The conductivity of the lower layer was assumed to be that of sea water (i.e. 5 mho/m); the conductivity of the upper layer was known to be about 10 mmho/m. It was then possible to fit the data to the theoretical curves [Wait, 1953, 1954], such as shown here in Fig. 10a, if the thickness hv of the upper layer was 9 m or about 30 ft. Depths of this order were confirmed by test drillings.

Appendix A

EVALUATION OF THE INTEGRAL P

The evaluation of the integral P from Section 11 is now discussed. It is

F_p(^M)e--»-> A Jo (wo + ^ Ο Δ ) Μ 0

as in Eq. (132). Before proceeding, certain conditions and restrictions must be clearly stated. At a later stage of the analysis these may be relaxed somewhat. For the moment

|Δ |<1 π

0 < arg Δ < -4 and

Since k0p> 1

J0(x) = KHtfXx) + H(0

2\x)2 and

where #01} and //0

2) are Hankel functions of the first and second kind, respectively, it follows that

il· A f °° 2 f>-uo(h-z) P=^f\ / ., Α, Η^Χλρ) άλ (150)

2 J_00(Wo + ifc0A)w0 " V

Introducing the substitutions

>i = /c0cosa and Δ = sin a0

it is seen that

4 J-.oo Ρ Β 3 Ι | cos a e - ^ - ' ^ ' f l y ^ f c o p cos a ) ^

" ' . a + a0 a — a0 sin —-— cos —-—

Essentially, P is now a spectrum of plane waves travelling away from the ground plane such that the vertical component of the propagation constant is k0 sin a. Again, complex angles are included in the spectrum. When |/:0pcosa| > 1, the first term of the asymptotic expansion of the Hankel function can be employed. That is,

[ 2 VA

jkop cos a e - i „ / 4 / 1 5 2 ) nk0p COS aj


When this is inserted into Eq. (151), the equation of P becomes

P = e ^ ^ (cosa)_e ^ 2\2np) J__ioo . a + a0 a - a0 sin —-— cos —-— 2 2

where Λ and 0 are defined by

h — z = K sin 0 and p = — R cos 0

In the usual case, the separation p between transmitter and receiver is large compared to their respective heights, h and — z, and consequently, 0 is slightly less than π. It can be seen that the exponential factor in the integrand of P is rapidly varying except for a region near a = 0 = π. This is, of course, the saddle point of the integrand or the point of stationary phase. The important part of the integrand is in the region near the saddle point. In fact, this is the justification for employing the asymptotic expansion for the Hankel functions Htf); the argument k0p cos a is always large in the region near the saddle point if k0p itself is large and h — z is less than p.

The integral for P is now in a form where the saddle point method of integration can be applied. The usual technique [Sommerfeld, 1949] is to deform the contour (which in the present case is along the negative imaginary axis, the real axis from 0 to π, and then along a line parallel to the positive imaginary axis) to a path of steepest descent. For example, if we let

cos(a — 0) = l — h2

and let τ range from — oo to + oo through real values, an integral of the type •π + ίοο

G(cos a) e-*0* cos(0["*) da (154) I ioo

where G(cos a) is slowly varying at a = 0, is transformed to

In this deformation of the contour, account must be taken of the singularities of the integrand that are crossed. In the case of poles, Ini times the sum of the residues is added to the new integral. The final step in this classical procedure is to expand G(cos oi)/^J(l — ιτ2/2) in a power series in τ enabling the integration to be carried out term by term. The leading term of this resulting asymptotic expansion is

(2TT01/2G(COS 0) e-ikoR(k0Ryv> (156)

and succeeding terms contain (k0R)~y2, (k0R) 5/2 and so on.


As is often the case, the contribution from the saddle point (which is the branch point in the original λ plane) cannot be separated from the pole(s) of the integrand. For example, in the present problem,

(cosa)1/2e"ifcoRcos(e"a)

G(cos a) = V — (157) . a + a0 a — a0 sin cos —-— 2 2 has a pole at α = π + a0 which is near the saddle point α = Θ since a0 is small and Θ is near π. In other words, the integrand is not slowly varying near the saddle point on account of the factor cos(a — a0)/2. The integral we have to contend with is then of the form

Λπ + ί«

J - t oo

n + ioo e ~ ik0R cos(0 - a)

a - a 0 cos—-— da (158)

after the slowly varying factors have been separated out. The necessary modification of the saddle point method to treat integrals of this type was devised by Van der Waerden [1950] and Clemmow [1950]. On making the usual deformation of the contour via the subsitution cos(0 — a) = 1 — /τ2, the integral now becomes

/ ~ - 2 e"iÄ/4 e - ^ V 2 - c o s ( ^ 2 ) I " V " 7 ? ΓΠ (159) / 0 - a o \ f + 0° e-*oKt2dT

It can readily be verified that no poles of the integrand are crossed in this deformation for the argument of a0(or Δ) less than 45 degrees. The integral to consider is now of the type

A = I zr-7- dx (160)

where, for real τ, χ = k0R and

Since | e ~ * t 2 d T = l - l it follows that J -a

^ . ^ " . î f * ^ . - * (161) «W cjo T2 + C

Now, it can be seen readily that d „ J - :cA=- (-) 2 e"*c (162) d.v \x/


and an integration with respect to x from x to oo leads to

-m Q-XC dx (163)

or, after a change of variable,

A = 2(TT/C),/2 QXC I e"z2 dz (164)

which holds for all c in the c plane with a cut along the negative real axis. This can be written

A = 4~e*c erfc J(xc) (165)

where erfc is the complement of the error integral given by

(166) erfc z0 = —r- e z2 dz ν π Λ ο

where the integration in the z plane is directed from z0 towards the right to + 00. Therefore

/ = - iln Q'koR e"w erfc(iw,/2) (167) where

w = -k0Rc = -i2ik0R cosi ""α° |

and finally, since R = r2,

P = ί °°, ^ ^ M e-^-^Vo^P) dA (168) Jo (w0 + ^oA)w0

S i(np)V2 e"w erfc(iw1/2) e-ifc°r2/r2

where

and - ( - ^ ) !

While the derivation was carried out for the argument of Δ lying in the range 0 to π/4 or b in the range 0 to — π/2, the formula is actually valid for all values of b if the above definition of the error function complement is used. The justification for this step is based on the principle of analytical continuation. If, in the initial derivation, the argument of Δ was allowed to exceed π/4, we would find that a pole would be crossed in the deformation of the contour in

the a plane. There would be a corresponding change in the integral A but, after a change of the sign of the variable τ, the expression yielded for / would be unchanged in form if we are consistent in the definition of the complex error function.

As mentioned at the beginning of the Appendix, the formula for P given by Eq. (168) is valid only if |Δ| is small compared with unity. When |Δ| is no longer small it is necessary to employ an alternative method. Actually, the approach is simpler since the branch point at λ = k0 is no longer close to the pole of the integrand. Therefore, the conventional saddle-point method may be used. To apply this method it is convenient to express P in the equivalent form

e - ikor2

where

where

P = + ik0Q(a, Δ) (170)

ρ(α, Δ) = e"i7coCa(C + Δ)"lJ0(k0pS)S dS (171) (α,Δ)=Γ

a = h-z and C = (1 - S2)v> On deforming the integration contour into the steepest descent path,

attention must be paid to the location of the pole at C = — Δ. The resulting asymptotic approximation is readily shown to be

cos Θ e ~ lfe°r2

β(α, Δ) S - r — j-η—, + qQs(«, Δ) (172) cos Θ + A { — ik0r2)

where a (h-z)

COS θ = —2 Ô7U = (a2 + p2)/2 r2

ρ4.(α, Δ) = - ϊπΔ e*°r2 cosA eH02\k0r2 sin θ^(ί - Δ2)] (173)

and q = 1 for arg sin - I - — Θ + arc sin ΔI > - (174)

= 0 for arg s i n - l - — 0 + arcsin ΔΙ < -

The firm term in Eq. (172) is the contribution from the saddle point, and the second term is the residue of the pole which may or may not be captured in the resulting deformation of the contour. The above expression for β(α, Δ) and the corresponding one for P are valid in the far field such that terms which vary as l/i£2, 1/R3, etc. may be neglected.

Rigorous expansions for the integral P have been given by Furutsu [1959]. His results would indicate that our asymptotic forms are adequate for most cases of practical interest.


Appendix B

NUMERICAL RESULTS FOR SURFACE IMPEDANCE OF A STRATIFIED CONDUCTOR

The surface impedance of a stratified conducting medium is seen to be an important quantity. In a large measure it determines the propagation constants for waves gliding along its upper surface. Furthermore, in many actual cases, the surface impedance may be measured directly, as indicated in Section 12 of this chapter.

To calculate the surface impedance, or a related parameter, for a stratified conductor is a tedious business. In many cases, however, the conduction currents are large compared with the displacement currents in each of the strata. Under this situation, the propagation constants yt(i = 1, 2, 3 ...) have a phase angle of π/4 rad. Thus

for each layer. It is understood that the magnetic permeabihty μι for each layer can be replaced by the constant μ. A model composed of two or three such homogeneous layers is often an adequate representation for the earth's crust.

To facilitate application of the theory outlined in the present chapter a three-layer model is assumed. The situation is depicted in Fig. 4 for M = 3 where the source field is a horizontal line current. For this case, and with the specializations mentioned above, the surface impedance at the upper boundary may be defined by

Z = — = - ^ 1 = Ely\ ^ι H0x]z=0 Hlxj

Explicitly, it follows from Eq. (50) et seq., that

where

and

In the above

1 Ni

_ UiQ + u2 tanh ulhl

u2 + utQ tanh u1h1

* _u2 + u3 tanh u2h2

u3 + u2 tanh u2h2

Ul = (λ2 + ίσφωΥ2

u2 = (Λ-2 + ΐσ2μώγ/2

"a = (>*2 + *>3μω)1/2


As mentioned previously, λ can be identified with k sin Θ where Θ is the (complex) angle of incidence. In many applications λ2 may be neglected compared with σμω and thus wf s yf for i = 1, 2, 3. Under this condition

z,-(£)"« where

Q = yi6 + yitanhy1h1

72 + Vie tanh y^ and

A = y2 + 73 tanh y2ft2 y3 + y2 tanh y2fc2

This expression for the surface impedance is exact for a normally incident plane wave and it is an approximation when λ is finite. The validity of the approximation is discussed below.

Calculations of the amplitude and phase of Q, for ?! = y3 and for various values of (yi/y2), have been carried out by programming the equation on a

M l 0.03 0.1 0.3 1.0 3.0 10.0 1

( 0 [ / i O J ) 2 h,

FIG. 13a. The amplitudes of the correction factor Q for a two-layer conducting medium (λ = 0).


(Cr; yLLCü) 2 h,

FIG. 13b. The phase of the correction factor Q for a two-layer conducting medium (λ = 0).

high-speed computer where the abscissa in each case is (σ1μω)^Α1. The results are shown in Figs. 13a-1 6b. In the first set (i.e. Figs. 13a and 13b) the thickness h2 is infinite, so the situation may be described as a two-layer model. It can be seen that when {σ1μωψιΗι is greater than about 3, the amplitude \Q\ is indistinguishable from unity and the phase of Q becomes zero.

The curves in Figs. 14a and 14b refer to a three-layer model where the conductivity of the bottom (semi-infinite) medium is the same as that of the upper stratum (i.e. at = σ3 or yt = y3). Furthermore, the thickness ht of the upper stratum is the same as the thickness h2 of the lower stratum. The shape of the curves is rather interesting. As (σίμω)ί/2Ηι becomes very small or very large, Q approaches unity. In the former case, the electrical thickness of the lower stratum is so small that its influence on the surface impedance is negligible.

In the latter case, the electrical thickness of the upper stratum is so large that the intervening stratum is not "seen". The curves in Figs. 15a and 15b, for h2 = 2hu as well as those in Figs. 16a and 16b, for h2 = hJ29 are very similar.

E

IO"4 IO"3 IO"2 0.1 0.3 1.0 3.0 10.0 1

(OJ/JLGU) 2 h, FIG. 14a. The amplitude at the correction factors Q for a three-layer

conducting medium (λ = 0).

{σ-{μω)ζ h, FIG. 14b. The phase of the correction factor Q for a three-layer


IO"4 IO"3 IO2 0.1 0.3 1.0 3.0 DJO

FIG. 15a. The amplitude at the correction factors Q for a three-layer conducting medium (λ = 0 ).

IO"4 IO"3 IO'2 O.l 0.3 IJO 3.0 I0.Ò I

(σ , /χω) 2 h, FIG. 15b. The phase of the correction factor Q for a three-layer



[σ[μωί2 h, FIG. 16a. The amplitude at the correction factors g for a three-layer


0.1 0.3 1.0 3.0 10.0

[σ\μω)ζ h, FIG. 16b. The phase of the correction factor Q for a three-layer conducting

medium (λ = 0).


Curves of the type shown in Figs. 13a and 16b are useful in a wide variety of problems. However, the source field must be such that the effective value of A (or k sin 0) is small compared with {σ^ωψ2. To indicate the significance of this approximation, calculations of Q were carried out for a two-layer model (i.e. hx = h and h2 = oo) when λ is finite. In this case

r ... Λ Μ * Q

where 6 = W „ 2 ) + tanhM a n d ßsm λ>

1 + (w1/w2)tanh uji σχμω

It is seen that when the thickness h of the upper stratum approaches infinity λ becomes unity and Zl has the value appropriate for a homogeneous half-space. Thus Q is again a measure of the influence of stratification. For convenience in the calculation, one may write

where

Ul = c\ l~iß Γ

u2 [l - ißC2]

li W Curves of the amplitude and phase of Q are shown in Figs. 17a-19b. In

the first pair, σ2 is effectively zero which corresponds to a conducting layer (of thickness h) lying on very poorly conducting substratum. It is apparent from these curves that a finite value of β leads to a major change in the behavior of the curves. This would indicate that one should be extremely cautious in interpreting experimental data on the surface impedance when the source field is not known. The same behavior is evident when the lower layer is finitely conducting as can be evidenced in Figs. 18a and 18b where σ1/σ2 = 25. However, in this case the effect is not as pronounced. When the lower layer is relatively highly conducting such that σ2/σ1 = 25 the influence of finite β is relatively small as can be seen in Figs. 19a and 19b.

The results shown in Figs. 17a to 19b are applicable to a source field variation which is periodic in the horizontal (x) direction. Thus, the variation may be of the form (±ikx\ cos λχ or sin λχ where the period is 2π/λ. For an actual source one must superimpose these solutions and integrate over all real values of λ. This is discussed in Section 5 for a line source of current.

The numerical results given in this appendix were obtained from a highspeed computer (CDC 1604) using a program devised by Mrs. Carolen Jackson and Mrs. Lillie Walters of the Boulder Laboratories of the U.S. National Bureau of Standards.


10 F

3p

i F

F L

l· r

F L

l·

r ^ r - 2 0 Q /

0.005

001

0.02

005 008 0.1

0.2

05

1

— Γ ~ Γ Τ

1 1

-TTTJ 1

. M i l 1

- ρ - 7 - τ - Γ τ τ - η - i

σ 2 Ό

1 1 -L--L M i l

1 ' 1 J

d

H

^!>-** ' Z

-

1 1

1

0.01 0.03 0.1 Û3

FIG. 17a

1 | 1 I 1 I

> = 0 _ 0.05 -—_

o.i ■ 0.2

- 0.5

1 1 1 1 1 1

i | ' l ' i

i l ! 1 1 1

1 M 1 1 !

A-25

1 1 1 1 1 1

1 -"-

-

~"Ή 0.01 0.03 0.1 0.3

(σμω)ι/ζ h FIG. 18a

0.3 h

0.1

-

—

' 1 '

^___Q5__ — I)

! 1 1

1 I I l | 1 |

of=25

1 1 M 1 1 1

1 1 1 1 1 1 |

1 M i l l i

1

""05

1

1

1 0.01 Q03 0.1 0.3

(σ;μω) h

FIG. 19a

FIGS. 17a, 18a, 19a. The amplitude of the correction factor ß for a two-layer medium showing the influence of finite λ.

[σμω) ch FIG. 17b

0.4

0.2

0

-Q2

-Û4

_ _ J I i I :

= ^ ^ \ ~~ ^^^v.

^ ^ ^ " ^ . — " ^ _ "" ""—^

" ^=25

i 1 i i i

I 1 1 | 1 | 1 1 ! 1

\ x xN^ \ Xy ^P/ X. X·

\X >\X. \^vx>-i j i 1 i L l l l l

ni 0.1

0

-0.1

i i l

I 1 I

ß2L~ ^ ^ ^ 5

ξ£°2 -7 1 1 j f 3

f

-

i 1 i 0.01 0.03 0.1 0.3

(σ;μω)[/2 h FIG. 18b

< Q <

FIGS. 17b, 18b, 19b. The phase of the correction factor ß for a two-layer medium showing the influence of finite λ.


REFERENCES

ATWOOD, S. (1951) Surface wave propagation over a coated plane conductor, / . Appi. Phys., 22, 504-509.

BARLOW, H. E. M. and CULLEN, A. L. (1953) Surface waves, Proc. I.E.E., 100, Pt. Ill, 321-331.

BARLOW, H. E. M. and FERNANDO, W. M. G. (1956) An investigation of the properties of radial surface waves launched over flat reactive surfaces, Proc. I.E.E., 103, 307-318.

BOUWKAMP, C. J. (1950) On Sommerfeld's surface wave, Phys. Rev., 80, 294. BRICK, D. B. (1954) The radiation of a Hertzian dipole over a coated conductor, Mono

graph No. 113, Institution of Electrical Engineers (London). CAMPBELL, G. and FOSTER, R. (1948) Fourier Integrals For Practical Application, Van

Nostrand, New York. CLEMMOW, P. C. (1950) Some extensions to the method of integration by steepest descents,

Quart. J. Mech., 3, 241-256. CULLEN, A. L. (1954) The excitation of plane surface waves, Proc. I.E.E., 101, Pt. IV,

225-235. ELIASSEN, K. E. (1957) A survey of ground conductivity and dielectric constant in Norway

within the frequency range 0.2-10 Mc/s, Geophys. Pubi, 19, No. 11, 1-30 (Norske Meteorologiske Institutt, Oslo).

ELLIOTT, R. S. (1954) On the theory of corrugated plane surfaces, Trans. I.R.E. AP-2,71-81. FURUTSU, K. (1959) On the excitation of the waves of proper solutions, Trans. I.R.E.,

Special Supplement, AP-7, 209-218. GOUBAU, G. (1950) Surface waves and their application to transmission lines, / . Appi.

Phys., 21, 1119-1128. GOUBAU, G. (1952) On the excitation of surface waves, Proc. I.R.E., 40, 865-868. GROSSKOPF, J. and VOGT, K. (1940) On the measurement of earth conductivity, Telegr.-u.

Fernspr. Techn., 29, 164-172. GROSSKOPF, J. and VOGT, K. (1941) The measurement of electrical conductivity for a strati

fied ground, Hochfrequenztech. u. Electroakust., 58, 52-57. HACK, F. (1908) The propagation of electromagnetic waves over a plane conductor, Ann.

Phy., 27, 43. JAHNKE, E. and EMDE, F. (1945) Tables of Functions, 4th ed., Dover Publications, New York. MARCUVITZ, N. (1952) Guided wave concept in electromagnetic theory, Polytechnic

Institute of Brooklyn, Microwave Research Institute, Res. Rpt. R-269-52 (P.I.B.-208), ONR.

NORTON, K. A. (1935) Propagation of radio waves over a plane earth, Nature, 135,954-955. NORTON, K. A. (1936) The propagation of radio waves over the surface of the earth and in

the upper atmosphere, Proc. I.R.E., 24, Pt. I, 1367-1387. NORTON, K. A. (1937) The propagation of radio waves over the surface of the earth and in

the upper atmosphere, Proc. I.R.E., 25, Pt. II, 1203-1236. NORTON, K. A. (1941) The calculation of ground-wave field intensity over a finitely con

ducting spherical earth, Proc. I.R.E., 29, 623-639. ROLF, B. (1930) Graphs to Prof. Sommerfeld's attenuation formula for radio waves,

Proc. I.R.E., 18' 391-402. ROTMAN, W. (1951) A study of single surface corrugated guides, Proc. I.R.E., 39, 952-959. SOMMERFELD, A. N. (1899) Über die Fortpfanzung electrodynamischer Wellen längs eines

Drahtes, Ann. Phys. u. Chem., 67, 233 . SOMMERFELD, A. N. (1909 and 1926) The propagation of waves in wireless telegraphy,

Ann. Phys., Series 4, 28, 665; 81, 1135. SOMMERFELD, A. N. (1949) Partial Differential Equations, Academic Press, New York. STANLEY, G. M. (1960) Layered earth propagation in the vicinity of Point Barrow, Alaska,

/ . Res. Nat. Bur. Stand., 64D, 95-99. TAI, C. T. (1951) The effect of a grounded slab on radiation from a line source, / . Appi.

Phys., 22, 405. VAN DER WAERDEN, B. L. (1950) On the method of saddle points, Appi. Sci. Res., B-2,

No. 7, 33-45.


WAIT, J. R. (1953) Fields of a line current source over a stratified conductor, Appi. Sci. Res., Sec. B, 3.

WATT, J. R. (1953 and 1954) Radiation from a vertical dipole over a stratified ground, Trans. I.R.E., AP-1, Pt. I, 9-12; AP-2, Pt. II, 144-146

WAIT, J. R. (1957) Excitation of surface waves on conducting dielectric clad, and corrugated surfaces, / . Res. Nat. Bur. Stand., 59, No. 6, 365-377.

ZENNECK, J. (1907) Über die Fortpflanzung ebener electromagnetischer wellen einer ebenen Leiterflache und ihre beziehung zur drahtlosen télégraphie, Ann. Phys., Series 4, 23, 846.

ZUCKER, F. J. (1954) The guiding and radiation of surface waves, Proc. Symposium on Modern Advances in Microwave Techniques, pp. 403-435, Polytechnic Institute of Brooklyn. (Many references are given in this comprehensive paper.)

Additional References EPSTEIN, P. S. (1950) Reflection of waves in an inhomogeneous absorbing medium, Proc.

Nat. Acad. Sci. Amer., 16 (new series) 129-133. FöRSTERLiNG, K. (1931 and 1950) On reflection in a non-homogeneous medium, Ann. Phys.,

11, 1-50; 8, (new series), 129-133. RICE, S. O. (1937) Series for the wave function of a radiating dipole at the earth's surface,

Bell System. Tech. J., 16, 101-109. RYDBECK, O. (1943) The reflection of electromagnetic waves from a parabolic ionized

layer, Phil. Mag., 34, 342. SOMMERFELD, A. N. (1920) Über die Ausbreitung der Wellen in der Drahtlosen télégraphie,

Ann. Phys., 62, 95. National Bureau of Standards, Applied Mathematics Branch. Tables ofBessel Functions of

Fractional Order (1949) Columbia University Press. WAIT, J. R. (1953) Propagation of radio waves over a stratified ground, Geophys., 18,

416-422. WAIT, J. R. (1956) Radiation from a vertical antenna over a curved stratified ground, / .

Res. Nat. Bur. Stand., 56, 232-239. WISE, W. H. (1937) The physical reality of the Zenneck surface wave, Bell System Tech. /.,

16, 35-44. ZUCKER, F. J. (1961) Surface and leaky wave antennas, Chapter in Antenna Engineering

Handbook (éd. H. Jasik), McGraw-Hill, New York.

Chapter III

REFLECTION OF ELECTROMAGNETIC WAVES FROM INHOMOGENEOUS MEDIA

WITH SPECIAL PROFILES

Abstract—The reflection of electromagnetic waves from planar stratified media is discussed in a relatively concise manner. Attention is confined to special forms of conductivity (or dielectric constant) profiles which lead to solutions in terms of Bessel functions. Most of the results, in equivalent forms, have already appeared in the literature. The present note is essentially a consolidation of known solutions and their (sometimes novel) applications to the determination of reflection coefficients.

1. INTRODUCTION

It was demonstrated in the previous chapter that the reflection coefficient from a stratified conductor could be effectively treated by sub-dividing the medium into a number of parallel homogeneous layers. Such a method may be used for an approximation to a continuously varying conductivity profile. In fact, by taking a sufficiently large number of such layers of vanishing thickness, any desired degree of precision may be obtained. While this approach is direct, it is not very elegant. Furthermore, the calculations usually require a large-scale automatic computer. The alternative is to take some special form of conductivity variation, which allows the solutions to be expressed in terms of tabulated functions. This aspect of the subject is discussed briefly here. Particular attention is paid to profiles which lead to solutions in terms of Bessel functions.

2. GENERAL CONSIDERATIONS

For the moment, attention will be confined to perpendicular (or horizontal) polarization wherein the electric vector is always parallel to the plane of stratification (i.e. z = 0). Following the earlier work, the inhomogeneous medium will be taken to occupy the space z > 0. Unless otherwise stated, the region z < 0 is taken to be free space with electrical constants ε0 and μ0. The conductivity σ(ζ) and dielectric constant ε(ζ) are to be specified in some definite fashion for z > 0. For convenience, the permeability for z > 0 is also taken to be μ0.

64

Reflection of Electromagnetic Waves from Inhomogeneous Media 65

The electric field, which has only a y component, for the space z < 0, is given conveniently by [Wait, 1958]

Ey = £ 0 (e"MoZ + Α±(λ)β"-) e~u* (1)

where λ = k0 sin 0, u0 = (A2 — kffi = ik0 cos Θ and 0 is the angle of incidence. As we have seen for the generally stratified half-space, the reflection coefficient has the form [Wait, 1958].

where N0 = w0/^o ω = c o s θ/ηθ9 η0 = ^/(μ0!ε0) = 120π. and Yt is the surface admittance at z = 0. In fact,

*~y (3) z = 0

The subscript 1 refers to the fact that the field is evaluated in the first layer of the multi-layer problem. Since we are dealing here with continuously stratified media, the subscript 1 may be dropped in what follows. Thus, in general,

where y--rl (5)

Alternately, we could write fao/cos Θ) - Z

1== (iy0/cosfl)+Z K)

The problem of reflection of plane waves from a (planar) inhomogeneous media then boils down to finding an expression for the surface impedance evaluated at the interface.

3. INVERSE SQUARE PROFILE

An example is now chosen which illustrates this approach. The propagation constant y(z) is taken to have the following form

y\z) = -k2(z) = -k 20 \a 2 - —^—21 = *>0ω[<τ(ζ) + Ιβ(ζ)ω] (7)

L (z + to) i

where a, b and z0 are constants. Now, remembering that the fields must vary

66 Electromagnetic Waves in Stratified Media everywhere as e"a* it is seen that Maxwell's equations in the inhomogeneous region can be written

dEv ίμ0ωΗχ = —> (8)

ίμ0ωΗζ = iXEy (9)

Combining these, it follows that

d2£,

O(z) + k(z)(o]Ey = -^ + ΐλΗ, (10)

"> _ι_/Ί.2ΛΛ 12-dz2

which can be written in the form

+ (k\z)-X2)Ey = 0 (11)

" ^ J E = 0 (12)

where Α:0αΐ = ^0α2 — λ2 and where the subscript on the E has been dropped. The objective is now to find a solution in terms of some known special

function. In the present case it turns out that the differential equation can be transformed easily to Bessel's equation. The latter is of the form

d2V dV ul-n +u—+(u2-v2)V = 0 (13) du du

where v is the order. Solutions of this equation are Bessel functions and may be generally designated Vy(u). We now try a solution of the form

E = const x ÌV2Vv(ße) where Ì = z + z0 (14)

Substituting this in (12) we see that

ßH2v:m+ßtvoßi) + Lk20a2ê2 - (k2b2+mvm) = o as)

where the primes indicate differentiations with respect to ßz. On comparing this with (13), it is seen that

ß=,k0ai = k0yJ(a2 - sin2 Θ) (16) and

v = (fcg62+i)% (17)

A solution equivalent to this was developed by Rytov and Yudkevich [1946]. Using

dE ίμωΗχ = — (18)

it readily follows that

Ey Ινν(β2)^2βί\μ0ω K }

Now, to specify the appropriate Bessel function, something more about the physical problem must be stated. The imaginary part of a0 (and y/(al — sin2 0)) is taken to be less than zero. Thus, if E ox H are to remain finite, the particular form of Vv(ße) must, at z = oo, be Hi2)(ße) which is the Hankel function of the second kind. This follows from the property

Lim (πζ/2)1/2#<2)08ζ) = e"1'"* ei(v*/2+*/4) -> 0 (20) z->oo

Therefore, in the present problem,

Y 1/2 \Η?)ΧβΖο) I 1 1 iß (21) Y 1 / Z - lH?Xßz0) + 2βΖο\ μ0ω ( 2 1 )

where β = k0J(a2 - sin2 Θ)

and v = (fc2i>2+i)^

As a partial check on this result we can set b = 0, in which case fc(z) = k0a = const.

The Hankel functions are now of order 1/2. These are known to be expressible in simple form. In fact

^^'iièÎ6'^ (22) On using this result in the above equation for Y, it is easily seen that

Y = l/z = -A. = - J(a2 - sin2 Θ) (23) μ0ω η0

ν

which is the expected result. Some further discussion of this solution is probably in order. The field is

proportional to the Hankel function (ße)i/2Hi2)(ße). For large values of |v| and \ße| this function can be conveniently approximated by the Debye asymptotic forms given, by Bremmer [1949], as

/ ie"a

(π/2)*Η<2>(χ) s {

(v2 - χψ ett

(v2 - x2)v* 2 eiff/4 COS(TT/4 + ia)

\ (v2 - x2)v*

(I)

(Π)

(III)

(24)

(25)

(26)

where v Cx,v du

a = (v2 - χψ> - v cosh"* - = v (1 - u2)v> — (27) * J i u

with Re(l - u2yA > 0 and Re cosh"1 v/x > 0. The 3 domains I, II and III are illustrated in Fig. 1, being somewhat loosely defined.

T X

Im. v

1 fö -I

FIG. 1. The Argand diagram for the complex quantity xjv showing the regions I, II, and III.

It can be noted immediately that if Re x/v > 1 and Im JC < 0 the field behaves as exp(—ißz) which is an outward travelling wave. Under other circumstances, particularly where the quantity {ai — b2/î2) becomes small, the field behaves as an oscillating function as indicated by the appropriate asymptotic form III. Physically this is related to the phenomenon of critical reflection. Further insight into this behavior is deferred until the linear profile is discussed.

4. PROFILE WITH AN EXPONENTIAL TRANSITION

Another profile which also leads to Bessel functions is the exponential variation of the propagation constant. For example,

-y2(z) = k\z) = (k\ - k\) Q~ßz + k\ (28)

for z > 0 and Re ß > 0. Thus

Lim fc(z) = kl z->0

and Lim k(z) = k2 Z-+0

Again assuming that the electric vector has only an Ef component, it is evident that Eq. (11) is now given by

^Λ + [(feî - fcl) *~ßz + (feS - kl sin2 0)]E„ = 0 (29) ûz

To convert this into the required form, a new variable v is introduced as follows

Then, we find that

provided

Solutions are then

where

*-Se-w a(k?-fc5)tt (30)

v2<^ + v ^ + (v2-v2)Ey = 0 (31) d i r di? ^ ' y

v2 = - \k\ - kl sin2 θ] (2/β)2. (32)

Ey = const J±v(i;) (33)

v = î[fe| - feg sin2 0]1/2(2/)S) (34)

To select the proper index for the Bessel function, we consider the behavior for large values of z. For example, if ßz > 1 it is seen that v <| 1, and thus

,±v J±y,(v) s const x v .±v

^ const | (fe? - fei)'71] ±V exp(+v(-/fc/2)] (35)

It is now noted that Re v > 0 since k2 has a finite negative imaginary part. Consequently, if fields are to be finite as z -* oo, the upper sign must be chosen.

Now, from Maxwell's equations we see that

„ d£ dv ίμ0ωΗχ = — —

dt; dz

= ~e -^ 2 ( fc 2 ~ fe 2 y /^ (36)

Then

(37) £y ΐμ0ωΙν(ν) Hx J*l\k\ - k\)*J'£v)

An immediate check on this result is to allow β to approach infinity. Then, using the small argument approximation for Jv(v), we see immediately that


which is the expected behavior for a homogeneous half-space of (complex) wave number k2. Another interesting check on this result is to allow ß to approach zero. Then we see that both v and v become indefinitely large. Here, it is convenient to employ the following asymptotic approximation

Jip(v) s {2π)-\ρ2 + v2)-* exp[- in/4 + ρπ\ΐ\ (39) x exp[i(p2 + v2y/z - ip sinhT^p/t;)]

where terms of order v"1 have been neglected. The above result is the first term of an expansion given by Bateman [1953] for purely imaginary order (i.e. p real). If the medium is a pure dielectric both k1 and k2 are real and v is purely imaginary. Thus

\>2VA [k2 - ki sin2 01 *

Thus m*v-?\ S,L *?-« J (40)

-|],.r4I-stalf (41) which again is the expected result.

The admittance at the boundary z = 0 is, of course,

(ki - fcfÄK) - r ] - i>œj;(t>0) where

fo = (2/)5)(k?-fci),/l

The reflection coefficient is then given explicitly by

(42)

cos Θ - Υη0

^ " c o s O + yifo C43)

5. OTHER EXPONENTIAL PROFILES

A closely related profile is an exponentially increasing propagation constant. For example, if

-y 2 (z) = fc2(z) = k?e", for β > 0 (44) we readily find that

1 ^ + (fcf e'z - fc2 sin2 fl)E, = 0 (45) ÛZ

Again, making the substitution

O = \jtl2kx (46) P


we then find d% dEy

dir dv ν2^Τ+ν^Γ + tf - v2)£y - 0 (47) where

v2 = k2(sin2 0)(2/j?)2 (48)

Independent solutions of this equation are still J±v(v)9 however, it is required that for z tending to + oo, the field should be non-infinite. If the imaginary part of kt is negative, the required form is the Hankel function of the second kind Η(ϊ\ν) which is really a special linear combination of J±v and /_v. Consequently,

Ey ίμ0ωΗ[2\ν) Hx εβζ/2^Η™'(υ) (49)

which is in analogy to Eq. (37). In this particular case v is taken as the positive or negative root of (48). However, for consistency, we shall write v = fco(sin0)(2/j?).

In certain physical problems the exponential increase may correspond to a conductivity which becomes indefinitely large. Thus, it is often more convenient to write

fc2(z)= -iBeß2kl (50)

where k% and B are essentially real when displacement currents can be neglected. Now, we choose the new variable to be

v = ^ J*I*. /(R\ cßzll ß

Thus, we easily find that

e^V(B) e"z/2 (51)

becomes

(j-2 - iko eßzB - fc02 sin2 flW = 0 (52)

[^+βέ-(ν2+φ-° (53)

where v = (ß/2)k0 sin Θ. This is the modified Bessel's equation of order v. The appropriate solution is of the form

Ey = E0Kv(v) (54)

where Kv(v) is the modified Bessel function of order v and argument v. Therefore,

-T,--^w°KI'°-wm (55)

and

or

where

K'V(VQ)

Kv(v0)

γ _ k(0)K'Xvo)_k(0)[Kv+i(vo) v j μ0ω üCv(t;o) μ0ω L £vOo) "oJ

2fc «o = - j e ^ V ( ß ) = i2fc(0)/ .

One should note that in terms of the conductivity σ(ζ), regarde das a function of z,

ϋο = 2[ίσ(0)/ιοω]1/ν)5 (58)

Inserting the above expression for y into Eq. (41) leads to an expression for the reflection coefficient at the interface z = 0 for a horizontally polarized plane wave incident in the free space region (i.e. z < 0). In the case of normal incidence (i.e. Θ = 0), it is seen that v becomes zero in this particular problem. Thus, for normal incidence

μ0ω Κ0(ν0)

This result had been given previously [Wait, I960]. The exponential profile is also well suited to a study of variable ionized

media. A special feature of such propagation is the possibility of critical reflection. At normal incidence this occurs in the region where the dielectric constant is zero. The special exponential forms chosen above are not suitable for investigating this particular phenomenon. The form adopted is

fc2(z) = kl[\ - b eaz] (60)

which is assumed to hold in the whole region — oo < z < oo. When b and a are both real, it is seen that

fc(z) = fe0 for z = — oo and

k(z) = 0 when z = -log(l/b)

For large positive values of z, k2(z) becomes an infinitely large negative real number.

The electric field (with only a y component) is taken to be parallel to the stratification. For large negative values of z, the field must be of the form

Ey = e-ikSxc±ikCz (61)

where C = cos Θ and Θ is again the angle of incidence. Now, in general, for any value of z

^ + fco2[C2-fce**]£, = 0 (62)

This equation is again reducible to Bessel's equation. Letting

v = (2Jfc0b,/2/a) ea2/2 (63) and

v = 2ifc0C/a (64)

the general solutions can again be made up in linear combinations of the functions Jv(v) and J-V(v). The particular combination of these suitable for the present problem is the Hankel function of the first kind defined by

„ i > ) a ■>-(-> - · - * < ■ » m

i sin νπ

This function, and only this function, has a non-infinite behavior as v tends to + 00.

The admittance at any point in the medium, looking in the positive z direction, is then found to be

Hx avH^Xv) Ey ~~2ίμωΗ^\ν) l ° ° '

For small values of v, corresponding to large negative values of z, it may be noted that

v W i sin νπ [Γ(1 - v) Γ(1 + v) J K }

where each term in the square bracket is the leading term of the power series expansion of J-V(v) and Jv(v), respectively. On replacing (av/2) by ik0C and changing the variable back to z, it then follows that (for z -> — co)

Ey s const x [e-ifcoCz + R e''*oCz] e~ikoSx (68) where

It is clear that R has the nature of a reflection coefficient referred to in the plane z = 0. However, it is not at all the same as the reflection coefficient at an interface z = 0, which is free space for z < 0. In the latter case, the reflection coefficient would be given

«_ |!· „0) C + Υη0


where αΡοΗ«"'^) (71)

where p0-(2ifco6'4/«) (72)

In the case of an ionized gas 6 = (1 — iZ)~l where Z is a real quantity which is equal to the ratio of the collisional frequency v to the angular frequency ω. We thus find Elias' [1930] and Budden's [1961] result that

|A| = e x P p ^ t a n - l z ] (73)

and argR = 7 t + ^ l o g P ) - ^ l o g ( l + Z 2 ) + (74) -(*)-¥■

+ 2 arg[r(l - 2ifc0C/a)]

Following Budden [1961], it is very interesting to note that the phase integra method leads to an expression which is an excellent approximation for k The phase integral formula* should read

k s ( Lim \i extfi2kCzi) exp[-î2 | ° [fc2(z) - fcgS2]*4 dzl (75)

where z0 is defined by k2(z0) = klS1. It is noted that, for z > z0, the integrand would be purely imaginary when k(z) is real. The quantity zi can be conveniently imagined as a negative value of z sufficiently large that k2(z) can be replaced by kl. Strictly in the case of an exponential profile defined by Eq. (60), the zl must be taken as —oo. The appearance of the exponential factor expQlkCzJ on the right-hand side of the above equation means that k is referred to the plane z = 0. Then, on using

k\z) = kl(i-^-^ (76)

the integration in Eq. (75) is carried out to give

|A| = exp[-(2fc0C/a) taif 'Z] (77) and

a r g £ = Ξ - *b£ (log 2C - 1) - ^£ log(l + Z2) (78) 2 α α

As Budden comments, it is really quite remarkable that the phase integral formula gives the correct (i.e. exact) value of |£|. However, the phase predicted

* The phase integral method is discussed in considerable detail in the next chapter.

by the phase integral method is only approximate. This is demonstrated by noting that arg R, given by Eq. (78), may be recovered from Eq. (74) since

π 4k C 2 arg[r ( l - 2ifc0C/a)] s - - - —°— [log(2fc0C/a) - 1] (79)

Z a when 2k0C/(x is somewhat greater than unity. The latter approximate relation is simply a consequence of using the first term in the Stirling formula for the gamma function.

6. LINEAR PROFILE

Another profile which leads to a convenient solution for horizontal polarization is the linear variation. For example, we let

k\z) = kl\l - uz] for z > 0

= k2 for z < 0 (80)

and, again, we assume that the permeability is everywhere equal to μ0. The equation for the electric field is thus

d2E *■ + kl(C2 - az)E, = 0 for z > 0 (81) dz2

ά2ΕΛ

dz + k\C2Ey = 0 for z < 0 (82)

As before, the electric vector is taken to have only a y component. By making the substitution

t = {k2<x)\z-C2loL) (83) it is seen that

^ r - i E v = 0 for z > 0 (84) at2 y

The parameter / should not be confused with time. Equation (84) is known as Stokes' differential equation and independent solutions are the 2 Airy functions u(t) and v(t). They may be defined in terms of definite integrals as follows [Miller, 1946].

"(0 = -ff V7CJo

{exp(- Jx 3 + tx) 4- sin(ix3 + tx)} dx (85)

1 f °° t<0 = - 7 - cos(£x3 4- tx) dx (86)

To choose the appropriate solution requires a knowledge of the asymptotic behavior for large positive or negative values of the argument. Following Miller [1946], we find that for |/| -> oo, |arg t\ < π/3

u(0 = ^exp($i%) and (87)

while for

and

KOss-^expi-if*) (88)

|i|-*oo, |arg(-t)|<27r/3

"W ~ (ZjjvS cos[ f ( - 0% + ]j] (89)

üW~(^s i n[ f ( -°% +3 <90> Actually, these are the first terms of asymptotic expansions for large values of the arguments. It is also worth noting that Airy Functions are closely related to Bessel functions. This is indicated by the following identities

/nt\i/z

<t) = (j) U-y&t*) + LS*3/2)] (91)

"( - 0 = ( y ) \j-%(*i%) - J-y3 (^3/2)] (92)

(nt)Vl

v(t) = ^ - [/-*(**) - '%<*'*)] (93)

(πίΥΔ

v(-t) = ^ - [J-%öl%) + J%»i%)] (94)

Further properties and relationships among Airy functions can be obtained from well-known relations in the theory of Bessel functions [Watson, 1944].

Now, if a is a real number greater than zero it is evident that, in view of Eq. (88), the desired solution is v(t). In fact, this is still true if a is complex, provided |arg α| < π/3. Thus

E,~E0v(t) (95)

where E0 is a constant. Furthermore,

dE //ιωΗ, = - ^ = (*?α)Μθ (96)

where the prime denotes a derivative with respect to t. The admittance of the wave looking in the positive direction is thus

JË-.,Αω*?™ (97) Ey μ0ω\^/ v(t)

for z > 0. The reflection coefficient at z = 0 is thus given by

where

where

Thus

y - l i f f î ^ or α£αΐ*<& (99)

η0 = μ0(οΙ^ο and i0 =-(fc0/a)%C2.

8 _ ( - , ^ ( , 0 ) - , „ ■ ( , . ) (-io)^(io) + iV(io)

It is important to note that if a is real, t0 is a negative real number which, in turn, indicates that the admittance Fis imaginary (i.e. it is a pure susceptance). Thus, in this case, \R\ = 1 and the reflection is complete. Of course, there is a phase shift on reflection which is determined by the argument of Y. It may be noted that if (—10) is greater than about 2, the asymptotic approximation for v(t) given by Eq. (90) may be used. Thus

^ * - i c o t [ * ( - f o ) % + | | (101)

and to this approximation the reflection coefficient is given by

K^iexp[~it(-i0)3^] (102)

^iexp[-if(fc0/a)C3] (103)

It is of interest to compare this result with the "phase-integral method" of Eckersley [1932]. This would predict that the reflection coefficient should be given by

R = i expi-2ifc0 |°[iV2(z) - S2]1/* dzl (104)

where N(z) = k(z)/k0 is the refractive index which varies from 1 at z = 0 to a smaller value at z0 such that N2(z0) — S2 = 0. In this particular problem

N2(z) -S2 = C2- Z/OL (105)

and z0 = C2/a. Therefore, it readily follows that Eq. (103) is regained. It might be of interest to note that pure geometrical optics would not give the multiplier /. This can be associated with a π/2 phase advance at the caustic of the ray system. The phase integral approach is discussed in the following chapter.

7. EXTENSION TO VERTICAL POLARIZATION

In previous sections attention has been confined to "horizontal polarization" or parallel incidence. The other case of equal importance occurs when the magnetic vector is perpendicular to the plane of incidence. Then, since the electric vector is contained in the plane of incidence, this is aptly called "parallel polarization." Sometimes this case is described as vertical polarization since for near grazing incidence the electric vector is almost vertical relative to the (horizontal) stratification.

Again, unless otherwise stated, the region z < 0 is taken to be free space with electrical constants ε0 and μ0. For the region z > 0, the conductivity and dielectric constant are denoted σ(ζ) and ε(ζ), respectively, and, as before, the permeability for z > 0 is also taken to be μ0.

The magnetic field, which is taken to have only a y component, for the space z < 0, is given by

Hy = tf0(e-M0Z + K„(A) eM°2) e"iA* (106)

where λ = k0sin 0 and u0 = (A2 — kffl2 = ik0 sin 0 as 0 is the usual angle of incidence. As shown before, for the discretely stratified medium, the reflection coefficient is given by

«■»-erf: (,o7) where

Κ0 = -^±- = η0 cos 0, (108) ιε0ω

and Zx is the surface admittance at z = 0 where

(109) <-tr] Again, since we are now dealing with continuously stratified media, the subscript 1 may be dropped in what follows. Thus, in general,

where

or Hy]z=o

ÔEX - ίμ0ωΗγ = ~âf + ik0SEz

E 1 dH> x σ(ζ) + ίε(ζ)ω dz

E Ìk°S H σ(ζ) + ΐε(ζ)ω "

From these we readily obtain

dz2 + [ k ( Z ) koSiH> k\z) ôz dz - = 0

«■»-gii < m ) where C = cos 0.

Noting that the fields must vary as exp(—ik0Sx), Maxwell's equations, for the space z > 0, are given by

(112)

(113)

(114)

(115) where

k2(z) = — ίμ0ω[σ(ζ) + ίε(ζ)ω]. (116) It can be seen that except for the latter term on the right-hand side, the equation for Hy is the same as Eq. (11) for Ey. The presence of this term for "vertical polarization" is a complicating feature. Often this term may be neglected when the medium is slowly varying such that derivatives with respect to z are small. Incertain cases, however, such as where k\z) — k^S2 is small, the latter term is very important. This means the physical processes are quite different for horizontal and vertical polarization when the medium is not slowly varying.

8. EXPONENTIAL PROFILE WITH VERTICAL POLARIZATION

For vertical polarization most of the special profiles mentioned above do not appear to lead to a standard form of the differential equations. Thus, in general, numerical or iterative procedures are required. However, one case does reduce again to a form of Bessel's equation. Apparently, this was first pointed out by Galejs [1961], in a related problem. The special profile is

k\z)=-iklBeßz (117) for ß > 0 and z > 0. When this is applied to a highly conducting medium where displacement currents are negligible, B is real.


For this exponential profile, the equation for Hy becomes

y^ - k&S2 + iB e*)ff, + β ρ = 0 (118) az dz

Letting u = — iB éz, it readily follows that

IS*iMÄ-£)]».-This equation has a solution of the form

Hy = H0vKv(v)e-ikSx (120) where H0 is a constant, if

v = i(2k0lß)u/2 = (2fc0//D eiÄ/4B1/2 eßz/2 (121) and

v = [l+(2fc0S/jS)2]1/2 (122) It is noted that Hy is non-infinite as z -* ex).

From Maxwell's equations 1 dHydv .,.,__.

ιε0ωΐί 3t; 3z This leads quickly to

d

Jff," , 0 B * e * ' 2 vKy(v) l '

It is interesting to compare this with the analogous formula for the admittance in the case of horizontal polarization. This is given by Eq. (55) where it can be seen that the argument v of the modified Bessel function is the same in the 2 cases. However, the argument v is defined differently.

Using the recurrence relation

£;= -K^^-K, (125)

it now follows that

Z-HyL0-Hu0« Kv(v0) ~2ik0 «o J (126)

u£ = *(0)/*o s e ^ ' V O ) / ^ ) *

»o = (2fco//0 e'*/4B* (127)

2 2 [ ί # ) μ ο « . ] ^

where

and where

Here σ(0) is the conductivity at z = 0. Also, it should be remembered that

v = [1 + (2fc0S/j8)2]1/2 for Z (128)

Inserting the above expression into Eq. (Il l ) , leads to an exact expression for the reflection coefficient R^ for a plane wave incident from the free space region (i.e. z < 0). At normal incidence (i.e. S = 0), the impedance formula reduces to

_ μ<*ω Κ0(ρ)

This is in agreement with Eq. (59) which is a check since the reflection coefficient at normal incidence is the same, regardless of the choice of coordinate axes.

It is important to note that Z given above is a good approximation for oblique incidence at vertical polarization provided (2k0S/ß)2 <ζ 1.

As a further check on the general formula given above for Z, we allow β to tend to zero corresponding to a homogeneous conducting half-space. In this limiting situation both v and v0 tend to infinity. Thus

d — v0Kv(v0) dv0

Therefore

This can be written

,WH)La-N/(i+§) im

^JK) Z = V(N2-S2) (132)

where N2 = — iB is the square of the refractive index of the homogeneous half-space occupying z > 0.

9. POWER LAW PROFILE FOR NORMAL INCIDENCE

A profile of considerable generality is given as follows

k\z) = k\(\+-^ " (133)

where ku a and ß are constants. Unfortunately, it does not seem possible to obtain closed-form solutions from Maxwell's equations in the general case of arbitrary values of a and ß and oblique incidence. However, at normal incidence the fields in such a medium can again be expressed in terms of Bessel

functions as first shown by Lahiri and Price [1939]. The analysis for this case is given in the following.

The electric vector is taken to have only a y component. Consequently, the magnetic vector has only an x component. One easily finds that

[έ+4+;Π^"° (134) and

ιμωΗχ = dEy/ôz As usual, μ is taken as a constant. Defining a new function by

Ey = i l + Z-\ V (135) it is seen that

-1 M, 1 / * \ ~ 2 . / *\-ß ♦;Kr^('+r*+4+;) ♦-'<«> Letting φ = Zv(w), it is found that this is a Bessel function of order v and argument w, if

-(a) (137) \* - PI

and Μ = 2ν (Μ) (1+- ) (138)

The appropriate Bessel function to employ is dictated by the required behavior as z tends to + oo. It is assumed that Im k1 < 0. For ß < 2, it is seen that

/ ζ\ί/2 Γ 2 / z\i2'ß)/2l Ey = £ 0 ( l + - j H $ 2 _ , ^ _ ( k i a ) ( l + Jj j (139)

where E0 is a constant and H^ is the Hankel function of the second kind. On the other hand, for ß > 2, it is found that

Ey = £0( l + - j JlKß_2)[—(kla)(l +-) j (140)

where / is the Bessel function of the first kind. The special case ß — 2 can be found from either of the above as limiting

cases. This process requires the use of the appropriate asymptotic expansions for the Hankel and Bessel functions when the order and the argument are both indefinitely large. More simply, one considers this case separately.

Reflection of Electromagnetic Waves from Inhomogeneous Media

Thus, for ß = 2,

d2

Letting

it follows that

Therefore

or

d 2 £ y _ m ( m - l ) / z\ dz2 " a2 \ + a )

Ey = 0

2„2 m(m — 1) = — k \a

2m = 1 ± (1 - 4fc2a2),/>

83

(141)

(142)

(143)

(144)

If Ey is to be finite as z tends to infinity, it is clear that the + sign must be rejected in the above radical.

Without further difficulty it is readily found that

—EJHX

Γ 2 / z\(2"")/21

- _ ^ [ 1 + ( 1 - 4 # , » ) * ] ( ΐ + ί ) , 0 - 2 ) 2feja \ a/

Γ 2 / z \ ( 2 -« / 2 l

= w 1 + £ ' r Ji /("-4^2(M)l1+â) J fci I a) Γ 2 / z\ ( 2-") / 2 l

Jmß-2)+1[JZ^(k1a)\[i+-J J

(β<2)

(145)

(146)

(β>2)

(147)

If the region z < 0 is free space, then the reflection coefficient at z = 0, for a plane wave incident from z = — oo, is

R = f o - Z >/o + Z

(148)

where

HlK2-ß)\2-ß) ιμω

h # 1 / ( 2 - « /2fc1q\'

^-a+ii-Akwn, 2k\a

ίμω ' 1/(0-2) [2/cta

U5-2J

Ί/(ί· [2fc1a]'

(β<2)

(ß = 2)

(ß>2)

(149)

(150)

(151)

As a partial check on these results, it can be seen that all 3 of the forms for Z approach μω/Α^ as a tends to infinity. This would correspond to a homogeneous half-space with wave number k^ Another check is to note that for 0 = 0,

z = H ( 2 ) [ 2 M 1

ίμω ,/2 1.2 - ft] fci

μω (152) „(2) Γ 2 Μ Ί fcl H-*\nl

which also corresponds to a homogeneous medium.

REFERENCES

BATEMAN, H. (1953) Higher Transcendental Functions, Vol. II, (edited by A. Erdelyi, et al.), McGraw-Hill, New York.

BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier, New York and Amsterdam. BUDDEN, K. G. (1961) Radiowaves in The Ionosphere, Cambridge University Press. ECKERSLEY, T. L. (1932) Radio transmission problems treated by phase integral methods,

Proc. Roy. Soc. A, 136, 499. ELIAS, G. J. (1930) Reflection of electromagnetic waves at ionized media with variable con

ductivity and dielectric constant, Proc. I.R.E., 19, 891-907. GALEJS, J. (1961) e.l.f. waves in the presence of exponential ionospheric conductivity

profiles, Trans. I.R.E., AP-9. LAHIRI, B. N., and PRICE, A. T. (1939) Electromagnetic induction in non-uniform con

ductors, and the determination of the conductivity of the earth from terrestrial magnetic variations, Phil. Trans. Roy. Soc, London, Ser. A, 237, 507-540.

MILLER, J. C. P. (1946) The Airy Integral, Giving Tables of Solutions of The Differential Equation y" = xy, Cambridge University Press.

RYTOV, S. M., and YUDEVICH F. S. (1946) Electromagnetic wave reflection from a layer with a negative dielectric constant, / . Exptl. Theor. Phys., 10, 285. (In Russian).

WAIT, J. R. (1958) Transmission and reflection of electromagnetic waves in the presence of stratified media, / . Res. Nat. Bur. Stand., 61, No. 3, 205-232.

WAIT, J. R. (1960) Terrestrial propagation of v.l.f. radio waves—a theoretical investigation, / . Res. Nat. Bur. Stand., 64D, 153-203.

WATSON, G. N. (1944) Theory of Bessel Functions, Cambridge University Press, 2nd ed.


Other References BOSSY, L., and DEVUYST, A. (1959) Relations entre les champs électriques et magnétique

d'une de peroid très longue induits dans un milieu de conductivité variable, Geofis. Pur. Appi, Milano, 44, Pt. Ill, 119-134.

BREKHOVSKIKH, L. M. (1960) Waves in Layered Media, Academic Press, New York. FÖRSTERLING, K. (1931) Über die ausbreitung des Lichtes in inhomogenen Medien, Ann.

Phys., (Series 5), 11, 1-39. GRAY, MARION (1934) Mutual impedance of grounded wires lying on the surface of the

earth when the conductivity varies exponentially with depth, Physics, 5, 76-80. JOHLER, J. R., and HARPER, J. D. Jr. (1962) Reflection and transmission of radio waves at a

continuously stratified plasma with arbitrary magnetic induction, / . Res. Nat. Bur. Stand., 66D (Radio Prop.) No. 1, 81-100.

PEKERIS, C. L. (1946) Theory of propagation of sound in a half-space of variable sound velocity, / . Opt. Soc. Amer., 18, 295-315.

SHMOYS, J. (1956) Long range propagation of low frequency radio waves between the earth and the ionosphere, Proc. I.R.E., 44, 163-170.

STANLEY, J. P. (1950) The absorption of long and very long waves in the ionosphere, / . Atmos. Terr. Phys., 1, 65.

TAYLOR, L. S. (1961) Electromagnetic propagation in an exponential ionization density, Trans. I.R.E., AP-9, 483-487.

WAIT, J. R. (1952) Reflection of electromagnetic waves obliquely from an inhomogeneous medium, / . Appi. Phys., 23, 1403-1404.

WAIT, J. R. (1962) On the propagation of v.l.f. and e.l.f. radio waves when the ionosphere is not sharply bounded, / . Res. Nat. Bur. Stand., 66D (Radio Prop.) No. 1, 53-62.

Chapter IV

APPROXIMATE METHODS FOR TREATING REFLECTIONS FROM INHOMOGENEOUS MEDIA

Abstract—The oblique reflection of plane electromagnetic waves from a continuously stratified medium is considered. Various approximate procedures are employed. For the slowly varying profiles, the WKB method and its extension are most suitable. However, certain modifications must be made when the ray has a turning point. It is shown that under this situation, the phase integral method is applicable. Finally, when the medium is rapidly varying, an alternative approach is adopted which becomes particularly suitable at low frequencies.

1. INTRODUCTION AND THE CONVENTIONAL WKB METHOD

In the previous chapter, the propagation in an inhomogeneous medium was treated by adopting special profiles for a continuously stratified medium. Here, approximate methods, which are not restricted to the form of the profile, are investigated.

To illustrate the central idea, the one dimensional wave equation is considered

^ + fc2(z)<A = 0 (1)

where φ(ζ) and k{z) are scalar quantities. Such an equation is appropriate for plane waves propagating normal to the stratification. To seek an approximate solution of this equation, we set

iA(z) = A(z)e~^(z) (2) where Λ and φ can be regarded as the amplitude and phase, respectively.

Substituting this into the wave equation, it follows that Λ" - Ιίφ'Α' - ίφ"Λ + [fc2(z) - (φ')2]Λ = 0 (3)

where the prime indicates a derivative with respect to z. An approximate solution of this latter equation is now obtained by using the following physical arguments.

The dimension / is taken to be the distance over which quantities Λ and φ vary by a significant amount. Thus, a slowly varying medium can be defined as one in which kl > 1.

86

Treating Reflections from Inhomogeneous Media 87

Order of magnitude estimates can now be made for the various terms in Eq. (3). Thus

Λ" - A/l2

is of second order of smallness and for a first crude approximation it may be replaced by zero. Then Λ exp( —ίφ) is an approximate solution of Eq. (3) if

Φ' = ±k(z) or φ = ± k{z) άζ (4)

and Λ' k' . const T = ~" ^τ 0 Γ ^ = / r i / xn (5) Λ 2fc V[fc(z>]

Then, the general solution is

^(z)^[l/fc(z)]1/2(aexpi-i | /c(z)dzl + fcexp[+i Γ*(*)<1ζ]) (6)

where a, b9 and z0 are arbitrary constants. This result is usually known as the WKB solution where the letters stand for Wentzel, Kramers and Brillouin, who used the method extensively in quantum mechanics in the mid-1920's. Sometimes it is called the WKBJ method where the last letter is for Sir Harold Jeffreys [1923], who described the method in a definitive paper. In most applications the lower limit of integration, z0, is set equal to zero since, in effect, this changes only the arbitrary constants a and b.

The physical meaning of Eq. (6) is fairly clear if k(z) is real. The exponent

Γ k(z) àz is the change of phase of a wave which has travelled from the

arbitrary point z0 to the point z. Obviously, the first exponential term in Eq. (6), which is preceded by an — i, corresponds to a wave propagating in the positive z direction. Conversely, the second exponential term, which is preceded by an + /, corresponds to a wave travelling in the negative direction.

The presence of the factor [l/k(z)]1/2, in Eq. (6), can be reconciled as the basis of energy flow. For example, if ψ is the electric field component Ey> then the magnetic field component, Hx9 must be proportional to δψ/dz or k(z)\j/. Then if the time average of the Poynting vector, in the z direction, is (1/2) Re EyH*9 which is constant for both upgoing and downcoming waves.

Actually, Eq. (6) is still valid for complex k(z\ corresponding to a medium with losses, but the physical interpretation is not as simple for the lossless case.

2. WKB METHOD FOR OBLIQUE INCIDENCE

The above derivation can be readily extended to oblique incidence. For example, if the electric vector has only a y component, Ey, it is necessary to

start with the equation

[Ï?+ME>=0 (7)

where q2(z) = k2(z) — λ2. Here, λ, which is a constant, describes the x variation of the fields (i.e. d/dx = — iX). For example, if a plane wave was incident from z = — oo and if k(z) -* k0 as z -> — oo, then λ = k0S = k0 sin θ0 where 0O is the conventional angle of incidence. Consequently, all field quantities must vary as exp(—ik0Sx) or exp(—ΐλχ). It thus follows that the WKB solution of Eq. (7) is given by

E, = 11ΙΦ)ΪΔ{Α0 expi - i Γ«(ζ) dz] + B0 expii Γβ(ζ) dxl j e~ik<>Sx (8)

when yl0 and B0 are also arbitrary constants. The WKB approximation for the magnetic field components are now found

from Maxwell's curl equation -ίμ0ωΗ=ο\χτΐΕ (9)

where μ0, the magnetic permeability, is assumed constant everywhere. Thus dE

ΐμ0ωΗχ = ^- (10)

or μ0ωΗχ = -A0q1/2 exp - i q dz + B0q'A exp + i q dz

+ ^ d l [ ^ o e x p [ - i £ « d z ] + B 0 « p [ + i J * i dzjje-**« (11)

where the z dependence on q is understood. Since it has already been assumed that the medium is slowly varying, the term containing àqjaz in the above equation can be neglected. The remaining field component is then obtained from

- W H Z 3 (12)

or simply μ0ωΗζ = k0SEy (13)

3. GENERALIZATION OF WKB METHOD

To investigate the true significance of the WKB method and to extend its usefulness, one should obtain correction terms. Only in this way, can one place a precise meaning on the term "slowly varying." Furthermore, following the ideas of Bremmer [1949], it is possible to show that the WKB approximation is indeed the first term of a rigorous series solution. The treatment used


here is more general than Bremmer's. The form of the WKB approximate solution for Ey9 given above, suggests that we write

HiiHhJ>H+râM']>*■]}·"■*(14) and

μ0ωΗχ = j - A(z)lq(z)-]1/2 exp[- i j «(z) dzj +

+ B(z)[i(z)]*expi+i i^(z)dzl)e-ifcoSjf (15)

and μ0ο)Η2 = k0SEy

The factors A and 5 are now allowed to be functions of z of a form yet to be determined. It is evident that the conventional WKB approximation is obtained when A and B are replaced by the constants A0 and B0.

Now, the fields must satisfy the Maxwellian equations

- q % = ιμ0ω dHJdz (16) dEyldz = ίμ0ωΗχ (17)

On substituting Ey and Hx given by Eqs. (14) and (15) into the latter pair of equations, one easily finds that A and B must satisfy

where the z dependence on A, B, and q is understood. For convenience of physical discussion, the latter 2 equations can be written

in the form dA(z)

and

dz dB(z)

dz

= sp(z)B(z) (20)

= *z(zM(z> (21)

where 6 is a small (constant) parameter. It is now assumed that a solution can be written as power series of the form

Α = Α0 + εΑί + ε2Α2 + ... (22) and

B = B0 + εΒχ + ε2Β2 + ... (23)


where A0 and B0 are constants. It is evident that if these are substituted into Eqs. (20) and (21), the successive approximations are related by

jzAm+i(z) = p(z)Bm(z) (24)

^zBm+i(z) = q(z)Am(z) (25)

It is now clear that the first terms of the series correspond to the conventional WKB solution, whereas the succeeding correction terms arise from internal reflection within the medium. For example, the εΒγ term corresponds to the first-order reflected wave propagating in the negative z direction resulting from a wave, A0, propagating in the positive z direction. Thus

*i(s) = f J o

q{z)AQ dz (26)

where the lower limit is chosen so that BB1 vanishes as z tends to +oo. In general,

Bm+1(z)=(Zq(z)Am(z)dz (27)

On the basis of identical reasoning, ζΑγ is the first-order reflected wave, propagating in the positive z direction, which arises from an incident wave, B0, propagating in the negative direction. Thus

or, in general,

Al(z)=\Z p(z)B0dz (28) J - 0 0

Λ1+ι(ζ)=Γ lfc)BJz)dz (29) J - oo

As an explicit example, the second-order reflected waves for the incidence wave A0 are written as a double integral of the form

A2(z) = [ p(zf) Γ q(z")A0 dz" dz' (30) J - oo J oo

where the primes have been used to distinguish between the 2 variables of integration.

4. GENERALIZED WKB METHOD FOR VERTICAL POLARIZATION

The formalism developed above can be readily extended to vertical polarization where the magnetic vector has only a component Hy (perpendicular to the plane of incidence). It is instructive to outline the derivation for this case also.

The field components are written in the following form

xexp(-ifc0Sx) (31)

> | - i q(z)dz\

- J* ( Zyff ) ] % e xP[+* j / z > d z ]} exp(-ifcoSx)

εοω^ = j^^)]^e x p | -_. ! e(2)dz| (32)

ε0ω£ζ = —-57^ H, (33)

and /c0o

where #(z) = k0[N2(z) — S2]%, as before. It is clear that if the new functions, A* and B* are replaced by constants, A* and B*, the solutions correspond to the conventional WKB form. In this case, upgoing and downcoming waves are uncoupled and the normal wave impedances are

±»io[N2(z) - S2]*/N2(z) in accordance with expected behavior.

To find the corrected form of the WKB solution, Eqs. (31) and (32) are substituted into the following Maxwellian equations

-$,*.--*■»£ <«> = — ΐε0ωΕχ (35)

N\z) dz This gives rise to the following coupled differential equations

and dB

where q* = q/N2. The z dependence of the quantities is understood. These equations have the same form as the pair given by Eqs. (24) and (25) and, therefore, the method of successive approximations is also valid here.

5. RELATION TO GEOMETRICAL OPTICS

We are now in a position to discuss the validity of geometrical optics which corresponds to the retention of only the first term of the series expansions for A and B. Except for special circumstances, it can be expected that the neglect

of partial or internal reflections is justified if εΒχ is small compared with A0. It is not difficult to see that, for horizontal polarization

On the other hand, for vertical polarization, the first square bracket term in the above is replaced by (l/#*)(d#*/dz).

Now, it is noted that the integrand in the integral is a rapidly oscillating function with a period approximately equal to n/q. Consequently, an order of magnitude estimate of the integral is simply

ldq 1 q dz q

Thus, for horizontal polarization, the condition for the vahdity of geometrical optics is

d 1 àz q <1

In terms of the refractive index N(z) and the angle 0(z) of the rayf, this condition is

d 1 ά~ζΝ(ζ)οο*θ(ζ)<λ° ( 3 9 )

In the case of vertical polarization, the order of magnitude estimate for SBJAQ is

1 dq* 1 q* dz q

Thus, the condition for the vahdity of geometrical optics is

1 dg* qq* dz <1 (40)

In terms of the angle θ(ζ) and the refractive index N(z), the restriction is

Ν^)^[^Μ\<λ° (41)

which is equivalent to

1 d [cos 0(z) cos2 0(z) dz

["cos 0(z)1 <λο (42)

t (Note that θ(ζ) may be defined by M^)sin 0(z) = S where λο is the free space wavelength).

The latter form was quoted by Brekhovskikh [1960] who, following Bremitìer [1949], treated only the case for vertical polarization.

It is quite apparent that internal reflections are particularly important at near grazing angles where cos 0(z) is small and is, itself, changing. Indeed, this is true for both polarizations.

6. APPLICATION TO TROPOSPHERIC PROPAGATION

An interesting application of the preceding development is to tropospheric propagation of radio waves. It is now known that large horizontal layers may exist in the troposphere. Order-of-magnitude estimates of the parameters of these layers have been given by du Castel et al. [1960], as follows

horizontal extent—kilometers vertical extent—tens of meters height (above ground)—hundreds of meters

The maximum change of the refractive index is of the order of 10" 5. The reflection coefficient for a plane wave incident on a troposphere layer

extending from z1 to z2 can be well approximated by

which is valid for horizontal polarization. Now, in the troposphere, the refractive index can be written as

N(z) = 1 + δΝ(ζ) (44) when |<5iV| <4 1. Then, since

q = k0(N2 - S2)v> s k0(C2 + 2δΝ)ν* (45) one finds that

and q^k0C

άζ * C dz y J C dz Here, C is the usual cosine of the angle of incidence as the latter approximations require that C2 > δΝ. Thus, the reflection coefficient is given by

^Mi&N]Q~2ikoczdz (46) This result has been obtained and used by many previous authors [SchelkunofF, 1948; Friis et al. 1957; de Castel et al. I960]. They have obtained it by starting with a discontinuous model. The idea of their derivation is given here, briefly.

The Fresnel reflection coefficient at horizontal polarization, for a homogeneous half-space of constant refraction index, 1 + δΝ9 is given by

^C-[2ÔN + (ÔN)2 + C2r> r C + 12δΝ + (δΝ)2 + C2]* ^ '

Now, if \δΝ\ < 1 and C2 > δΝ, this can be well approximated by

δΝ 2C2 r=-—2 (48)

Then, using somewhat heuristic reasoning, the reflection coefficient, dr, resulting from a change, d(5N)/dz, within a layer is given by

2C dz

where the phase 2k0Cz accounts for the two-way path traversed by the ray. Then it is supposed that the total reflection coefficient is obtained by integrating over the vertical extent of the layer. Thus

~ 2 C 2 J Z 2 L ' ^ ] e - « * » d z (50)

where z2 > zv This result is identical to Eq. (46) derived from the WKB series solution. It should be emphasized that this simple formula neglects all internal reflections. It is not just the total change δΝ in a layer, but the derivative d(ôN)/dz which must be small to justify the use of such a simple formula.

It is not difficult to show that the reflection coefficient for a tropospheric layer at vertical polarization is essentially the same as for horizontal polarization. This follows from the approximations

q* = q2IN2(z) S k0C and

d^*/dz s (ho/C) d(ÔN)/dz when

\ÔN\ <ζ 1 and C2 > \SN\

The integral in Eq. (46) can be readily evaluated in closed form when certain analytical forms for the derivative of the refractive index are assumed. Four examples, given by du Castel et al. [1960], are as follows:

(1) Linear layer,

where χ = k0Ch; (2) Parabolic layer,

diV(z)/dz

IH

= 9

9h 2C2

for 0

sin* 1

<z <h,

(51)

i < z < h dN(z)ldz = g(i-^\ for 0

(3) Sinusoidal layer,

( n z\ - - I for 0 < z < f t

2 C 2 1 - Z

(4) HyperboUc layer,

dN(z)ldz = gcosh'2(^\ for 0

I 9h c o s * ,CK

< Z < 00

ΙΓΙ = 2 ^ ϋ τ ^ (54)

It is seen that for each of these cases the total change of the refractive index across the layer is gh. In each case, as χ approaches zero, \r\ tends to ghjlC2

which is the value appropriate to a sharp discontinuity. It is apparent that when χ is large the reflection coefficient is greatly reduced in magnitude.

7. THE PHASE INTEGRAL APPROACH

When the medium is slowly varying, the WKB method and its extended form are very convenient. However, when the quantity (l/q)(dq/dz) is not small, the method fails. In particular, if q becomes zero at some point in the medium some further modification is required as pointed out by Langer [1937] and Budden [1961]. The approach used here is adapted from their work.

To illustrate the method, N2(z) is allowed to be a real and a smooth monotonically decreasing function of z. For z -» — oo, N(z) approaches unity and therefore q(z) approaches k0C. At z = z0, q(z0) = 0 or N(z0) = S.

q 2 (z )

FIG. 1. Sketch of q\z) = ko2[N2(z) - S2]

If attention is restricted to horizontal polarization, the electric field Ey is a solution of

*L*l+q\z)E, = 0 (55)

The medium outside the interval z_i < z *ζ Zi is assumed to be sufficiently slowly varying that the usual WKB solutions are valid. Now, in the region z < z_i9 the appropriate form of the WKB solution is

Ey = 4 [ e x p ( - i I « dzj + K expii Π β dzjl e"*05* (56)

where P and i? are arbitrary constants. It is noted that if z -> — oo, this could be written

£y = 7Γ7ΛΥ2 le~ikoCz + R eifcoCz] e"**** (57)

which is the superposition of 2 plane waves. The constant R can thus be regarded as a reflection coefficient which has yet to be determined.

In the region z > zu the WKB solution has the form

where now M 'l (58)

iq, or {-q2)Vl

is essentially a real quantity. 7* is another unknown constant which is also to be determined.


For the whole region extending from z_2 to z2, which includes the portion z_ ! to zu it is assumed that the profile can be approximated by

i2(z)£-fcgo(z-zo) (59)

where a is a constant. Thus Eyy in this region, satisfies

^ - f c f c ( z - z 0 ) E y = 0 (60)

Introducing the new variable t = (fcfc)*(z - z0) (61)

it readily follows that à2E, at

-tEy = 0 (62)

As pointed out in the previous chapter, solutions of this equation are the Airy functions u(i) and v{t). On inspection of the asymptotic forms it is found that only v(t) gives rise to an exponentially damped wave for large positive values of z. Specifically, the asymptotic forms for v(t) may be written

v(t) s ■ i e x p[ | (_ i ) %]e x p^)_e x p[_|- (_0 %]e x p(_|)

(63) (-t)'/4 ■■ li

when t is a large negative number, while

t>(i) = 2^4exp(-$i%) (64)

when t is a large positive number. These can be rewritten in terms of q and z. Thus

v(t) s y ^ - [exp(-M 4 dz)exp(^) - exp(H « dz)exp("T)j (65)

when (z0 - z) > (fcga)"1/3, and

ü(,) s 2F?F' «p[-jv«a)* H (66)

when (z-z0)>(kga)-1/3

It is immediately apparent that these are the WKB forms. It is assumed that they are valid in the respective intervals, z_2 to z_x and zx to z2. Thus

Eqs. (65) and (66) should be proportional to Eqs. (56) and (57), respectively, in the intervals of common validity. Therefore, it is easily found that

) —2i q{z) dz\ R = i exp | - 2i I q(z) àz | (67)

and

T = ei n / 4expi-î I °qdz\ (68)

which completes the solution.

8. A GENERALIZATION OF THE PHASE INTEGRAL METHOD

It is evident that the successive steps in the previous derivation are equivalent to using the following substitutions

'=ϊ4ί\ω<ΙζΓ (69) when z < z0, and

'-[IjViW'fc]'4 (70) when z > z0. Here, z0 is defined by the condition q(z0) = 0. Thus, the electric field is given by

E = E0v(t) (71) throughout the whole range of z. This can be verified by observing that the WKB solutions given by Eqs. (56) and (58) are regained when — / and t are respectively large compared with unity. In fact, this is true even when q2(z) is not a Unear function of z in the transition region. However, it is important that no additional zeroes or other singularities of q2(z) are near the principal zero z0. Implicit in this latter statement is the requirement that the curve q2(z) vs. z has a small curvature at z = z0.

The substitutions (69) and (70), used in conjunction with (71), are sometimes called the "extended WKB approximation." It has been used by Langer [1937], Pekeris [1946], Bremmer [1960], and possibly others.

In the preceding discussion of the phase integral method and its relation to the extended WKB approximation, the parameter q2(z) was regarded as a positive (or negative) real quantity. The root z0 was then always also a real quantity and corresponded to the "level of reflection." If the square of the refractive index N2{z) is complex, it is evident that q2{z) is also complex and, consequently, the root z0 is complex. Provided q2(z) is still approximated by a linear function of (z — z0) in the vicinity of z0, the previous arguments still hold. Furthermore, the reflection coefficient still has the form

K = iexpi-2i I °q(z)dz\ (72)


Here, the contour of integration extending from the origin to z0 may be chosen in any convenient way. It must be remembered, of course, that account must be taken of any singularities that are crossed in the formation of the contour. An interesting discussion of the physical significance of the contour used in the phase integral formula has been given by Budden[1961].

9. PHASE INTEGRAL FOR VERTICAL POLARIZATION

The phase integral method or "extended" WKB approximation has been discussed only in relation to horizontal polarization. In considering vertical polarization, at oblique incidence, where the magnetic field has only a y component, Hy9 the differential equation is

where

This is identical to Eq. (115) discussed in the previous chapter. By introducing a new function Φ(ζ), defined by

Φ(ζ) = Η,/ΛΓ(ζ) (74) the differential equation may be written in the form

[-^ + β2(ζ)]φ(ζ) = 0 (75) where

Λ , 2 1 d2JV2 3 /dN2\2

The previous arguments for horizontal polarization may now be carried over without modification provided Q2(z) is an approximate linear function in the vicinity of its principal zero ê0 defined by Q2(ê0) = 0. In this case, the reflection coefficient R^ defined in the usual way for vertical polarization, is given by the phase integral

R s i expi-2i °Q(z) dzl (77)

It is of interest to examine briefly the significance of the phase integral formula for vertical polarization when q2(z), rather than Q2(z)9 is a linear function of z in the vicinity of the reflection level. For example, if

N\z) = 1 - ^ ^ (78)


where a and b are constants, it is seen that

q\z) = kl(c2-Z-^ (79)

and

**>-#*-«,Λ-sf (80)

Now, since q2(z) = 0 when z = z0 = a + bC2, it is desirable to express Q2(z) as a Taylor expansion about z0. Thus

ß2(z) - -{4W + (Z - Z o ) ( f + 2W) + (Z - Z 0 ) 2 ( 4 W ) + - } (81)

It is readily apparent that Q2(z) has a zero near z = z0 if k0b > 1 and S is not small compared with unity. Therefore, at sufficiently high frequencies and oblique incidence, Q(z) may simply be replaced by q(z) in the phase integral formula given above. When these conditions are not met, the phase integral method for vertical polarization fails.

For additional remarks concerning the validity of the phase integral method at vertical polarization, the reader is referred to Budden's [1961] treatise.

10. RAPIDLY VARYING TRANSITION REGION

10.1 Introduction When the properties of the medium change rapidly the WKB and the phase

integral methods are unsuitable. For example, the theoretical treatment of v.l.f. radio waves* from the ionosphere requires a different approach. Here, although precise information is not yet available, the refractive index changes significantly in a distance small compared with a wavelength [Wait, 1960, 1962]. A first, and often a very good approximation, is to represent the lower edge by a sharply bounded homogeneous medium of constant refractive index Nt. Then, for a vertically polarized plane wave incident from below the reflection coefficient, R9 is given by the well-known Fresnel coefficient

JVfCo-(Ni-Sg)*

where C0 = (1 — SQYÀ = cos θ0 in terms of the angle of incidence 0O. (To simplify the discussion, the earth's magnetic field has been neglected.) We are now interested in obtaining a correction which will account for the finite thickness of a transition region between the free space below and the isotropie ionosphere above.

* v.l.f. = very low frequency (the range 3-30 kc/s).


To start with, we shall assume an inhomogeneously stratified medium. With respect to the usual rectangular coordinate system (x9 y, z), the refractive index is assumed to be only a function of z. As z tends to positive and negative infinity the refractive index is assumed to approach constant values. That is

N(z) ] 2 . + 0 0=iV1

ΛΓ(ζ)1—β = 1 At z = —oo, a plane wave is incident at an angle 0O with respect to the positive z direction.

10.2 Differential Equation for the Reflection Coefficient Without any loss of generahty, the H vector is taken to have only a y

component. Thus, for a time factor, eia>i, Maxwell's equations are given by

3EX dEz .

iecoEx = — —2, and iecoEz = —^ (83) oz ox

where μ and ε are the magnetic permeability and permittivity of the medium. For sake of generahty, both μ and ε can be regarded as functions of z. Of course, in applications to the ionosphere μ can be replaced by its free space value μ0· The retention of a variable μ in the theory permits one to readily adapt the results to arbitrary polarization. Furthermore, the analogy in acoustics is readily brought out.

Now the sum of the incident and reflected waves is defined in the following manner. The sole magnetic field component is written

Hy = \_A(z) + JB(z)]exp[ - ί(εμ)1/2ω5χ] (84)

while the 2 electric field components are written

Ex = lA(z) - 5(ζ)](7(μ/ε)1/2 e x p [ - ί(εμ)1/2ω5χ] (85) and

Ez = - \_A{z) + 5(ζ)]5(μ/ε)1/2 exp[ - ί(εμ)1/2ω5χ] (86)

where C and S are also functions of z. In this analysis, k = (ε0μ0)Υ2ω where ε0 and μ0 are the constants of

free space, while A(z) and B(z) are not yet defined. Since these field components are to satisfy Maxwell's equations, it is required that

(εμ)ί/2Ξ = constant

and since S = sin 0O, ε = ε0, μ = μ0> at z = — oo, it follows that

(εμΥ^Ξ = (ε0μ0),/2 sin θ0 (87)


which is just a statement of SnelPs law. Furthermore, C2 + S2 = 1

Again, as a consequence of Maxwell's equations, A(z) and B(z) must satisfy

^ + id A + T(A - B) = 0 (88) dz

^ - iôB + Γ(β - i4) = 0 (89) dz

where δ = (εμ)ν2ωΟ

and

2δάζ\ε)

Equations (88) and (89) are easily combined into a single equation for the ratio B/A = R(z). Thus

^ = HÖR + Γ(1 - R2) (90) dz

where .K(z) is, by definition, a reflection coefficient. Results, more or less equivalent to (90) have been given by Budden [1961] and Brekhovskikh [I960].

10.3 Iterative Solution To obtain a solution, R(z) is written in terms of a new function v(z).

Thus R_em*)-9i (91)

where

and

g(z)v(z) + gt

0(z) = 775 (W2 - S2-)* « = μΐμο, S0 = sin 0O N2

9l = timg(z) - £ § ( * ? - Sg)*, JC1 = /ij/io

Now, since lim^«, i?(z) = 0, it follows that l im^^ v(z) = 1. The differential equation for v(z) is obtained by substituting (91) into (90). This can be written in the relatively simple form

dv_ikN2g1/ "2

"dz~ K Ί-5-) (92)

where k = (ε0μ0)1/2ω and N, K and g are functions of z. Using a method of successive approximations, the solution can be expressed as an ascending


series in powers of A:. For example, the zero'th approximation is to replace the right-hand side of (92) by zero, thus v is a constant which must be unity to satisfy the limiting condition at z -> oo. The first approximation is obtained by replacing v2 on the right-hand side by unity. Thus

v = l + ik9ij°Ç(l-^jdz (93) where the limits of the integration are chosen so that v satisfies the limiting condition at z = oo. The second approximation is then obtained by substituting the latter result for v into the right-hand side of (92). In general, the «'th approximation, vn9 can be found from the (n — l)'th approximation, !>„-!, by using

vn = 1 + tkgx £ ° ^ ( l - 9^ vl^ dz (94)

10.4 Some Simple Extensions of the Solution While these results have been developed with specific reference to an incident

wave with the electric vector in the plane of incidence, the results are also applicable to the other polarization. If the magnetic vector of the incident wave is in the plane of incidence (i.e. horizontal polarization), the results are still valid if the following transformations are made,

Hy-+Ey9 Ex-+—Hx, E2-+—Hzi μ-»ε, and ε->μ Thus, the formula for the reflection coefficient R, given by (91), is still valid if Kis replaced by ε/ε0 and iVis not changed.

There is also a well defined acoustic analogy to the problem being discussed. In this case N(z) = c0/c(z) where c(z) is the velocity of sound and c0 is the limiting value of c at z = — oo. Thus the velocity is varying from a constant value CQ to a differing constant value ci at z = + oo. Also, K{z) = p(z)/p0 where p{z) is the density and p0 is its limiting value at z = oo. The component Hy is then analogous to the acoustic pressure and Ex and Ez are analogous to the x and z components, respectively, of the particle velocity.

10.5 Discussion of the Form of the Solution It is interesting to note that for the zero'th approximation (corresponding

to v = 1), the reflection coefiicient may be written

This is the Fresnel reflection coefficient for the reflection of a plane wave at oblique incidence from a sharply bounded and homogeneous medium. Thus the higher terms in the ascending k series account for the "gradualness" of the boundary.

H

In the general case, the reflection coefficient may be written

K{Zo) - C0 + (KJNÌXNÌ - So)Xzo) K ' where z0 is some convenient level, below which K(z) and N(z) may be regarded as unity. Thus the total field in the region z < z0 can be written

Hy = H0 Q-**So[Q-ik(z-zo)Co + R ( Z o ) e + ifc(z + ro)Co-| ( 9 7 )

where H0 is the value of the incident wave at the fictitious interface, z = z0. In the first approximation, neglecting terms in k3 and higher,

Succeeding terms quickly become more complicated. It can be seen that the integrands contain the factor 1 — (glg^f in each of these terms. The presence of this factor permits one to replace the upper limit of each of these integrals by zu where zt is the level, above which N(z) and K(z) may be replaced by Nx and Kt. Thus the transition region may be defined as the interval z0 < z < zx. On this basis, it is apparent that the n'th term in the series for v(z0) is the order of [k(zt — z0)]n. Consequently, the series converges rapidly when the electrical thickness of the transition layer is small.

As an interesting check, the transition is replaced by a homogeneous slab. Thus,

N(z) = ft and K{z) = R when z0< z < zt

In this case, the reflection coefficient is given exactly by [Wait, 1958]

Covizoì + iNl-SDHKjNl) K }

provided t K^Nl-SD* t r . f / , / / Λ 2 „2vl

1 + ^ ( y _ ^ tanhE,^ - z ^ ( ^ - Sg)] »(Ζο)=\ ÈNUfr-sfr ur.„ w/fta c2vi

(100)

1 + g^Vî - si)*tanh[,fc(Zl - zo>V(^2 - sly] If [Afo — z0)yj(ft2 — *S0)| < 1, it is seen that to a first order,

Φ0) s i + — ^ [i - sìa*;jffTsjl,k^ -z°> F (101)

which is consistent with the first 2 terms of the series given by (93).

For some applications to v.l.f. propagation, it is desirable to express the reflection coefficient in the following form

ßC0 + i where

R_ Afofro) P K,{N\ - S*)*

Then, if ßC0 is regarded as a small parameter, the following expansion results R = -e-2iCo + Kj8Co)3 _ fflçj* + ... ( 1 0 3 a )

Thus if \ßC0\3 < 1, R s _ e - 2 i c 0 ( 1 0 3 b )

which is a convenient form when the incidence is highly oblique and the frequency is not too low. On the other hand, if ßC0 is regarded as a large parameter, it is convenient to use the expansion

κ-""(-έ)-5(^)3 + 5(^+ - <1 0 4> which can be approximated by the first term if \ßC0\3 > 1.

REFERENCES

BREKHOVSKIKH, L. (1960) Waves in Layered Media, Academic Press, New York and London. BREMMER, H. (1949) The propagation of electromagnetic waves through a stratified medium

and its WKB approximation for oblique incidence, Physica, 15, 593-608. BREMMER, H. (1960) On the theory of wave propagation through a concentrically stratified

troposphere with a smooth profile, / . Res. Nat. Bur. Stand., 64D (Radio Prop.), No. 5, 467-482.

BUDDEN, K. G. (1961) Radio Waves in The Ionosphere, Cambridge University Press, Cambridge.

DU CASTEL, F., MISME, P. and VOGE, J. (1960) Sur le rôle des phénomènes de réflexion dans la propagation lointaine des ondes ultracourtes, in Electromagnetic Wave Propagation, 670-683, Academic Press, London and New York.

FRIIS, H. T., CRAWFORD, A. B., and HOGG, D. C. (1957) A reflection theory for propagation beyond the horizon, Bell Syst. Tech. J. 36, 627-644.

JEFFREYS, H. (1923) On certain approximate solutions of linear differential equations of the second order, Proc. Lond. Math. Soc, 23, 428.

LANGER, R. E. (1937) On the connection formulas and the solution of the wave equation, Phys. Rev., 51, 669-676.

PEKERIS, C. L. (1946) Theory of propagation of sound in a half-space of variable sound velocity under conditions of formation of a shadow zone, / . Acoust. Soc. Amer., 18, 295-315.

SCHELKUNOFF, S. A. (1948) Applied Mathematics for Engineers and Scientists, p. 212, Van Nostrand, New York.

WAIT, J. R. (1958) Transmission and reflection of electromagnetic waves in the presence of stratified media, / . Res. Nat. Bur. Stand., 61, No. 3, 205-232.

WAIT, J. R. (1962) On the propagation of v.l.f. and e.Lf. radio waves when the ionosphere is not sharply bounded, / . Res. Nat. Bur. Stand., 65D (Radio Prop.), No. 1, 53-62

(102a)

(102b)


Other References

NORTHOVER, F. H. (1962) The reflection of electromagnetic waves from thin ionized gaseous layers, / . Res. Nat. Bur. Stand., 65D (Radio Prop.), No. 1, 73-80.

RYDBECK, O. E. H. (1961) On the coupling of waves in inhomogeneous systems, Report No. 56, Research Lab. of Electronics, Chalmers University, Gothenburg, Sweden.

WAIT, J. R. (1960) Terrestrial propagation of v.l.f. radio waves, / . Res. Nat. Bur. Stand., 64D (Radio Prop.), No. 2, 153-204.

Chapter V

PROPAGATION ALONG A SPHERICAL SURFACE

Abstract—The basic theory of wave propagation around a sphere is given in outline. By utilizing the concept of surface impedance, the derivations are greatly simplified. The formal solution in the form of a slowly convergent series is transformed to a more useful form by following the method of G. N. Watson. A further transformation is made in order to obtain a formula which is suitable for very small curvature of the surface.

1. BASIC FORMULATION

In previous chapters the curvature of the stratifications has been neglected. In many applications to field strength computations it is necessary to consider the influence of sphericity. In this chapter, the theory of propagation over a sphere with concentric stratifications is developed. The results have particular application to the propagation of radio waves over the surface of the earth.

The method of solution is a direct extension of Watson's method [1918, 1919]. He treated the (radially oriented) dipole in the presence of a homogeneous sphere. To simplify the analysis, the surface impedance concept is exploited here. Furthermore, in the initial formulation, the atmosphere is assumed to be homogeneous.

The source of the field is considered to be an electric dipole oriented in the radial direction to the sphere of radius a. Choosing a spherical coordinate system (r, θ, φ\ the spherical surface is then defined by r = a, and the dipole is located at r — b and 0 = 0. It is understood that b > a. Due to the spherical nature of the problem and because of the assumed azimuthal symmetry, the fields can be derived from a Hertz vector, which has only a radial component Ur. Therefore, the fields are obtained from

E = curl curl (rU) (la)

and

/ / = ίεω curl (rU) (lb)

107

where r is a radially oriented vector of magnitude r. In component form these become

£ ' = ( k 2 + ê ) ( r l 7 )

d . ndU

r sin Θ

r ordv

„ ■ δ υ

for r > a

(2a)

(2b)

(2c)

(2d)

and Εφ = Η, = Ηθ = 0 (3)

where k = 27r/wavelength and ε is the dielectric constant in the exterior region. The function U satisfies

(V2 + k2)U = 0 (4) in the exterior region r > a except at the source dipole. In the immediate vicinity of the source, U must have the following singularity

bR where R = (r2 + b2 — Ibrcos 0)1/2 and c0 = —Ids/4nieco. This condition is readily verified by noting that the primary fields of the dipole are obtained when the operations are carried out.

The total field U is now written as the sum of 2 parts, Ue + Us9 where Ue has the proper singularity as R -> 0, and Us remains finite. Us is now expressed as a superposition of suitable solutions of the homogeneous wave equation. These have the form

A<2>(fcr)Pn(cos0) where h{

n2\kr) is a spherical Hankel function of the second kind, which

assures outgoing waves at infinity, and Pn(cos Θ) is a Legendre polynomial. The index n takes integral values. Now, making use of the known fact that exp(—ikR)/R can be expressed as an expansion in spherical functions,! t Such an expansion is

-ikR = i £ {In + l)Pn(cos Θ) Uhn^Kkr) + hn< \[hnW(kb) + hn

.<8>(*r)]A»<2>(A£) ; r b which is well known [Morse and Feshbach, 1953]. The spherical Hankel functions are defined as follows:

Propagation Along a Spherical Surface 109

it readily follows that a suitable expansion for Ue is given by

and

ike °

1/« = - ^ Σ ( 2 » + iy£Xkr)l£Xkb)PJjx» Θ), for r (fefc)PII(cos0), for r>b (6) where h^\kr) is the spherical Hankel function of the first kind. A corresponding expansion for Us is

ike °° 1/, = ^ Σ ( 2 » + l)Anhi2\kr)Pn(cos Θ) (7)

where An is an unknown coefficient. Having Ue and C/s expressed in this form enables the boundary conditions

to be applied in a straightforward manner. The important simplification to the solution is that we impose the single boundary condition

Εθ = -ΖΗφ at r = a (8)

By virtue of Eqs. (2c) and (2d), this can be rewritten

-—Γί/ = Ζΐεω17 ( 9 )

r or In other words, it is assumed that the influence of the earth can be described adequately by its surface impedance Z. In this problem, Z is taken to be equal to the ratio of the tangential electric and magnetic fields for a vertically polarized plane wave at grazing incidence on a plane stratified earth.

By application of Eq. (9), it readily follows that

" ~ h?\ka)

f4-logxh<tiXx)-iA ax

. - ^ logx^ 2 ) ( x ) - /A Lax

hfXkb) (10)

J x=ka

where Δ = εωΖ/k = Ζ/η. The total field is then of the form

U= f ( 2 n + l)/(n)Pn(cos0) (11) n = 0

where f(n) is a known function. The solution for the problem of the radially oriented magnetic dipole may

be obtained from the above solution very simply. In this case, U is proportional to the radial magnetic field component Hr9 provided Δ = Υη0 where Y is the appropriate value of the surface admittance.


2. THE WATSON TRANSFORMATION

The expansions of the field as developed above are not particularly useful in radio propagation problems, since an enormous number of terms would be required in the summations over n. Instead, following the ideas of Watson and others, the summation is transformed into the following contour integral

v.,\ J! an cos ηπ

/ ( η - * ) Ρ , - % [ Μ 8 ( π - β ) ] (12)

where n is now regarded as a continuous variable. The contour Ci + C2 encloses the real axis as illustrated in Fig. 1. Noting that the poles of the

Ci

Q yK

FIG. 1. The contour in the complex n plane showing the location of the real and complex poles.

integrand are located at n = £, f, -§-, ... etc. it can be readily verified by the theorem of residues, that this integral is equivalent to Eq. (11).

Now since f(n — J) is an even function of n, the part of the contour Q above the real axis can be replaced by the contour C[ which is located just below the negative real axis. The contour, C[ + C2, is now entirely equivalent to L, a straight line running along just below the real axis. Replacing n — \^ by v, the contour integral representation for U takes the form

JL δΐηνπ /(v)Pv[cos^ - 0)] dv (13)

The next step in the analysis is to close L by an infinite semi-circle in the negative half-plane. The contribution from this portion of the contour vanishes as the radius of the semi-circle approaches infinity. The value of the integral for U along the contour L is now equal to (-2πί) times the sum of the residues of the poles of the integrand in the lower half-plane.

The poles are located at the points v = vs which are solutions of M(v) = 0

where

M(v) = — log xh\2\x) - ΐΔ dx (14)

(15)

with x = ka. It follows then that U is proportional to

Σ (v, + mV(kb)h<ZXkr)Pv£coS(n - Ö)]

where the summation extends over the complex poles vs. The location of these poles in the complex plane is one of the major problems in radio propagation. The difficulty stems from the fact that v5 is of the same order of magnitude as ka. Thus, care must be exercised in approximating the spherical Hankel functions, since their order and argument are not appreciably different. Watson and others [Langer, 1937] have shown that in this region the spherical Hankel functions may be represented by Hankel functions of order 1/3. For the present problem this amounts to using the result

xh£\/2(x) * e -ίΛ/6(-2τ/3)1/^1/6Η^>[α)(-2τ)%] (16) where v = x + xvH. This relation is often called the Hankel approximation and it is valid when |v — x\ 4. x9 and x > 1. Apparently, it was first used by Lorenz [1890]. The root-determining equation can thus be well approxi-ma,edby

3l„B9m-*m , : o - , (17)

where < 5 = - i and τ, = ka

(ka)v*Z 5 (fca)1/3

Equation (15) for U can now be considerably simplified; the Legendre function can be replaced by the leading term in its asymptotic expansion, and hl2\kb) and h{

v2)(kr) are replaced by their Hankel approximations. This

results in U = 2U0(2nX)i/2 e ■ i B / 4 y / ^ l P 2 ) c "

r 2τ5-1Ιο2 irsX

where

Λ ^ - | . -2τ . J H(J>[Ì(-2TS)%]

läse-""* Un =

hi = r- a, 4πίεωα2(θ sin 9)t/2

h2 = b-a, X = (1<α)ν>θ

(18)

(19)

(20)


and Xi~{ka)*CLhJa)* for i = 1, 2

The function/^) can be called a height-gain function, since it becomes unity as ht approaches zero.

The preceding equations are identical in form to those obtained by van der Pol and Bremmer [1937] for a homogeneous spherical earth. Their results are obtained as a special case by letting

Z = η0Α (21)

with Δ = (y0/yi)(l — yolyìY/2> where γ0 = ik and where yt is the propagation constant of the homogeneous ground. It should be emphasized that the present results are only valid if Ai and h2 are both much less than a.

The residue series formula given by Eq. (18) has also been obtained by Fock [1945]. Despite the fact that the notation is very different, his final results are very similar. There are a number of reasons why Fock's notation is to be preferred. The main advantage is that it does away with the awkward Hankel functions of order one-third. In their place Airy integrals are used. Now, Fock's solution may be written (for a time dependence βχρ(ιωί)) as follows

U = U0V(x, yu y29 q) (22) where

V = -*-^2(πχ)^;~1*'^-^ ^àZp}. ( 2 3 ) where

>i«[£]V and ^ = [ ä / 3 f c Ä 2

x = (fca/2)1/30, q = - i(fca/2),/3A

and Wi(t) is an Airy integral. The roots ts are solutions of the equations

wi(0~^w1(0 = 0 (24)

where the prime indicates a derivative with respect to t. Since

Wl(0 = β-^3(-π/30%Η^[(*Χ-0%] (25) it is not difficult to see that Eq. (23) is identical to Eq. (18). Furthermore, the root-determining Eqs. (24) and (17) are also equivalent to one another. The Airy integral function w^t) is defined by a contour integral as follows

"m'U, Wi(0 = -y exp(si - s3/3) ds (26)

where Γ\ may be taken as a straight line segment from oo cin% to the origin and out along the real axis to oo. The associated Airy function w2(t) also occurs frequently and it is defined by

* *Jr w2(0 = - 7 exp(si - s3/3) ds (27)

where Γ2 is the straight Une segment from oo e~in2/* to the origin and out along the real axis to oo.* In terms of the functions u(t) and v(t\ introduced in an earlier chapter, w^t) = u(t) — iv(t) and w2(t) = u(t) + iv(t).

For certain applications it is desirable to express F as a contour integral in the complex t plane. This has the form

V = e1«'4^] 2 f cixtF(t, yu y2, q) dt (28)

F = iWl(i - y2)lw2(t - >>i) + *(0wi(* ~ Ji)] (29) where

where Brrt _ K(Q - gw2(ty\ ^-"LwMO-ew.coJ (30 )

When the contour is closed by a circular arc (of infinite radius) in the lower half-plane it is possible to verify that —lui times the sum of the residues at the poles / = ts leads back to Eq. (23). To demonstrate the equivalence, use is made of the fact that wx{t) and w2(t) satisfy the (Airy) differential equation

w"(0 - tw(t) = 0 (31) and that

WA) - qwx(ts) = 0 (32)

The form of Eq. (28) is particularly suitable for deriving approximate forms which are useful in special cases.

3. FORMULA FOR SMALL CURVATURE

The residue series formula given by Eq. (18) allows field strength calculations for propagation around large spheres to be carried out. However, at short distances and in some cases at low frequencies, the residue series formula becomes quite poorly convergent. A new type of expansion has recently been developed in which the first term corresponds to the radiation of a dipole over a horizontal stratified medium [Bremmer, 1958; Wait, 1956]. Succeeding terms are then proportional to inverse powers of ka.

* Since Fock [1945] employs an c~i<ùt time factor, his definitions of wi(t) and W2(t) correspond to our definitions of wz{t) and wi(t\ respectively.

The first step is to re-express U in the form of a contour integral or inverse Laplace transform, as follows

iU(g) 1 C<+"°n*f(P, htfjP, hz)ePa AP m ,

where c is some small positive constant,

g = ÎXK2Ô2) (34)

MF) jäcr (35)

and

A i - , f t J - [ ( p /^2 ) j H<j>[KP/S2)%] ( 3 6 )

It can be noted that the poles Ps of the integrand are determined by the solution of

H%Ìw)+e'nl3p*H%Ìw) = (37)

If Ps is replaced by 2ô2xsein this equation is identical to Eq. (17). It can be verified that the sum of the residues evaluated at the poles Ps leads back to Eq. (19).

The procedure is now to expand the integrand in powers of l/p. Each term is then a fairly simple inverse Laplace transform. Inversion of these can be carried out if certain results from the operational calculus are employed. The details of this rather tedious process are given elsewhere [Wait, 1956]. The final formula is given as follows

J - * G{F(P) - y [1 - Knp)V> - (1 + 2p)F(p)] (38)

+ (56[1 - KnpfHX - P) - 2p + \p2 + fe - I)F(P)]

+ term s in δ9, <512, etc.!

where

F(p) = 1 - i(np)* «fctf**) e~', <53 = - ^ (39a)

p = -ifca0A2/2 (39b) and

G s (1 + ifcfc^Xl + ikh2A) (39c)

It can be noted that when a -> oo, the right-hand side of Eq. (38) is simply proportional to F(p). This is in agreement with the theory for a flat surface. Thus, the remaining terms in the expansion can be regarded as corrections for curvature. This particular series formula is valid for relatively small heights such that kh\ and kh\ < 2αθ.

At distances greater than a few wavelengths from the source the quantity U is proportional to the vertical component of the actual electric field. On the other hand, 2U0 is proportional to the field of an identical source over a flat perfectly conducting surface. For apphcations to the earth, the field strength E9 in millivolts per meter at a distance dkm in kilometers, is then given by

(40) 300 «km 2U<

for a dipole whose strength is such that it would radiate one kilowatt over a perfectly conducting ground.

4. INFLUENCE OF AN INHOMOGENEOUS ATMOSPHERE

In the preceding sections the atmosphere has been assumed to be homogeneous. An attempt is now made to treat an inhomogeneous atmosphere. Also, it is shown that "the distance to the horizon" is a valid concept in diffraction theory even when the atmosphere is not homogeneous.

The earth is to be represented by a homogeneous sphere of radius a and the source is to be a vertical electric dipole which for the moment is located near the earth's surface. The usual spherical coordinates (r, 0, φ) are chosen with the source oriented along the polar axis and the surface of the earth is defined by r = a. The dielectric constant relative to free space is denoted by K and it is understood to be a function of r and decreases smoothly and monotonically with increasing r. The analysis given here follows the work of Fock [1945], Eckersley and Millington [1939], and Bremmer [I960].

The resultant magnetic field has only a single component, Ηφ, because of the assumed polar symmetry in the problem. As a further consequence of the symmetry, the electric field component Εφ is zero. It is convenient to express the remaining field components in terms of a single scalar function U which is actually the radial component of a Hertz vector. For a time factor exp[icot]f this leads to

■£î£ οθ

Ht=-ik0K— (41)

and

where k0 = 27r/wave length in vacuo, and η0 = 120π. When Kis independent of z these reduce to Eqs. (2a), (2c), and (2d), apart from a constant factor. These field components satisfy Maxwell's equations if the function U satisfies

| - [ i ! - ( £ r l / ) l + —^^\*hie^ + k$KrU = 0 (44) drlKdr \] rsin0 50[_ οθ \ ° v

Following the work of Fock [1945], it now proves to be convenient to introduce an auxiliary quantity U' which is defined by

17' = e-ika9Kr j(sm θ)ϋ (45)

and, consequently, the equation for U' satisfies

d2U' 2ikdU' ^ r + k or2 a ΰθ

K'dU' K dr

where K' = dK/dh, K0 is the value of K at r = a, and k = koy/KQ is the effective wave number at the surface of the earth.

Since EQ S -ΖΗΦ at r = a, it follows that

dU' — = ik(K0)*ZU' at r = a (47)

where Z is the effective value of the surface impedance. The right-hand side of Eq. (46) is now replaced by zero. The justification

for doing this is based on the fact that

dO'jdr= Order of (fcl/'/M) and

eU'ldO = Order oÎ(kaUf/M2)

where M = (ka/2)i/3 is a large quantity. It is also possible to simplify Eq. (46) by noting that

(1 - a2/r2) Ä 2h/a Consequently, U' satisfies

d2U' dh2 2lk?E+e(^+

2-ï)u·.» <48) ds \ K0 a)

where s = αθ is the great-circle distance measured along the earth's surface. The boundary condition at the surface of the earth then becomes

dU'/dh = + ik(K0)lAZO' (at h = 0) (49)

The function U', which must satisfy Eqs. (48) and (49), should also have the proper singularity at the source. That is, it should behave as

e- ikR U = - ( l + R,) R

near the source, where R, is a Fresnel reflection coefficient of the ground for vertically polarized waves. For h < s, and assuming that our approximate boundary condition is valid, R E' s + (h2/2s), and thus

h/s - Z R, E - h/s + Z

Consequently, the required behavior of U' near the origin is

U' E' Ko(a/s)%(2h/s)(h/s + Z)-' e-i(kh2/2s) (52)

5. EQUIVALENT EARTH RADIUS CONCEPT

The equivalent earth radius a, will now be defined in the following manner

1 1 Kb a , a 2 K o e ) h = f j

= - + - where Kb = - (53)

This equation is usually justified on the basis of geometrical optics. It can also be demonstrated in a more rigorous fashion by rewriting Eq. (48) in the form

au' 2k2 a2 ui ah2 as a,

2ik - +- h(l + g)U' = 0 -- (54)

where

9=- ~- 2KO a, (Yo Now, if the atmosphere is linear, by definition KA = (K - Ko)/h and g = 0, and consequently

a v 2k2 a2u'

ahz as a, 2ik- +- hU' = 0 --

But, on the other hand, if K = KO for all h then from Eqs. (48) or (54),

aut 2k2 azu' ah2 as a 2ik - + - hU' = 0 --

(55 )

f 56)

This strongly suggests that propagation in a linear atmosphere of radius a is equivalent to propagation over an earth of modified radius a, with a homogeneous atmosphere. It is, therefore, concluded that the effective earth

radius is indeed a valid concept. This conclusion has also been reached by many other authors, including van der Pol and Bremmer [1937-1939, 1947], Bremmer [1949], Fock [1945], Leontovich and Fock [1946], and Miller [1952].

6. EXTENSION TO NON-LINEAR ATMOSPHERE

In many practical cases of interest, however, K is not a linear function of height, particularly at great heights. Then the function g is non-zero and is itself, in general, a function of height. To further analyze this case, the relevant equations will be put in dimensionless form. The following parameters are first introduced

Η¥)Χ¥Γ» ΗέΓω

Ika \1/3

< Z = - / ( - y j (^ο),/2(Ζ/120π), and

1 / 2 \I/6

The equation for W is d2W ÔW _ _ i _ + , ( 1 + ^ = 0 (57)

and the boundary condition at the earth's surface now becomes

dW — +qW = 0 at y = 0 (58) dy Furthermore, as x tends to zero, W should behave as (2/x/^)exp(—iy2/4x).

It is now desirable to consider first the solution of Eq. (57) when g can be neglected compared to unity. This corresponds to an atmosphere which is linear or near linear. The relevant equation is now

d2W ,dW dy2 ~ l "fo

This can be solved by the method of separation of variables. That is, assuming

W = X(x)Y(y) one finds

1 d2Y(y) . 1 dX(x)

j - ' — + yw - ° (59>

= i—-—^ = t (60) Y(y) ay2 X(x) dx


where t is the separation parameter. Thus dX dx ^ + UX = 0 (61)

and à1 Y -£2+(y-t)Y = 0 (62)

Solutions of the first equation are

X(x) = Q±itx (63) and the second equation has solutions of the form

Y(y) = Mf-y) (64) where w(t) satisfies Airy's equation

d2w(0 at2 - tw(t) = 0 (65)

The appropriate solution for the x dependence is chosen such that outgoing waves only are retained. Therefore

Z(x) = e"iix (66)

Furthermore, the radial part of the solution (i.e. y dependence) is chosen so that the phase of Y(y) is a decreasing function with height (i.e. phase lag increases with height). At this stage, it should be noted that Airy's equation is closely related to Bessel's equation of order 1/3. In fact,

w(0 = const x *1/2Z1/3|B)(-0%] (67)

where Zi/3 is any cylindrical Bessel function or linear combinations thereof. In order to satisfy our requirement that the phase lag be an increasing function with height, the Hankel function H{£> of the second kind must be used. A particular solution which is suitable from the above standpoint is

W(t) = Wl(t) = e-i2*/3 (-ntpy> H™ [(2/3)( - /)*] (68)

where the constant is chosen to be consistent with the work of Fock and Leontovich [1946]. In fact, the latter authors represent w^t) as an integral of the Airy type

w1(0 = - ^ | 0 0 e'r-z3/3dz (69)

where the contour is a straight line from infinity along the straight arg. / = 2π/3 to the origin and then out to oo along the real axis. (i.e. arg t = 0.) This is equivalent to Eq. (26) above.

I


To satisfy the boundary condition at the earth's surface, one must choose

M(t-y)-qwi(t-y)l,=o = 0 (70)

A numerical study of this equation indicates that the roots t = ts (s = 0, 1, 2 ...) are in the fourth quadrant of the t plane. The solution is then made up of a linear combination of terms of the type Q~ixts H^ (ts — y).

It can be seen immediately that an equivalent representation is

W = f e - ^ d - y )

where C is some contour which encloses the poles of the integrand (i.e. at t = ts). The factor γ, assumed constant, is to be chosen such that W has the proper singularity as x approaches zero. It is to be expected that behavior of W for small x is to be determined by the behavior of the integrand for large values of t because of the factor Q~lxt. But, when t is very large, it follows from the known asymptotic behavior of the Hankel functions, that

Wl(* " y) s ±(0",/2 exp[±y(0%] (72) wi(i) - qwt(t)

where the upper signs are to be employed for 2π/3 ^ arg t > - π/3 and the lower signs for — π/3 > arg t > 4π/3. The contour C is now drawn such that one branch is along the negative imaginary axis (—/oo to 0) and the other along the real axis (0 to oo). For small values of x, then

W~y\ Q~ixt -j- at + y\ Q~ixt —r àt (73) Jo V* Jo V*

These integrals can be transformed to the form

i: (74)

and thus

and therefore, if

as x approaches zero.

W~2yi?-\ ' e ' ^ 4 " (75)

y = (i/B)*

WïÉlx-He-W**) (76)

The required solution can thus be written

W=(-\ 1V Ç-dt (77)

In this particular case the solution for U becomes Q-ikau e - '*"'

where V = x*fV

υ-^β^ψνχ^Γν <78>

F - ( 2 ) * f £ £ * r 2 U (79) \ W Jcwi (0-ewi (0

This is equivalent to Eq. (28) if yl = 0, y2 = y, and a = ae. Attention is now returned to the case of an inhomogeneous atmosphere

wherein g(y) is no longer negligible compared to unity. The more general equation for W must then be retained

d2W dW jp: - i-fo + y[i + g(y)iw = o (80)

Proceeding in an analogous fashion as in the linear case, it is readily found that solutions are of the form

W = Q'ixtF(y91) (81) where F(y, t) must satisfy

^ ρ ^ + [y - * + yg(y)lF(y, 0 = 0 (82)

The solution is then expressed as an integral

=N/%H.. which is constructed so the poles of the integrand are the roots, t = ts, of the equation

and the constant term outside the integral is chosen so that it reduces to the corresponding expression for W in the linear atmosphere (i.e. g = 0). The corresponding (formal) solution for the function U is thus written

e-ikae υ=Φ^Ψν (85)

Electromagnetic Waves in Stratified Media

b+4.„ 7. ASYMPTOTIC FORM OF THE SOLUTION

Various asymptotic approximations are now investigated. For large values of the real part of (y — t + yg\ the function F(y91) of Eq. (86) is given approximately by the WKB solution of Eq. (82). This is

«y. 0 « iy _ t + ygmy. «*[-l\lu-t + ugiuW duj (87)

where C is a constant and the lower limit of the integral is arbitrary. It should be noticed that if g = 0

F(y, t) = Cw^t - y)

which for large t — y behaves asymptotically like

F(y, 0 s ^ — ^ e x p [ - i J (II - 0% duj (88)

in accordance with Eq. (69). Equation (86) for V in the case of an inhomogeneous atmosphere can

now be simplified by noting F(y, t), where it occurs in the denominator, can be replaced by wx{t — y) and, therefore,

{fy + qF)=-W'lit) + qWl (0

In the numerator, F can be replaced by the asymptotic form given by Eq. (86) where y is large. Thus

y, _j(*)" r r—' l* — Î ^ _ d, m

W J c h - t + yg(y)l wl(0 - qw^t)

Ω = xt -f yj[u - ί + ug(u)~] du The saddle point, determined by άΩ/dt = 0, is thus given by t = —p2 where

du

where

J. o VC" + p2 + Μ 0 ( " ) ] - 2x = 0 (90)

Thus, following the prescription of the saddle-point method

*/2

where

and

K ~ e - W 2 x ^ ) V(-p) (91)

^-^ i /^ i raô <92)

Ω0 = -xp2 + [u + p2 + «g(u)]'/2 d« (93)

The integral F(—p) has been discussed on previous occasions and numerical results are available [Wait and Conda, 1958]. For large negative values of the argument (i.e. p > 0) it was noted that

n-PÌ^-^-expi-ip3/!) (94) p + iq

and therefore y **'*{* ffljh**-*® (95)

must be the geometrical optical approximation and, consequently, the factor 2p/(p + iq) can be identified as 1 + R(<x) where R(<x) is the Fresnel reflection coefficient for a plane wave incident at an angle a on the boundary. The parameter p is then related to a by

p = (kaJ2)v> cos a

In the case of the linear atmosphere (i.e. g(u) = 0)

Ω 0 = -χρ2 + (u+p2)i/2du (96)

s p3/3 + 2/ / 2 /3

for p small compared to x and y . In this same case

2x and consequently

eP\*

v — x2

psV (97)

(*S« When #(w) = 0, the formulae thus reduce to those for a homogeneous atmosphere with a modified earth radius.

In the case of the inhomogeneous atmosphere Eq. (90) for p can be solved approximately to yield

_ 1 Cy du _ Ρ~2]0Λ/ΐη+η9(η)ΓΧ ( '

under the condition that x and y are large but the difference x — yjy is finite. Thus

χΔ

S 1 (99) ( *8 - iy + yg(y)yA

and

Ω0 s ρ3β + Γ [M + ugf(u)]1/2 du (100)

For application to an actual atmosphere, it is necessary to evaluate the integral

P + x = ^\ ir d " , v <101) 2 j o V [ " + ««(«)]

where

. _ 1 f ' -2 joV[M

ae (K-K0 \

When the dielectric constant varies in an arbitrary, but monotonie, way with height, the integral can be evaluated readily by numerical methods. Some simplicity is obtained if it can be assumed that the variation is exponential in form. For example, if

K-K0 = (K'0lc)(l-e-ch) (102)

it is seen that K decays exponentially from the value K0 at the surface (A = 0). The constant c is of the order 0.13 km" 1 for an average atmosphere and c measures the rate of decay of the dielectric constant with increasing height [Bean and Thayer, 1959]. Before proceeding further in this direction, it is probably preferable to show the connection of the above results with those of geometrical optics.

8. DISTANCE TO THE HORIZON

The distance s to the horizon from an elevated point in the atmosphere is often used conveniently as a parameter in comparing one type of vertical refractive index profile with another. Using a method based on a stationary phase principle, it readily follows from Bremmer [1949] (and others) that

Cha2

= n0\ — Uh (103) J{Lrn(r)Y - n0a2}

where r = a + h, n(r) is the refractive index which is a function of r and nQ is the surface value [i.e. n0 = n(a)].

In the case of exponential atmosphere

n = 1 + b Q'CH = yjK (104) so that

"o = n]h=o = 1 + b = yjK0

The term under the radical can be written

a2l(l+bc-ch)2(l+ï) -(l+f>)2]

^ a 2 [ l + 2 6 e " c , , + 2 - ~ l - 2 f c ] a

Sfl2[^ + 26(e-rt-l)l

under the assumption that b < 1 and A/a < 1. To the same approximation n0/r in the integrand can be replaced by l/a. Consequently, the integral becomes

dk (105) V[(2/I/a) + 2ft(e-cA-l)]

which is equivalent to Eq. (101) derived from the full theory (with p = 0). Now, it is convenient to break the integration into 3 parts, such that

ΛΛο Ch' Çh" [ho=h'" = + + (106)

Jo J0 JA' Jh"

and thus

s = s' +s" +s'" In the first range (0 < h < s')

ah <"i [(2A/a)+2&(-(cfc)+^)] (107)

which is valid if h' is chosen such that the (cA)3/6 term in the expansion of e"c* is negligible compared to ch. This is satisfied if

(cft')2/6 < 1

(For example, if ch' = \ then (ch')2/6 = 1/24). Writing

.' - 1 Γ dh aom - VK2Ä/0) - 2f)c] J o V[A(1 + α/ί)] l ;

we find

V(2/« where

be2

, = i 1 Μ - . //j*L) (109) - 2bc) yj(x V U + a*7

a = (2/a) - 2fec

If the effective earth's radius is defined by

where

and

1 = 1 + -*°. ae a 2K0

K0 = (1 + b)2 s 1 + 2ft

d

it is seen that

Therefore

* 0 = ^ ( 1 + 6e~CA)2

QË - 2 f c c

Λ = 0

where

IfaA' « 1

^vökj'"""'^^) (1I0)

a = V(2/'

''«^^-^O (1U)

which is the usual approximation for the distance to the horizon from a height hf in a linear atmosphere. The hyperbolic tangent formula above is actually the distance to the horizon from a height h! in an atmosphere defined by

n = 1 + b(l - ch + (cA)2/2) (112)

for 0 < A < A'. This is an approximation to

n = 1 + b Q~ch

as mentioned above.

We shall turn our attention to the third portion (i.e. h" < h < A'" = h0); %ho dft

(113) -I I». V[(2A/a) + 2 i> (e - c A - l ) ]

Now the term in the radical is written in the following form

2h e~ch

{Ihja - 2b) + 2b e~ch = (2h/a -2b) i+w£ä) (ii4> Letting u = (2A/<J - 26) it follows that

1 1_ v . (2b)n

Ui+™Ç\r >-o.fe.3... * «"

-^TTT-J <1 1 5> where

A - 1 A - - * A - — / l 0 — 1, A i - — Î , Λ 2 —

1 x 3 x 5 1 x 3 x 5 x 7 3 2 x 4 x 6 ' 4 2 x 4 x 6 x 8

and so on provided 2be~ch

Λ < 1.

This is satisfied if 2b < h/a < h"\a or if h" > 2ba s 2 x 3 x 10~4 x 6360 £ 3.9 km for any value of c. Actually, this is more stringent than necessary since e~c* < 1. Thus

s'"= Σ An(2b)"e-°»b" —j^duiaß) (116) π = 0,1,2... Ju» U /2

Now consider the integral which occurs in the above expression,

F^") = i c ° ^ d M (117> and integrate it by parts successively to yield the recurrence relations


and so on, until 1 [e~ßu Ί

V2(ßu) = —\-u-ßVi(ßu)\

V1(ßu) = I - ^ - du = (-J erfcCV^«)] (119)

erfc[V(j?u)] = 4 - | e-*2 dx (120) is the complement of the error integral which is extensively tabulated [Pierce, 1929]. Finally,

where

where

s" = az Σ An(2b)n e-"6cTO/iu") - Vn(ßu0)2 (121)

where

and

It would appear that in most cases h' can be taken equal to h" and chosen so that h' = h" = 4 km which is a convenient upper limit for the quadratic profile and a lower limit for the expansion formula. Thus s = s' + s'".

An alternative is to return to the integral s" and expand e"CÄ to terms of third order. This leads to

1 fh' an 5 " = /r/.ix u n / » „ . , ^ . g „ , (122)

dfe ' V[(2/a) - 2fcc] J„, V[*(l + yh)(l + 5ft)]

where be2 . „ be

y = — and δ = a-) a-) This integral is expressible in terms of elliptic integrals (see Pierce [1929], No. 535 for example).

9. CONCLUDING REMARKS

The present analysis provides a strong justification for the validity of an equivalent earth radius concept, when the refractive index of the atmosphere varies in a linear manner with height in agreement with the conclusions of Bremmer [I960], Norton [1941, 1959], Miller [1952], and Northover [1955]. The effect of the non-linearity becomes important at great heights as pointed


out by Millington [1958] and others [Wait and Conda, 1958; Personal communications from K. A. Norton and H. Bremmer], who have suggested that the theory for a linear atmosphere could be modified by simply relocating the optical horizon point. In the present work a complete mathematical justification is given for this step provided that the variation of the refractive index with height is monotonie.

The idea that the structure of the diffraction fields in a monotonically varying or tapered atmosphere is similar to that for a homogeneous or airless atmosphere is in apparent contradiction to Carroll's theory [1952] of "radio twilight". However, the model he uses is not smoothly tapered since there are discontinuities in the gradients of the refractive index. In the bilinear model, for example, the atmosphere is represented by a layer of dielectric with a constant gradient up to about 30,000 ft after which the gradient is zero. Carroll and Ring [1955] obtain their solution by matching the appropriate wave functions at the atmosphere-free space interface.! In the free space region (above 30,000 ft for the bilinear model) only outgoing waves are allowed, but at the top of the dielectric graded layer the two types of wave functions which are used can be identified as upgoing and downcoming waves. In the present treatment which uses a modified WKB method, the solution in this region does not have this double character.

Other models used by Carroll and Ring [1955, 1957], such as the inverse-square and trilinear profiles can be objected to on similar grounds. It is of interest to note that the inverse-square profile is somewhat better in that the discontinuity is in the second derivative at the bounding surface and the computed attenuation, as expected, is greater. The trilinear model has 2 piece-wise linear sections with interfaces at 15,000 and 90,000 ft and the mode computations show less attenuation, which is probably a consequence of the strong reflections from the lower interface.

It is the opinion of this writer that the Carroll and Ring computations are correct for the models specified. However, they cannot be regarded as valid solutions for propagation in a monotonically tapered layer. It is probable if the model were chosen so that the higher-order derivatives of the refractive index are continuous at the interfaces, the mode solutions and the modified WKB solutions should approach one another. Unfortunately, the poor convergence of the resulting mode solutions will make this ultimate comparison very difficult.

It is of interest to note that Bremmer [1962] has recently obtained higher order corrections to the WKB solutions for a refractive index structure which decreases smoothly with height. His results support the contention that modes of low attenuation are not possible in a normal gravitationally stratified atmosphere.

t Wendell Furry was apparently the first one to use the bilinear profile (see Chapter XII).


REFERENCES

BEAN, B. R., and THAYER, G. D. (1959) Models of the atmospheric radio refractive index, Proc. I.R.E., 47, No. 5, 740-753.

BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier, New York. BREMMER, H. (1958) Applications of operational calculus to ground wave propagation, par

ticularly for long waves, Trans. I.R.E., AP-6, No. 3, 267-272. BREMMER, H. (1960) On the theory of wave propagation through a concentrically stratified

troposphere with a smooth profile, / . Res. Nat. Bur. Stand., 64D (Radio Prop.), No. 5, 467-482.

BREMMER, H. (1962) On the theory of wave propagation through a concentrically stratified troposphere with a smooth profile, Pt. II, / . Res. Nat. Bur. Stand., 65D (Radio Prop.).

CARROLL, T. J. (1952) Internal reflection in the troposphere and propagation well beyond the horizon, Trans. PGAP, 4, 19.

CARROLL, T. J., and RING, R. M. (1955) Propagation of short radio waves in a normally stratified troposphere, Proc. I.R.E., 43, 1384-1390.

CARROLL, T. J., and RING, R. M. (1957) Twilight region propagation of micro-waves by a monotonically tapered layer of air dielectric, Vonde Electrique, 37, 471-479 (in French).

ECKERSLEY, T. L., and MILLINGTON, G. (1939) Application of the phase integral method to the analysis of the diffraction and refraction of wireless waves around the earth, Phil. Trans. Roy. Soc, No. 778, 237, 273-309.

FOCK, V. A. (1945) Diffraction of radio waves around the earth's surface, / . Phys., U.S.S.R., 9, 256-266.

LANGER, R. E. (1937) On the connection formulas and the solutions of the wave equation, Phys. Rev., 51, 669.

LEONTOVICH, M. A., and FOCK, V. A. (1946) Propagation of electromagnetic waves along the earth's surface, / . Phys. U.S.S.R., 10, 13-24.

LORENZ, L. V. (1890) Lysbevoegelsen i og uden for en plane Lysbolger belyst Kugle, Vid. Selsk. Skr. (6), 6, 403.

MILLER, W. E. (1952) Effective earth's radius for radio wave propagation beyond the horizon, / . Appi. Phys., 22, 55-62.

MILLINGTON, G. (1958) Propagation at great heights in the atmosphere, The Marconi Review, 21, 143-159.

MORSE, P. M., and FESHBACH, H. (1953) Methods of Theoretical Physics, McGraw-Hill, New York.

NORTHOVER, F. H. (1955) Long distance v.h.f. fields, Canad. J. Phys., 33, 241-256. NORTON, K. A. (1941) The calculation of ground-wave field intensity over a finitely con

ducting spherical earth, Proc. I.R.E., 29, 623-639. NORTON, K. A. (1959), System loss in radio wave propagation, / . Res. Nat. Bur. Stand.,

63D (Radio Prop.), No. 1, 53-73. PIERCE, B. O. (1929) A Short Table of Integrals, 3rd rev. ed., Ginn. VAN DER POL, B., andBREMMER, H. (1937,1938 and 1939) The diffraction of electromagnetic

waves from an electrical point source round a finitely conducting sphere, with applications to radio-telegraphy and the theory of the rainbow; the propagation of radio waves over a finitely conducting spherical earth, Phil. Mag., S.7, 24, No. 159,141-176. ; S. 7,24, No. 164, 825-864 (November, 1947); S. 7, 25, No. 171, 817-834 S. 7, 27, No. 182, 261-275.

WAIT, J. R. (1956), Radiation from a vertical antenna over a curved stratified ground, / . Res. Nat. Bur. Stand., Radio Propagation, 56, No. 4, 237-244.

WAIT, J. R., and CONDA, A. M. (1958) Radiation pattern of an antenna on a curved lossy surface, Trans. I.R.E., AP-6, 348-359.

WATSON, G. N. (1918) The diffraction of radio waves by the earth, Proc. Roy. Soc, A95, 83-99.

WATSON, G. N. (1919) The transmission of electric waves around the earth, Proc. Roy. Soc, A95, 546-563.


Additional References ANDERSON, L. V. (1958) Tropospheric bending of radio waves, Trans. AGO, 39,

No. 2, 208-212. LEONTOVICH, M. A. (1944) Approximate boundary conditions for the electromagnetic

field on the surface of a good conductor, Bull. Acad. Sci., U.S.S.R., serie physique, 9, 16 (in Russian).

MILLINGTON, G. (1939) The diffraction of wireless waves round the earth (a summary of the diffraction analysis, with a comparison between the various methods), Phil. Mag., S. 7, 27, No. 184, 517-542.

MONTEATH, G. D. (1951) Application of the compensation theorem to certain radiation and propagation problems, Proc. I.E.E., 98, Pt. IV, 23-30.

VVEDENSKY, B. (1935, 1936 and 1937) The diffractive propagation of radio waves, Techn. Phys., U.S.S.R., 2, 624^639; 3, 915-925; 4, 579-591.

WAIT, J. R., and SURTEES, W. J. (1954) Impedance of a top-loaded antenna of arbitrary length over a circular grounded screen, / . Appi. Phys., 25, No. 5, 553-555.

WAIT, J R. (1959) Radio wave propagation in an inhomogeneous atmosphere, Nat. Bur. Stand Technical Note No. 24.

Chapter VI

FUNDAMENTALS OF MODE THEORY OF WAVE PROPAGATION

Abstract—A self-contained treatment of the waveguide mode theory of propagation is presented. The model of a flat earth with a sharply bounded and homogeneous isotropie ionosphere is treated for both vertical and horizontal dipole excitation. The properties of the modes are discussed in considerable detail. The influence of earth curvature is also considered by reformulating the problem using spherical wave functions of complex order. The modes in such a curved guide are investigated and despite the initial complexity of the general solution, many interesting and limiting cases may be treated in simple fashion to yield useful and convenient formulas for calculation.

1. INTRODUCTION

In previous chapters attention has been restricted mainly to reflection from the upper smooth surface of a homogeneous or stratified medium. Of great practical interest is the case when the source is located in the space between 2 smooth boundaries. Such a region is often called a waveguide since the waves are channeled or guided between the bounding surfaces. In this chapter, the subject is developed with special attention to two-dimensional waveguides where the energy is constrained in a parallel plate region. Much of the discussion will be phrased in terms of the waveguide formed by the earth and the lower boundary of the ionosphere. To simplify the discussion, the influence of earth's main magnetic field is neglected, and therefore the ionosphere can be approximated as a sharply bounded homogeneous conductor. The influence of the earth's magnetic field and other extensions will be considered in a later chapter.

The waveguide concept of terrestrial radio wave propagation has been particularly useful in the very low frequency range (i.e. 3-30 kc/s). Since theoretical investigations of this subject have extended over many years, it is desirable to present a brief historical sketch of various developments.

Many years ago G. N. Watson [1919] employed a waveguide approach when he considered, at least in a formal way, the propagation of electromagnetic waves between an idealized homogeneous spherical earth and a concentric reflecting layer. Because of the extremely poor convergence of the exact series solutions, Watson devised a technique to convert this to a more

132

Fundamentals of Mode Theory of Wave Propagation 133

rapidly converging series using function-theoretic means. The new representation corresponds to the sum of residues at poles in the complex plane and hence the name "residue series". The waves associated with these poles are the waveguide modes. Watson studied the numerical properties of these models for the case of long waves or low frequencies and he assumed a very highly conducting shell. This particular aspect of his investigation was prompted by the recent discoveries of Marconi that radio waves decay much more slowly with distance than predicted on the basis of classical diffraction theory in the absence of a reflecting shell.

Watson found that the modes of low attenuation behaved like

(sin d\d)x/*

where d is the great circle of distance,/is the frequency, σ is the conductivity of the reflecting ionosphere, a is the radius of the earth, and a is a constant. For frequencies in the range from about 20 to 40 kc/s, observed field strengths behaved more or less in this fashion if the effective ionospheric conductivity was taken to be about 10~4 mho/m or a conductivity of the same order as "tap water". Actually, for frequencies in this range some 10-30 modes would be excited and if the complete mode sum were considered, the calculated field strength vs. distance curve, using such a model, would show many rapid and violent undulations. Such a behavior is not observed under normal conditions and this fact alone is sufficient cause to reject this model even from a phenomenological viewpoint. The same model with certain refinements has been discussed more recently by Rydbeck [1944] in a monograph, Bremmer [1949] in his book, Schumann [1952, 1954a and 1954b] in a series of papers, and quite recently by Kaden [1957].

From the frequency analysis of atmospheric wave forms [Taylor and Lange, 1959; Chapman and Macario, 1956] it is known that the attenuation rate does not vary like y/f except possibly at frequencies near 1 kc. Actually, the attenuation rate decreases with increasing frequency in the range from about 2-18 kc and thereafter increases. A behavior of this kind is highly suggestive of a Brewster angle effect. Such a proposal was first made by Namba [1933] as far as this writer can ascertain. It thus appears that the ionosphere at v.l.f. is behaving more like a magnetic wall (tangential H near zero) rather than an electric wall (tangential E near zero) as postulated by Watson, Bremmer, and Schumann.

Contributions to the waveguide mode concept have also been made by Budden [1952, 1953, and 1957] who unlike the early workers did not assume a highly conducting reflective layer at the outset. His model was a vertical electric dipole source in the space between the surface of a flat perfectly conducting ground and a sharply bounded homogeneous ionosphere. Various


extensions and generalizations have been made by Al'pert [1955, 1956], Liebermann [1957], Wait [1957a, 1957b, 1958a, 1958b], Wait and Howe [1957], Wait and Perry [1957], Howe and Wait [1957], and Friedman [1958]; also, Wait and Spies [1960] and Spies and Wait [1961]. Their work is referred to from time to time in what follows.

2. BASIC CONCEPTS

To introduce the subject a very simple model is chosen. The earth and the ionosphere are represented by perfectly conducting planes. In terms of a cylindrical coordinate system (p, φ9 z) the ground surface is the plane z = 0 and the lower boundary of the ionosphere is the plane z = h. The source is now considered to be a vertical electric dipole located on the ground. The electric field observed at some other point on the ground plane has only a vertical component and can be deduced by considering the images of the source dipole. These images are located at z = + 2A, + 4A, + 6A, etc., and all have equal sign and magnitude, because of the assumed perfect conductivity of the bounding walls. These images will always direct a wave broadside since the radiation from each image is in phase. At a distance which is large compared to A, this field can be calculated by replacing the line of dipole images by a continuous line source carrying an equivalent uniform current 7a. This current is the average current along the z axis and is given by

ds I = — I a h

in terms of the height as of the dipole and its current /. Now the field of a line source of current is well known and thus

E.J^mi*)-^BV\kp) (2.1)

where H(02\kp) is the Hankel function of the second kind of argument kp,

μ is the permeability of the space, ω is the angular frequency, and k = 2π/λ. When p > A, the Hankel function can be replaced by the first term of its asymptotic expansion and this leads readily to

where η = (μ/ε)Υ2 = 120 π. As mentioned above, this field corresponds to the radiation directed broadside so the rays are parallel to the bounding walls. However, there will be other angles where the rays emanating from each of


the dipoles in the line of images are also in phase. Such a resonance condition exists when

2hC = ηλ (2.3) where C is the cosine of the angle subtended by the rays and the z axis and n is an integer (see Fig. 1). It is seen that for each value of n there are 2 families of rays which have the same radial phase velocity (i.e. = c/S) but with opposing vertical phase velocities (i.e. = ±c/C). Again the radiation of these sets

z = o

FIG. 1. Depicting ray-geometry corresponding to the first mode between parallel plates; for resonance, λ = 2hCi.

of waves (i.e. modes) can be imagined to originate from an equivalent Une source. The strength of this fine source is IS where S is the sine of the angle subtended by the rays and the vertical direction. To obtain the resultant vertical field, this must be again multipUed by S. Consequently, the resultant field of all the families of rays or modes is obtained by summing over integral values of n from 0 to oo to give

*. = **Tjr Σ tâH«XkSnP) (2.4)

where ε0 = 1, e. = 2(n = 1, 2, 3 ... ) and Sn = (1 - CM2) and C„ = ηλ/2!ι.

The term n = 0, corresponding to mode zero discussed above, is only included once in the summation, whereas the higher modes are included twice. In the far field, this expression for the field reads

til as °° **-2Ηλρ)* n=o ì 2 , . . . A 6 ( 2 · 5 )

Up to this point the bounding walls have been assumed to be of perfect conductivity: The reflection coefficients for the rays are always +1. Another simple case is when the upper boundary has a reflection coefficient of — 1 corresponding to a perfect magnetic conductor, and the lower boundary still has a reflection coefficient of +1 corresponding to the more common perfect electrical conductor. For this situation the images are now located atz = ±2A, ±4A, ... but now they alternate in sign. It may be observed that there is no coherent family of rays directed broadside. This would have been the zero-order mode. The resonance condition for the modes is now

2hC = (n - i)A (2.6) where n = 1, 2, 3, ... . The corresponding expression for the vertical electrical field is thus given by

E^mlas g slH<i\kSnp) (2.7) ^rl „= 1,2,3,...

where the summation starts at n = 1 and includes all positive integers. In the far field T - œ

E , S - 2 i * e ' « ' * Σ S ? e - ^ (2.8) h(Xp)/2 η=ΐί?,3,...

In the foregoing discussion the observer and the source, which is a vertical electric dipole, are located on the ground plane. The above results are easily generalized to a finite source height, z0, and a finite observer height, z, by inserting the factor cos(kz0Cn)'COs(kzCn) inside the summations of Eqs. (4), (5), (7), and (8). This can be verified by returning to the image picture and noting that they are located at z = - z 0 , ±(2h + z0), ±(2h - z0), ±(4A + z0), ±(4A — z0), .... It may also be observed that the cos(£zCn) when replaced by [ex$(ikzCn) + exp(—ikzCn)]/2 can be identified as a family of upgoing and downcoming rays within the guide.

The important modifications of the preceding formulas as a result of imperfect reflection can be obtained by a rather simple physical argument. The complete treatment requires a more mathematical approach which is to be described later on.

The reflection coefficient for a ray incident on the ground plane at an angle (whose cosine is C) is denoted Rg(C). The corresponding reflection coefficient for the upper boundary which is the lower edge of the ionosphere is denoted jRf(C). The resonance condition now has the form

RgiQRtiQ c'i2khC = e-'2nn (2.9) which reduces to Eq. (3) if the reflection coefficients are both +1 and reduces to Eq. (6) if one reflection coefficient is +1 and in other — 1. Physically, the more general form above can be the condition for a ray to traverse the guide twice, be reflected at each boundary, and yet still suffer a net phase shift of


Inn rad where n is an integer. Since Rg(C) and R&C) may be complex and less than unity the value of C (i.e. C„) which satisfies the resonance equation may also be complex. The angle of incidence of these rays in the guide are thus complex. The corresponding value of Sn is also complex and this results in attenuation of the wave in the radial direction. In fact, the attenuation constant is minus the imaginary part of kSn in nepers per unit distance.

When the angle or its cosine C must be complex in order to satisfy a resonance equation, the resulting waves are damped. The numerical solution of such a complex resonance equation is quite difficult, in general, since it is not usually possible to obtain an explicit expression for C in terms of known parameters. This aspect of the problem is discussed in a later section.

3. FORMULATION FOR FLAT EARTH

3.1 Vertical Dipole Excitation It is now desirable to formulate the problem in a more definite fashion.

A vertical electric dipole of moment / d.s is placed in a homogeneous plane layer bounded by 2 plane interfaces (see Fig. 2). The lower interface is at z = 0 corresponding to the surface of a homogeneous ground of conductivity

//////

z0

M^MgW^/AsW

I (ionosphere)

/ / / / / / / / / / s / s / s S s / /

(air)

*o μο

(ground) z=o

FIG. 2. Cylindrical coordinate system for the vertical dipole between the 2 plane interfaces.

σβ and dielectric constant, sg. The upper interface at z = h is the lower edge of a homogeneous ionosphere which for the moment is assumed to be isotropie and has effective electrical constants at and ε(. The fields in these regions can be derived from a Hertz vector which has only a z component, Πζ. Thus, for h ^ z ^ 0,

Μ'2+έ)π· k = (εμ^ω = 2π/λ.

H2 = 0

(3.1)

Similar expressions are applicable for the regions z > h and z < 0 where the z directed Hertz vectors are denoted Π<° and U(

z9\ respectively, and the

corresponding wave numbers are k{ and kgi respectively. The formal solution of this problem is obtained in a straightforward fashion by requiring that the tangential field components Ep and Ηφ are continuous across the 2 plane interfaces. An equivalent statement of these matching conditions is as follows :

(3.2) k2Uz = fc2n<»>~

δΠζ _ δΥΙζβ)

dz dz .

k2nz = kfn«"

dUz _ dUf dz dz .

for 2 = 0

for z = h (3.3)

To facilitate the solution, the primary excitation resulting from the source dipole is represented as a spectrum of plane waves. This well known representation, for the primary Hertz function, is given as follows [Sommerfeld, 1926] :

_ Mexp{-»fc[p2 + (z-z 0 ) 2 ]^} [ρ2 + ( ζ - ζ 0 ) 2 ] ' Λ

Mik f = — H(

02XkSp)expl-ikC\z - z0|] dC (3.4)

where M = Ids/(4niœs) and S = (1 — C2)1/2. The integration variable Ccan be regarded as the cosine of the angle of incidence of the plane waves in the spectrum. Γ is the contour of integration and it extends in the S plane, from — oo along the negative real axis to the origin, then out along the real axis to -f oo. It should be noted that, since C can be greater than unity, complex angles in the spectrum occur. The above form for the primary excitation then suggests that the resultant Hertz function for the 3 regions can be written in the respective forms

(I) Π2 = Π<ρ) + I [.1(C) Q~ikCz + B(C) e+ikCz]H(0

2\kSp) dC (3.5)

for 0 < z < A;

(II) Π ^ = | G(C)e+ik°c°2H(0

2XkSp)dC (3.6)

for z < 0; and

(III) Π ° = ί /(C) Q-ikiCi2H(0

2\kSp) dC (3.7)


for z > h. In the above it can be verified that these Hertz functions satisfy the appropriate wave equation subject to the conditions that

Ng(i - C])* = (1 - Οψ = N{1 - C?)* (3.8) where

9 \ ieco ) \ ιεω / Terms containing Qxp(—ikgCgz) and exp^Qz) are not permitted since they would violate the Sommerfeld radiation condition at \z\ -> oo.

The form of the unknown function A(C), B(C), G(C) and 1(C) can be obtained explicitly by using the 4 equations of continuity. This purely algebraic process is easily carried out and further details are omitted. The resultant Hertz function for the air region is explicitly given by

where

and

ikM f Π, = — I r F(QH«XkSp) àC (3.9)

SçikCz _|_ g Q-ikCz\/QikC(h-z0) _j_ ^ Q-ikC(h-z0)\ F ( C ) = ' i*Cfc/i » r» „-likhL ( 3 · 1 0 )

R»

Ri

eikCh(l -

= RS(.Q ■

= Ri(Q ■■

-RgRie~ -2ikhC\

HgC - Cg

NgC + Cg

Nfi-~~ AT Γ* i

^1

(3.11)

(3.12) NtC + Ci

It can be immediately noted that the integrand has poles where 1 - RgiQRiiQ Q~2ikhC = 0

This is the (complex) resonance equation obtained in the previous section by intuitive reasoning.

The integral may be evaluated by using function-theoretic means. The contour is transformed to the S plane. Thus Eq. (9) becomes

ikM n . - - 2

-f F(QHtfXkSp)^ (3.13)

where the contour Γ may still be taken as the real axis from — oo to +00 in the S plane.

The contour is now closed by semi-circles in the lower half-plane as indicated in Fig. 3. Because of the branch point at S = +1 and its associated branch line drawn vertically downward, the closing contour runs from one Riemann sheet to the other in the manner indicated. After making 2 circuits the contour closes on itself. The contours are indented at other branch points in the manner shown for B on the figure. These branch points are located well


below the real axis (i.e. imaginary part of S > 1) and the corresponding branch cut integrations lead to waves which are heavily damped provided

h I 1 - Nf and kp\Nf-l\$>l (3.14)

Now the line integral around the complete circuit in the two-sheeted Riemann surface is equal to — Ani times the sum of the residues of the poles of the integrand*. The poles which occur in pairs are located on both Riemann sheets.

FIG. 3. S plane. Branch points; x, poles.

For highly conducting walls, a number (at least one) is located just below the real axis between the origin and the branch point at +1. The remainder are located along or near the negative imaginary axis. The contribution from these latter poles is very small and they correspond to the waveguide modes beyond "cut-off".

The contribution along the semi-circles is seen to vanish if the radius R approaches infinity. This is assured by the presence of the Hankel function HtfXkSp) which is exponentially decreasing in the lower half-plane of S. Consequently each of the 2 integrations along the real axis is approximately equal to —2ni times the sum of the residues at S = Sn. When the integral is expressed in the original C plane the residue series may be regarded as the contributions from the poles at C = Cn which for n = 0, 1, 2, ... are in the first quadrant and for n = —1, —2, —3, ... are in the third quadrant. This leads tof

Π2 = nkM n

* This follows from the identity F(- C) = -F(C).

00

' Σ n = — oo

1 ] d 1

JdCF(Cj\ H(o2\kSnp) (3.15)

t Actually, Eq. (3.15) and other such expansions are asymptotic. Some of the higher order poles, corresponding to highly attenuated waves, are improper and strictly speaking they should not be included in the summation [Karbowiak, 1959]. In the present case, this fact is of no practical consequence.

where the square-bracket term is the residue of the function F(C) at the pole C = Cn. Carrying out the differentiation and making use of the resonance condition

Rg{Cn)RiCn)^2ikhC^l (3.16) leads readily to

inM + °° n z s — - Σ H<0

2XkSnp)Mz0)Mz)ôa(Q (3.17) Λ w = —oo

where

where

«q-in-^^^T (3.18)

eikCnz i Ώ (Ci ë~ikCnZ m= £$& (3"> and/n(z0) has exactly the same form.

When the walls are perfectly conducting Ri(C) = Rg(C) = 1, the factor <5„(C) becomes unity if n = 1, 2, 3 ... and becomes \ if n = 0, and/„(z) = cos kCnz. The above expression can then be written

inM °° Πζ = — Σ enHi2\kS„p)cos(kCnz0)coS(kCnz) (3.20)

Λ n = 0, l ,2 , . . .

where ε0 = 1, εη = 2(« > 2). The corresponding value of the electric field component Ez can be expressed

3 2 '

μωΐ ds Ah Σ e^iif^^fcS^cosifeQzoicosifcQz) (3.21)

which is in agreement with Eq. (2.4) obtained from physical intuitive reasoning.

The extension to the case when the source is a vertical magnetic dipole is simple. Formally the above results are still valid if / is replaced by the magnetic current K. Ez then becomes Hz and the field is essentially horizontally polarized. The reflection coefficients Rg(C) and Ri(C) are to be replaced by their counterparts for horizontal polarization.

Explicitly these are given by

R*C>~ÏTW, <3 Ώ> and

It may be noted, since R( — C)R(C) = 1 where R is any of the reflection coefficients Rh Rg, R* or Rg9 that for negative values of n

C-i = — C0

C_2 = — Ci

It then follows that the summations £ ί * ... may be replaced by 2 £ £ ··· everywhere since <5„(C) = <5n(—C).

For convenience in numerical computation it is desirable to express the field components as a ratio to the quantity

E0 = ί(ηΙλ)Ι ds(Q-ikp)lp (η s 120π) (3.24)

£0 is the field of the source at a distance p on a perfectly conducting ground. Thus for both the source and the observer near the ground it is not difficult to show by means of Eqs. (1), (17) and (19) that

Ez = WE0 where

Ws-Ιπζcik" f ônS2nH<0

2\kS„p), (3.25) n «=0

(3.26)

where

and

where

Ep = SE0

S^^-^ik"t^S„H\2\kS„p) Jygn n=o

ΗΦ = ΤΕ0/η

Ts-nïei'"'î,ànSnH?KkSnp) n n=o

In the above it has been assumed that \Ng\2 > 1.

(3.27)


When kp > 1, corresponding to the "far-zone", the above expressions may be simplified since the Hankel functions may be replaced by the first term of their asymptotic expansion«! This leads to the compact result

W S T

: (ftM) , ί [ (2*ρ/λ)- (π/4) ]

1 = 0

s* ■S*IN.

S*

- ilnSnplk (3.28)

which is valid for p > A. As expected, the ratio of Wto Tfor a given mode is Sn which for low order or grazing modes is of the order of unity. The ratio of S to T for a given mode is - i/Ng which is very small compared to unity; in fact, it vanishes for a perfectly conducting ground as it must.

3.2 Horizontal Dipole Excitation The previous section contains the formulation for a vertical dipole source.

The corresponding treatment for a horizontal dipole is also quite straightforward although the lack of symmetry increases the complexity. Often in radio wave literature the statement is made that the fields of a horizontal electric dipole are the same as, or proportional to, the fields of a vertical magnetic dipole at the same location. This is only true broadside to the

Z F h - //////

z=o /hsfflsMsM-ws/A

C;

/ / /

€i Mo / / / / / / / / / / / /

H-i

~/f/^///^ar^ww//^///^y//^//sf//sfiJ=///~/i/=///^^/^^/=i'f=///s//r£///j=

Cr

Fio. 4. Rectangular coordinate system for the horizontal dipole between two plane interfaces.

horizontal dipole where the field is purely T.E. (transverse electric) or horizontally polarized. For other directions, the field has a T.M. (transverse magnetic) component corresponding to vertical polarization. These modes corresponding to the T.M. waves may have much smaller attenuation than

t The relevant expansions are

\TTXJ L %ix 2(Six)2

Hii*Kx) ( - ly / 2e - i*e i 3*/4[ l - — —— ...1

W L 8/* 2(8/x)2 - J

+ ; ·■]

the modes of the T.E. type and thus it is desirable to formulate the problem directly with a horizontal dipole source.

As in the previous section the earth and the ionosphere are assumed to be bounded by parallel planes separated by a distance h. Choosing a rectangular coordinate system (x, y, z), the dipole is located at z = z0 and is parallel to the x axis. (See Fig. 4.)

The solution for a horizontal dipole over a homogeneous flat earth with no ionosphere (i.e. h -► oo) was obtained by Sommerfeld many years ago. The generalization for the 2 interfaces is quite straightforward. A Hertz vector is introduced which has both an Λ: component TLX and a z component Πζ. The fields in terms of these are

Ex = k2nx + — —^ + — ^ x dx[dx dz\

y dy[dx + dz\

Hz= - ί ε ω ^ (3.29)

As before, a subscript g or / is added to these quantities when specific reference is made to the ground or the ionosphere, respectively.

The boundary conditions at the interfaces z = 0 and z = h are that tangential components of the fields are continuous. This, in turn, requires that k2Tlx9 dUJdx + dHJdz, ιεωΤίζ and ιεω(3ΤΙχ/3ζ) are each continuous at these interfaces. Integral representations of IT . and Πζ which are suitable for matching, are

— ikR I*

Π, = MQ—— + [C/(C) Q~ikCz + V(C) e+i*Cz]^2)(feSp) dC (3.30) R Jr π--ίί IX(C) e-ikCz + Y(C) e+ikCz]H(

02\kSp) dC (3.31)

for 0 < z < h. Similar expressions are used for the x and z components of the Hertz vector in the spaces z < 0 and z > h. Applying the boundary


conditions involving Hx only, leads directly to the following solutions for the unknown coefficients in Tlx.

= TR) + RhgRÌ

[ 1 - RhgRÌ

{R) + flffi exp[-2ifcCz0]] >hexp(-2ifcCfc) ^(O = i pZJL·-/ „■,„,; e xPl>l f c c(2 f t - *o)] (3.33)

where Rhg and R* are the complex reflections given by Eqs. (22) and (23) and

they are also functions of C. The remaining 2 boundary conditions, namely, the continuity of ΐεωΤ12 and

dUJdz + dUJdx enable the coefficients X(C) and Y(C) to be found in terms of U(C) and V(C). The connecting relations are

and

where

and

x(0 p-QR*™P(-ikch) ( 3 3 4 ) v ' l-RiReexp(-2ikCh) y ' '

PR, exp(-2»fcCfe) - Q exp(-ifcCft) 1 ' l-RiR^xvi-ilkCh) l ' ;

P«[e---+tf+n(^j£) (3-36)

ρ = [e-i k cc- «> + 1/ e -,tc* + V e«*] ft^r) (3.37)

It is understood that Äf, Rg are functions of C and are defined by Eqs. (11) and (12).

The integral for Π^ can be observed to have precisely the same form as the z component of a (magnetic) Hertz vector for a vertical magnetic dipole. This in turn has the same general form as the z component of the (electric) Hertz vector for a vertical electric dipole. The residue series representation for Hx is given by

Π , S ^ Σ H(0

2\kSmP) fm(z0) fhJz) ô"m(C) (3.38) h

where

- I* S*m(C)= l + i , d[J?1(C)R*(C)]/dC-|

Rf\C)Rhe{C)2khc\c=Cm

where the summation is over the poles of the integrand at C = Cm of the integral in Eq. (30). These are solutions of

RhiÇC)Rh

g{C)Q-2mc = t~l2Km (3.39) for integral values of m.

The height functions have the form

2/Ì(z) = [^(Q]-1 / 2 c+ikCz + lRhg{Cy\v> e~ikCz (3.40)

Of particular interest is the vertical magnetic field component. It is given by

Hz= — ιεω -T— = — ιεω —— sin φ dy dp

= sin φ l-^ Σ SmH[2) (kSmP) fhm(z0) fh

m(z) (3.41)

When \kSmp\ > 1 or when ρ ί> λ, the first term of the asymptotic expansion of the Hankel function H[2)(kSmp) may be employed. This leads to

Hz~ _sMEo^^ZS%fl(z0)fl(z)^-s^okQ (3.42) η \nJÄ) L m

The other field components involve integrals which may be treated in the same way. Also of great interest is the vertical electric field. It is not difficult to show that

Ez = ~L pTT ? H°){ks»p) 9niZo) /n(z) s-(c)] (3>43)

where

2gn(z0) = Cn\_Rg{Cn)Y* exp(i/cC„z) - Cn[_Rg{Cn)^ exp(- ikCnz) (3.44)

It may be noted that

kgn(z0)=-i^-fn(z0) (3.45)

dz0

The summation is now over the roots C = Cn of the equation

RiiQRgiQ e~i2"kCh = e"'2™ (3.46)

When p > λ the above expression simplifies to

Ez s cos φ E0 KjLL- — Σ S*m gn(z0) e'«1 -s->" i,(C) (3.47)

When I C Zol <ζ 1 and \kCnz\ < 1, the preceding simplifies even further to Ez s cos φ E0 ± J g i L î j - Ç S? e'**1 "** « O (3.48)

4. PROPERTIES OF THE MODES FOR FLAT EARTH CASE

4.1 Vertical Polarization Much has been written in the literature on the numerical characteristics of

the modes. Controversy concerning the method of numbering the modes has also arisen. It is our opinion that much of this discussion has been unnecessarily involved. The important thing is to sum over all modes which are excited by the dipole. Consequently only these modes need be considered. Because of the form of the integration contour Γ the relevant solutions must satisfy

RJiQRAC) t-2ikhC = e~i2nn (4.1)

and have their real and imaginary parts positive. That is, Cn is located in the first quadrant. The numbering is then assigned in such a way that there is continuity in the limiting case of perfect conductivity (i.e. Rg = R{ = 1).

Cn = TZ w i t h n = 0 ,1 ,2 ,3 , ... kh

As Dr. H. H. Howe points out this is not quite unambiguous when both the ground and the ionosphere are both imperfectly conducting. The more general statement of the rule is [Howe, 1957]:

For a fixed value of kh, determine n on the assumption of perfectly conducting walls, then ag and ai9 in turn are to decrease continuously to their prescribed values while C varies continuously. For walls of high but finite conductivity this means that mode 0 has a minimum attenuation and successively higher attenuation as n increases. For poorly conducting walls, this is not necessarily so, and in fact, in cases of most practical interest for the v.l.f. band the mode of lowest attenuation is of order one.|

Numerical values for C„ are available and will not be quoted here. Some properties of the modes, however, may be simply obtained without resorting to a full numerical solution. For example, if the walls are highly conducting the reflection coefficients may be approximated as follows:

*-(c>=^a'-^H-^c) <42)

„ /rn NtC - Q Λ 2Ci / 2 \ R^=w^rl--m^A-^) (4·3)

tThe mode numbering system described above is somewhat different from Budden. For a fixed value of σ< he starts with a very small value of kh, increases it continuously and requires that C varies continuously for the same n value [Budden, 1957].

subject to | C\2 > \Ng\ " 2 and \Nt\ " 2. Then the resonance equation is simplified to

Δ khC = nn + i-; (4.4)

where

U + K) u Regarding Δ/C as a small quantity, this can be solved to give

«■«[»-©Τ-Μ'-ΟΓ ™ where

ε0 = 1, εη = 2, (n Φ 0). The magnitude of the second term must be small compared to the first

term for the above perturbation method to be valid. This restriction and the previous one are both met if simultaneously

kh\A\ < 1 and

kh \khf

Now for highly conducting walls σβ ρ εβω and at > ε(ω and thus

W.-W .,/.·[(=/♦ (=n Consequently

»ι«['-©Τ+Α-©Τ" *» and

\i-(-)V 2V(2)kft L \kh} J

ImS.£-,_^Lr1_«TK («) The influence of finite conductivity is thus to increase the real part of Sn and consequently the phase velocity c/Re Sn is decreased relative to the free space value c. As expected the finite conductivity produces damping and the resulting attenuation factor is — k Im Sn in nepers per unit distance, to this approximation.

The above approximate formulas for the real and imaginary parts of Sn are the ones usually encountered. They have been quoted by Schumann [1954a] for example. It is not always appreciated that they are not applicable for a

mode which is near cut off. This should be evident, however, from the second inequality given above. To relax this restriction, the resonance equation

khC = nn + i^ (valid for \A\kh < 1) (4.9)

is solved as a quadratic in C to yield

The positive sign before the radical is chosen since it reduces to C = (nn/kh) when Δ approaches zero as it must. The corresponding form for Sn is then given by

H-(snNH#)if <-> When n = 0, this simplifies to

which of course is the same as Eq. (5) when n = 0 and |Δ| < kh. Now since \A\kh < 1 the radical can be expanded for n > 0 to yield

«■"[' -©'- 'ΒΓ for " - 1 · 2 ' 3 · - < 4 · ' 2 ) The preceding discussion concerns walls which are highly conducting.

The approximate solution obtained would indicate that the attenuation increases indefinitely as the conductivity of the walls decreases. Such is true as long as |Δ| <| 1. For very poor conductivities this condition becomes violated. When |Δ| kh is of the order of unity it is apparently necessary to solve the resonance equation by numerical or graphical means. This approach is described briefly in a later section. As it turns out, for a given value of n, the attenuation reaches a maximum value as |Δ| is continuously increased and thereafter diminishes and approaches a broad minimum. To illustrate this interesting phenomenon the resonance equation

RJiQRtiC) Q-2ikhC = e~2nin (4.13) is solved approximately under the condition that

\NiC\<l and \NgC\>l Thus

A,(Ç)s-e"2 N | C / C | and Rg(C) s Q~2C^C (4.14) and therefore

Cg ; + ^β + ikhC = ni(n - i) (4.15) Njc a

The zero-order approximation is obtained by replacing R£C) by — 1 and Rg(C) by + 1 . This would yield

C = C„ = {n - *)π/(ΛΑ), n = 1,2,3,... (4.16)

as mentioned in Section (2). For the first-order perturbation

sV1 i C, = (l - | â ) 2 S I - [N2 - 1 + (C„)2]* (4.17)

/ S2 \* and

since \NgC\ > 1. With these simplifications it readily follows that

< n - j ) + >7(NgCn) fcft - ijv?[jv? - 1 + ( 0 2 Γ ν

and S„ = ( l - C 2 ^

When the upper medium is an ionized region, it is convenient to write

It may be shown that, for v.l.f., L is approximately real and has a magnitude of the order of unity. Furthermore for a highly conducting ground

1/Nf s G1/2 β/π/4

where

Then

c - - fcft _ /(l - //L)[(C„)2 - ί / L ] - * ^ - i y ;

Assuming (C„)2 < L (which is true for low-order modes), and that L is real, the real and imaginary parts of Sn can be written

lmS-'m^m,i(C^L+7i)+^] <4-21)

where = π(η - j) = (n - j)

kh (2fc/A) and

5n = [ l - ( C n ) 2 ] ^ It may be observed that for a fixed value of h/λ and ^/G the attenuation

factor, — k Im 5,,, has a broad minimum when L = 1. For L somewhat less than unity, the attenuation factor varies as L"% or directly as the square root of the effective conductivity of the ionosphere. On the other hand, for L somewhat greater than unity, the attenuation factor varies as L% or inversely as the square root of the effective conductivity.

4.2 Properties of the Modes for Horizontal Polarization When the excitation is by a vertical magnetic dipole or horizontal electric

dipole the modes excited may be of a transverse electric (T.E.) type. The appropriate modal equation is

Rhg(QRÎ(C) e~i2khC = e-i2*(m-1} (4.22)

where m = 1, 2, 3, . . . Now if \C/Ng\ and \C/Nt\ < 1

Rhg(C) s -exp(-2C/JVfCf) (4.23)

and Rhi(C) s -exp(-2C/iViCi) (4.24)

The modal equation is thus simplified to

cfe+^] + iÄfcc=inm (4·25) remembering that

s2\i/2 I sV 2 M1-*?)· c--(isò ' 9'

and S2 = 1 - C2. A first-order solution is obtained by replacing S2 in the expressions for Cf

and Cg by the zero-order value, e.g.

The approximate solution of the mode equation is then given by

Cm s (4.27)

L

where

and Sm = (l- Clf. When

|N?| and | ^ | > l - ( ^ ) 2

it is seen that

For|A?| <kh,

and

Δ!" S 1 = Δ " N,Nt

nmlA . Δ \ ' aX*l1 + ,J*) (4.29)

s.4-OT-<â( i )Mi)T <«■» which is valid when the modulus of the second term is small compared to the first.

For highly conducting walls

ww-v.[(=J"-(=f] and therefore

and T „ |Δ| Γ« (iun\2Y*/nm\2 , , „

In summary, these are valid when

Μ«Η»-(£Γ1 and

nm Th<h

It is rather interesting to note that the above expressions for Re Sm and Im Sm are very similar to the corresponding expressions derived for Re Sn and Im S„ in the case of vertical polarization. [For example compare with Eqs. (7) and (8)]. In the present case, of course, there is no zero-order mode but apart from this, the perturbation term involving |Δ| now has an additional factor (nm/kh)2 which is less than unity if the mode is above "cut-off".

Thus, everything else considered equal, the attenuation factor of the T.E. mode is decreased relative to the T.M. mode in the earth-ionosphere waveguide with walls assumed to be of high conductivity.

In the earlier notation it was convenient to represent the refractive index in the form

Hi-l-L

where L is a real number which may be comparable to or much less than unity. Thus

*-[(5)'-ä'"4 (4.34)

The corresponding solution for the modal equation is obtained from 2-ΛΥΖ

«-[-GÄC)T When L < 1 this reduces to Eq. (30).

(4.35)

5. INFLUENCE OF EARTH CURVATURE

The curvature of the earth has been neglected up to this point. The problem is now formulated in terms of spherical coordinates (r, 0, φ), with the earth idealized as a homogeneous sphere of radius a, of conductivity σβ and dielectric

(r,0,.*)

FIG. 5. Spherical coordinate system for vertical electric dipole between concentric spherical interfaces.

constant eg. The lower edge of the assumed homogeneous ionosphere is located at r = a + h. The source vertical electric dipole is then located at r = a + z0 and the observer is at r — a + z.f In view ofthe intrinsic spherical

t See Fig. 5.

symmetry the fields can be represented in terms of a single scalar function, φ9 as follows [Sommerfeld, 1949]:

Er =

E„ =

Ηφ =

ΕΦ =

' 1 d(- c r ' sin Θ δθ\

i d2

kd9

Hr = He = 0

50/

(5.1) οσ and

η = (μ/ε),/2 and k2 = ω2(ε — ι'σ/ω)μ. As usual, the permeability is taken to be the same as free space for all the regions (μ = 4π χ 10"7). A subscript # is affixed to σ, ε, etc. when reference is made to the ground and a subscript i for the ionosphere. Since φ is a solution of the wave equation appropriate for the regions, the solution may be represented in terms of spherical wave functions,

aPhYXktfPJi-costy for r<a Ibi^Kk^ + bl^Xkr^PX-cose) for (a + h)>r>a

42)^2)(V)^v(-cos0) for r>(a + h) In the above . Vl

h{*U1){kr)=Wr) H^kr) (5'2)

where H^^JJcr) is the Hankel function of the first or second kind of order v + \ with argument kr. Pv( —cos 0) is the hyper-geometric series which is a special case of the hyper-geometric function Ρ(α, β, y, z) namely

Pv(-cos 0) = F(-V, v + 1,1,1 + ™s0j (5.3)

The reason Pv(—cos 0) is employed rather than Pv(+cos 0) is due to the fact that φ must be regular on a ray 0 = π whereas 0 = 0 is to contain the singularity which is the source of the field. Sommerfeld [1949] has pointed out that

i· „ / * < l + cos0\ sinv7r ~ hm F[ - v, v + 1, 1, -* log 02 (5.4)

which illustrates the singular nature of φ along the polar axis. The quantity v is to be found from the boundary conditions that the fields

Ee and Ηφ are continuous at r = a and a + h. This, in turn, requires that (η/r) 3(ΓΦ)/ΘΓ and Ιϊφ are continuous. Thus, 4 linear equations in the coefficients a(J}, b(l\ b(l\ and c{2) are obtained. In order that these yield a non-trivial solution the four by four determinant of the coefficients should vanish.

This

requ

irem

ent i

s exp

licitl

y gi

ven

by E

q. (5

.5) a

s fo

llow

s:

ing\l

drhi

l Xkgr

n

k gh^

\kga

)

0 0

h\\d

rh[l Kk

r)l

\a)[

èr

\ a

-kh^

Xka)

/ η

XÎdr

hïXk

rK

\a +

h)[

dr

\ a+

h

fc/j<

l) [k(a

+ ft

)]

/η\ t

drhï

Kkr)

-]

(à)E

Or

-khi

2 \ka)

drh^

Xkr)

Ò

r

khi2 Xk

(a +

h)-

] a +

h \a

+ h

)

0

L dr

J,

k^XH

a +

h)}

Stat

emen

t of

Eq.

(5.5

) for

the

Det

erm

inat

ion

of th

e Ei

gen-

Val

ues

v.

Such an equation as this was obtained by Watson in 1919. To solve it for v without approximation does not seem to be possible, although if the general spherical Hankel functions of complex order and argument could be programmed for a computer, an exact numerical solution might be obtained. In view of the idealizations of the model and the uncertainty of the effective electrical constants of the lower ionosphere, however, it does not seem warranted to expend too much effort in this direction. As is so often desirable in physical problems, asymptotic approximations to the rigorous wave functions are introduced which greatly simplify the problem but at the same time lose some generality.

The Debye-Watson representation of the Hankel functions are [Sommerfeld, 1949]:

""Ί1!!1-^"*] <5'6) when |(v + i)/kr\ < 1 but not near 1. Also |v + | | and kr must be large compared to unity. The upper (and lower) signs are to be considered together. This is really a WKB approximation to the radial part of the wave equation. It is not difficult to show that the resonance equation involving spherical Hankel functions can now be expressed in the equivalent form

/ Çk(a + h) Γ (v . l \2- i i/2 x RgRi exp - ill 1 - v {' dx = exp( - ilnn) (5.7)

where n = 0, 1, 2, ... .

t h _ (v + *)2 V2 _ J i _ fr + i)2 V2

' ut* ("+j)2 V2, ,r, (y+i)2!'7' p

^L1 k2(a + h)2\ +T kf(a + h)2\ The functions Rg and Rt quoted above can readily be identified as Fresnel

reflection coefficients for complex angles of incidence cos_1C and cos^C, respectively. Furthermore

C = [ l - S 2 ] , / 2 where S = Ì ^ Ì ka

and

and

C = [1 - (S')2]1/2 where S' = V + * k(a + h)

The resonance equation can thus be written f

where

and

RjLQRtCyexpl-ak Γ i c 2 + — 521 * dzj = e-2™ (5.10)

NgC - Cg ... ^ [\ S2VA

NeC+Cg W = =£ r? with ^ = [ 1 " ^ ] / 2 (5'n)

iVfC - c; r (s')2lVi

«n-Sëri ™ith ^-['-Ί^] < 5 · 1 2 )

It can be seen that, in view of the relation

(a + h)S' = aS

the resonance equation reduces to its flat earth counterpart as hja tends to zero. In fact, it appears that, if \C\ > (h/a)V2 = ^ , the effect of curvature can be disregarded. This condition is violated for most of the numerical results given by Al'pert in the region from 15 to 30 kc. He assumed that the modes could be calculated on the basis of a flat earth in all his work [1955, 1956].

The resonance equation quoted above for a curved earth is only valid if the WKB or second-order approximations to the spherical wave functions are valid. In a later section the corresponding form of the mode equation based on the Airy or third-order approximation is developed, following the work of Rydbeck. It is indicated from this more involved analysis that the second-order approximation is valid if (fca/2)1/3C > 1. As will be seen, this is met for most cases of practical interest if the frequency is less than about 15 kc.

6. MODE SERIES FOR A CURVED EARTH

Following the suggestion of Sommerfeld, the field is written as a sum of modes. Thus

« r , Ö) = Σ D A C k i ^ - c o s 0) (6.1)

t Note that 5 2

~ C2 H— S2\ bebause - < 1 ; also solutions of equation

4

2zt

(5.10) exhibit the property Cn = — C-(n + i> for n = 0,1, 2, 3 . .

for a < r < a + A, where

Zv(kr) = b\l)h[l\kr) + b{2)hS2\kr) (6.2)

The factor Dy is to be determined by insisting that the function \j/(r9 Θ) has the proper behavior at the source. The summation is over all integral values of n and the corresponding (complex) values of v are obtained from the resonance Eq. (5.5) as described in the previous section.

Invoking the WKB or second-order approximation for the spherical wave functions, it follows that

zy(kr) s const x \R-\C») expi + i/c i V c 2 + ~ S*\* dzl

+ R*{Cn) exp[-ifc j'Jci + ~ SljV2 dzj j (6.3)

This can be identified immediately as a combination of a downcoming and an upgoing wave. The ratio of these 2 at the earth's surface (z = 0) is Rg(Cn). An alternate representation is

zv(kr) s const x inf \Cn) exp Γ + ifc f (c2n + ^ S2) ^ dzj

+ mcn) exp[-ifc [(c2n + | S2)jA dz]J (6.4)

which is a combination of an upgoing wave and a downcoming wave. The ratio of these 2 at the lower edge of the ionosphere (z = h) is Ri(C'n). It should be noted that

c'^fo + stf since (a + h)Sn = a*Sn and h/a < 1. The internal consistency of these 2 representations at z = 0 and h for zv(&r) can be readily demonstrated from the relation

1 = Rlf(Cn)RHC'n) exp[-ifc He2 + | Sî)% dz] (6.5)

which also indicates that the multiplicative constant is the same for the 2 representations. In what follows the constant can be absorbed into the factor

To study the orthogonality properties of the modes, the following integral is considered

= Jka

zv(p)z„(p)dp (6.6)


where v and μ are 2 sets of modes. Now quite generally the function zv(p) satisfies

P ^ 2 0*zv) + [P2 - v(v + l)]zv = 0 (6.7)

and there is a similar relation for ζμ. These 2 equations are now multiplied by ζμ and zv respectively, and integrated over the domain ka to kb, to obtain

1 = Ρ ^ Η ρ ί Ρ ^ - Ζ ν ^ ί Ρ Ζ μ ) ]

d_

' d p '

ito

v(v + 1) - μ(μ + 1) (6.8)

For the important modes, the right hand side of Eq. (8) is negligibly small if μ Φ v since the numerator vanishes at the limits ka and kb when Rg and R{ approach ± l.f For the important case for μ = v, a normalization factor is defined by

Çkb Ny = lim ζμ\

μ-*ν Jka

Ί c

Jka

(p)zv(p) dp

= 1 W Ä r * . f , «here ^ , - „.' (6.9) 1 +

2khCn

It should be remarked, at this point, that the modes are not strictly orthogonal since the right hand side of Eq. (8) does not vanish identically although it is small compared to Νν. As the conductivity of the bounding walls approaches infinity the modes would be completely orthogonal.

Multiplying both sides of Eq. (1) by zv(p) and then integrating with respect to p from ka to kb, leads to the following formula for Dv:

-coSf>).LZ'(

To actually evaluate Dv, it is desirable to let r -► r0 and Θ -> 0, in which case \l/(r, Θ) -+ il/0(r, Θ) where ψ0 is the primary influence which is singular at (r0, 0). For a vertical electric dipole consisting of an infinitesimal element of length d^ and carrying a current, /, it is well known that

Ids e~ifcK

where R = (rl + r2 - 2rr0 cos 0)1/a

t It is also of interest to note that the numerator of Eq. (6.8) vanishes identically if the fields satisfy impedance-type boundary conditions. In this case (llzv)(dpzvldp) is assumed to be independent of v. (Compare with Section 10 of this chapter.)

Following the process suggested by Sommerfeld for the determination of the Green's function for the perfectly conducting sphere the integration in Eq. (10) is carried out in the immediate neighborhood of the source. For example, r = r0(l + ή) where -ε < η < +ε, ε < 1 and dr = r0 άη, zv(kr) s zv(kr0), e~ikR s 1 while

J « i [a+#+1 - 2 a + 4 - £)] "K » j p i ^ («e Therefore

"■»■"^S brâ>C(?rM (613) Now

0->O " ~ 71

^~ , Λν-, sin νπ # A-hm [Pv(-cos 0)] -4 log 02

and

l i m SFTTFiS = I i m 0og[e + V(*2 + ö2)] e-»o J - e w + 0 ;/2 Θ-+Ο

- l0g[ -8 + V(fi2 + 02)]} ~» lQg 0 2 (6·14) It then follows that

1 z ^ ) / d _ S v 2fcft sin νπ 4r0 " v

The final form of the function ψ is thus given by

K r . i D * ^ f Z # r ° ) Z - ( f e r ) P - ( " C O S g ) ^ (6.16) 2khr0 „=ΟΛ,2,·.. zv(ka) zv(ka) sin νπ

where the second-order or WKB representations may be used for radial functions zv(kr)9 etc. As can be seen from Eq. (4) these can be greatly simplified if z/a < |C2|, for then

zv(kr) ^ eikCnZ + Rg(Cn) Q~ikCnZ

zv(ka) = 2 C Ä / 0 ] * =/„(*) (6.17)

which is the same height-gain function obtained for the flat earth case. For heights even as great as 10 km and frequencies less than 20 kc, this is an excellent approximation. Similarly,

M"M (6i8) where z0 = r0 — a is the height of the source dipole.

The radial field component is of most practical interest and, for the moment, attention will be confined to it. Since

i I d i d\l/\ Er^-η -r-r ^ sin ^ (6.19)

r r ' s ino M de) v ' and

it follows that I as n °° v(v 4- 1Λ

Er S ^ τ - Σ Κ U*o) Λ(ζ) - V ^ P v ( - c o s 0) (6.21) 2khr0r π=0 sin νπ

with ν 4- £ £ fcaSn. This is the final solution of the problem being valid for the air space between the earth and the ionosphere.

For purposes of computation several simplifications can be made. The asymptotic expansion for the Legendre function, given by

Pv(-cos 0) s ( - 4 - τ ) 1 / 2 COSf(v + *Χπ - 0) - £1 (6.22) \πν sin v/ [_ 4J

is valid if |v| > 1 and Θ not near 0 or π. Since the imaginary part of ν(π — Θ) is also large for π — Θ greater than about 10 or 20 degrees, it follows that

\2πν sin 0/ P v ( - c o s 0 ) £ — exp[i(v + i X « - e ) - i W / 4 ] (6.23)

\2πν sin 0/

Furthermore, the source and observer heights are usually sufficiently low that kh0Cn and kh^ <ζ 1 and r0 ^ r ^ a.

The simplified form of the field can now be written

Er = E0(J^-Y2 (4r£reiC2*<d/A>-^4>] Y SmS* e^2*5«*™ (6.24) \sin d\aj (η/λ) „=0

where d = <z0, the arc length between the source and the observer, λ is the free-space wavelength and Sn = (1 — C2)ya. E0 is the field of the source at a distance d on a flat perfectly-conducting earth. For d/λ i> 1,

E0 = i(i//A)/ d5(e-i2Äd/A)/ii (6.25)

As the radius, a, of the earth tends to infinity it is immediately evident that the flat earth formula given by Eq. (2.5) is recovered.

7. ANTI-PODAL EFFECTS

The general form for the field in the space a<r<a + h for a, vertical dipole source has the form

Jdsq ^ v ( v + l ) p , m , 7 ,x E* = ^ΓΓϊ Σ δη — Pv(-cos 0) (7.1)

2kha „=o sin νπ where δ0 = J, <>„ = l(n # 0), and

v(v + 1) s (v + i ) 2 s fcaS2 (7.2)

Now as mentioned above when 0 is not near 0 or π, the Legendre function may be replaced by the first term of its asymptotic expansion. This result quoted above is valid if

π « ( π - 0 ) and ΓΊ<Θ (7.3)

In this region, the modes are simply proportional to

(^wHkas"(n-e)-ï] ( 7 · 4 )

which apart from a constant factor can be identified as the linear combination of 2 peripheral waves of the form

* Q-ikaSnQ (sin 0),/2

and 1

Q-ikaSn(2n-e) Qin/2 (sin θ)ν>

where 0 < π. These waves are travelling in opposing directions along the 2 respective

great-circle paths αθ and α(2π — 0) from the source to the observer. It is noticed that there is a π/2 phase advance which the wave travelling on the long great-circle path picked up as it went through the pole 0 = π. The linear combination of these 2 travelling waves is to form a standing wave pattern whose distance Δ/Μ between minima is approximately given by

kAm Re Sn = π or Am = λ/(2 Re S„) subject to

Im Sn <ζ Re Sn

As one approaches the pole 0 = π, the first term in the asymptotic expansion for the Legendre function is inadequate. A more general form is the asymptotic series [Magnus and Oberhettinger, 1949].


P^^JLCiii^ me ι·">Γ(ν + ί ) ΐ ( ν + ί),/!

cosi X (2sin0)I+1/2

where (a), = a(a + l)(a + 2), ... (a + i - 1)

for example, a0 = 1, a! = a, a2 = a(a + 1), a3 = a(a + l)(a + 2), etc.

Since |v| > 1 the factorial functions may be replaced by the first 2 terms of their asymptotic expansions ; this leads to

Γ(ν + 1) ÀK-G)) The preceding asymptotic expansion for Pv(—cos 0) is not usable at and in

the vicinity of the pole 0 = π. In this region a suitable representation is given by [Magnus and Oberhettinger, 1949]:

Pv(-cos0) = J0(n) + s i n 2 ( ^ ) [ ^ - J2{fÙ + \ J3(»i)]

+ terms of order sin41 —— 1 (7.7)

where η = (2v + 1)8Ϊη[(π - 0)/2]. JJsi)9 for m = 0, 1, 2 and 3, is the Bessel function of first type of argument η and order m. When π — Θ is small the first term is usually sufficient and furthermore

η s (v + £>(π ~ Θ) s kaSn(n - Θ)

Thus for mode n, the field in the neighborhood of the pole is proportional to the Bessel function

J0ikaSn(n - 0)]

It is then not surprising to see that the first term of the asymptotic expansion of J0 is the same as that of Pv(-cos 0).

8. RESONATOR-TYPE OSCILLATIONS BETWEEN EARTH AND THE IONOSPHERE

At extremely low frequencies (e.l.f .), where the wavelength is large compared to the height of the ionospheric reflecting layer, the electric field is essentially

radial and only one waveguide-type mode is significant. The field is thus expressed by the first term of the mode series which reads

/ d ^ H v + i) Akhar βιηνπ

where v + \ ^ kaS0 and

<?. ~ 1 _ 2kh « . » ' - ΐ ί ι ^ + ϊζ) <Η>

in terms of the relative refractive indices Nt and Ng of the homogeneous ionosphere and the homogeneous ground, respectively. Now at e.l.f. \Ng\ > \N\ and furthermore,

„„(ϋϋϋΙ',Ιί)'' \ icoe / \ιωε/

Thus fA

2(a^co)l/2h

As mentioned in Section (7), the factor Pv(—cos Θ) may be replaced by an asymptotic expansion if kaO or ka(n — Θ) is somewhat greater than unity. The field in this case may be regarded as 2 azimuthal-type travelling waves. Furthermore, at the pole (0 near π) where the second of these restrictions is violated, it is possible to use an equivalent representation which correctly accounts for the axial focussing. An alternate viewpoint which is suitable at e.l.f. is to consider the field as a superposition of cavity-resonator type modes. It is expected that such a representation would be very good when the circumference of the earth is becoming comparable to the wavelength. A suggestion of this kind was apparently first put forth by Schumann [1957].

The starting point is the expansion formula

PX~X) 1ΥΡ(χί 2η + 1 (S3) sin νπ ~ π „ΐΌ η(η + 1) - v(v + 1) ^ }

where the summation is over integral values of n. This result follows directly from a formula given by Magnus and Oberhettinger [1949 (p. 57)] which is valid for v # 0, ±1, ±2, ... and 0 < θ < π. The electric field, for h/a < 1, is thus written

Jdsv(v + 1 ) " 2n + l r~ Anahœh n% nW n(» + 1) - v(v + 1) K*A)


where x = cos Θ. The early terms of the series are then proportional to

Po(*) = l Pt(x) = cos 0

P2(x) = i(3 cos2 Θ - 1) (8.5)

and so on. The configuration of the electric field in the first three cavity modes is depicted in Fig. 6.

FIG. 6. Depicting electric field lines in first three cavity resonator modes.

Retaining just the first term it is seen that

Ids 1 Er = £r]rt = 0 !

Ana2eh ico (8.6)

which is independent of Θ. Clearly this corresponds to a concentric spherical capacitor energized by a current Ids/h resulting in a constant voltage hE% between the plates. On rewriting Eq. (6) in the form

fc£? =

it is seen that

Ce =

(/ ds/ft) icoCe

4πα2ε

(8.7)

which can be identified as the capacity between the spherical surfaces whose areas are both Ana2 within the approximation h/a <ζ 1.

It has often been suggested that the omni-present constant voltage gradient in the atmosphere results from the accumulated action of lightning strokes which impart a charge to this earth-ionosphere condenser. For example, when a current surge flows for 10"3 sec with an average strength of IO3 A with an average column height of 3 km (i.e. ds = 3 x 103), and for h £ 70 km, it readily follows that

hE°r s 1.3 V or £r° s 2 x 10"5 V/m

Presumably, many such charges are required to build the field up to its observed value.

Of somewhat more interest are the cavity-resonator oscillations which may be excited. Using the notation of the operational calculus, iœ is formally replaced by p then

v(v + l ) s - j p 2 - p % a (8.8)

where a = l/[h(a^)l/2]. The source dipole moment / ds is in general a function of time. For purposes of illustration, consider

/ ds = (/ ds)ow(0 (8.9)

where u(t) is the unit step function at t = 0. The Laplace transform of the source moment is given by

J: IdsQ-ptdt = ^k ( 8 e l 0 )

p

The Laplace transform of the field is given by

where ω2 = (a/c)2n(n + 1). The actual time response of the electric field is denoted er{t) and is zero for t < 0. It is related to Er(p) by

f*00

UP) = er\ (t)Q"ptdt (8.12)

The inversion of this integral equation is a standard problem in operational calculus and has been carried out explicitly by Schumann [1957] for a transform which has the form of Eq. (11). In the present discussion a much simpler approach is used which is justified when the damping is small. It should be noted that ocp1/2 has already been assumed small compared top, thus a perturbation method is in order.

^(p) = ^Pn(x)^^ï (8.13)

4"\t) = W 2 Ρ»(*Χ2" + Dc o s <°J (8 ·1 4)

When a = 0 corresponding to no dissipation (i.e. perfectly conducting boundaries)

(Jd*)o„, v(2« + l)p

The poles in the /? plane are thus at p = ±icon. The inversion to the time domain gives

qds)0

h2Ce

which may be verified by noting that the above expressions for E^(p) and e{;\t) satisfy Eq. (12).

A step-function dipole source thus excites the static field (i.e. co0 = 0) and the cavity-resonator modes (n = 1, 2, 3 ...). For a = 6400 km

(ojln = 10.6 c/s ω2/2π = 18.3 c/s ω3/2π = 25.9 c/s

To account for finite conductivity, it is necessary to solve the equation p2 + pK(x + co2

n = Q (8.15) which gives the poles for the function Er(p) in the case when α Φ 0. Remembering that apVi » - "» (8-16) where

is the resonant frequency and

Ω„ = αω!/2

23/2

is the damping coefficient. It then easily follows that cos ωηί is to be replaced y e~"ni coseni

To this approximation, the effect of finite conductivity is to exponentially damp the oscillations with time. For a = 6400 km, h ~ 100 km, at ~ 10"4

mho/m, the time constant is given by

K= Iv A i v i s e c (Λ = 1 . 2 , 3 , . . . ) (8.17) Qn] j[n(n + 1)]

which is rather interesting.

The total field is then given by

(I as) °° er(t) s ^-ζ-μ Σ P„(x)(2n + 1) e"»»' cos <o'nt (8.18)

n W « = 0

which is valid for a2t > 1 or t > l/(h2a^).

9. EXCITATION BY HORIZONTAL DIPOLES FOR THE CURVED EARTH

The formulation of the theory for a horizontal dipole is similar to that for a vertical dipole. The complexity of the equations is greater, however, because of the non-symmetry of the problem. Schumann [1954b] uses this approach in his analysis but his results are not complete as discussed below. The deficiency arises when the eigen-function series is matched to the source singularity. In the case of the vertical dipole as outlined in the previous sections, this process is relatively straightforward but in the case of the horizontal dipole there is coupling between T.E. and T.M. modes which apparently is not accounted for using this technique. An alternative is to set up the problem in terms of a harmonic series representation wherein the summation is over integral values of n, the index of the spherical wave functions. This series is poorly convergent, however, and the Watson technique must be used to transform it to a series of residues of the complex poles v. Such a procedure was used by Wait [1956] for a horizontal dipole over an earth with a homogeneous atmosphere. It would not be difficult to generalize these results to include the influence of the ionospheric reflecting layer. In the present work, however, it seems more instructive to use a different method which makes use of the reciprocity theorem and the results for vertical electric and vertical magnetic dipoles.

For the first part of the problem, a vertical magnetic dipole of moment K às is considered. It is located at r = rQ on the polar axis. Due again to the intrinsic symmetry of the problem, the fields can be obtained from a single scalar function \j/h as follows:

__ i l ô / . # * \ Hr = : — ( Sin 0 -rrr

r νη sin θδθ\ δθ )

and Ηφ = Er = Ηθ = 0. Such fields are purely of the T.E. type whereas they were of the T.M. type for a vertical electric dipole source.


The solution proceeds in the same manner as for the vertical electric dipole. Now, however, the boundary conditions are that (Ι/η) d(nl/h)/dr and k\l/h are continuous at the concentric spherical interfaces. In a form suitable for application to v.l.f. propagation, the final result is given byf

# . _ £ * £ « f a j ^ - c o s » » (9.2,

where 2/c/ir0m=1^2>3... sinvn

ôm ~ sin 2khCm ( 9 ' 3 )

2khCm

2fm(z) = [^(Cm)]-^exp[ifeCmz] + [^(Cm)]^exp[-ifcCmz] (9.4)

and similarly for/^(z0). The modal equation has the form

where

and

Kj(C)li?(C0exp[-î2fc ί ( c 2 + - S 2 ) / 2 d z l =e"2 n i m (9.5)

|V ' C + Nfii I Nf \

The electric field component Εφ is thus given by

E = ^ ds fhm(z0)fh

m(z) aPv-(-cos fl) * 2ftr0 h m sin νπ 30 l ' '

Now when the source is a small loop of area da carrying an average circulating current / it follows that

K ds = ίμωΐ da (9.7)

Furthermore if the receiving antenna is a horizontal electric antenna of effective length d/, the voltage at the terminals is given by

v = Εφ dl sin φ (9.8)

where φ is the angle subtended by the receiving antenna and the arc joining

t To conform with standard waveguide practice, the T.E. mode of lowest attenuation is denoted m = 1.

the 2 antennas.j Thus the mutual impedance Zm between the source loop of area άα and the horizontal receiving antenna of length al is

v ίμω da d/ sin φ ^ Ζ"—Γ 2 / ^ i ' " (9*9)

where the summand is the same as in Eq. (6). /

/ source M a n view) / dipoles ^ V / Λ φ -®- A ^

FIG. 7. Source vertical magnetic and electric dipoles and receiving (horizontal) electric dipole for mutual impedance calculation.

The mutual impedance Ze between a vertical electric dipole source at 0 = 0, r = r0 and the horizontal receiving antenna is also required. This may be obtained from the scalar function φ previously obtained. In particular

Ε°=-;η&^ψ)^-ίη-&ζφ ( 9 · 1 0 )

IdsΣ fj^gMd p ( _ c o s ö ) ( 9 1 1 ) 2hr0j? smvn δθ v v '

where

2<7„(z) = [C2 + ^ S 2 ] * a

x \ R ; a exP Γ + i/c fZ i c 2 + ^ s2„)A dzl

- Ä*exp[- i/c Γ (c 2 + ^ S2j ] <fc} (9.12)

When zia < \S2n\

2g„(z) S C„[R;1/2 e**7" - Ä» e"'*^] (9.13)

Furthermore if \kC„z\ < 1 which is the usual case g„(z)*Anfn(z) (9.14)

where ?2 \ '4

t See Fig. 7.


The mutual impedance Ze between the vertical electric dipole and the horizontal receiving antenna d/ is thus given by

η ds dl cos φ ^ Ze^ 2hr0

(9.15)

where the summand is the same as in Eq. (11). It is now a simple matter to write down the field expressions when the

source is a horizontal electric antenna carrying a current / of length dl. The antenna or dipole now is considered to be located at r = r0 and 0 = 0 and oriented in the direction φ = 0. The vertical magnetic field at (r, 0, φ) is obtained from the relation

i>a>tfrda = /Zm (9.16) which relates the total magnetic flux in a small loop of area da at (r, 0, φ) and the vertical magnetic field at the same point. Using Eq. (9) it is seen that

Idlrskf&z0)f%z)dPA-cose)_. sin νπ 30

(9.17)

In a similar fashion the vertical electric field at (r, 0, φ) is obtained from the r d a t i 0 n Erds = IZe (9.18) which relates the voltage in the small vertical antenna of length d$ at (r, 0, φ) and the vertical electric field at the same point. Thus

i d / ^ z ^ X z o ) Ô 2hr „ sin νπ οθ

(Γ,ο,ψ)

FIG. 8. Spherical coordinate system for horizontal electric dipole between concentric spherical interfaces.

The other field components can be found from the above expressions for Er and Hr. Quite generally the field components in spherical coordinatesf can

t See Fig. 8.

be written in terms of a set of purely T.E. and T.M. modes derivable from scalar functions U and V. Thus

- r » · « " ) or

+ k2ru\ (9.20)

£ e = i J L ( r t / ) - ^ - ^ (9.21) β rdQdr ' rsinfl Οφ Κ '

ψ r sin 0 δφΟΓ r δθ

Hr-^jP + khV (9.23)

Β.-~ίΠΤ) + - ^ ^ ϋ ) (9.24) r drôO r sin θ δφ

JL_JL(rF)_^A, r sin θδΓδφκ } r 30

ΗΦ = Γ3Γ7Ϊ ΤΞΙ ( ^ ) ~—Τλ (rU>> (925)

Since £/ and V satisfy the equations

(V2 + k2fv = 0 (9.26)

in the space a < r < a 4- h they must be made up of solutions of the form

h[l\kr) cos q<$> P?(cos 0) (9.27)

h(v2\kr) sin q$

where q is an integer. Since the field for Er and Hr has already been prescribed, q = 1. With some consideration it is seen that

υ = ΣΑηΑ(Ζ)δ-^^οο5φ (9.28)

and

F = Z ^ / ^ ) g P v ( " a ; O S Ö ) s i n ^ (9.29)

Further, on noting

•1 ir/i<v1,2)(fcr)l = 0 (9.30) ,, + £_.*+l>l

L dr2 r2

it is seen that

Er = 1±iAAnUz)sJA^lC0^ (9.31)

and v(v + l) h dP,(- cos Θ)

Hr = Σ 1 Bm Jm(Z) ™ Sill φ (9.32) m T OU

On comparing Eqs. (30) and (31) with (17) and (19), it follows that

A» = j!dlJn9n(Z0) (9-33) 2hv(y + 1) sin νπ v J

and B= Idl * # « « > ( 9 3 4 )

" 2Λν(ν+1) sin m y }

When |v| > 1 and Θ is not near 0 or π the following asymptotic expansion is valid:

1 d 2 e"in/4 /a\i/2

P v (- cos Θ) s , . m y T S* e"**5" (9.35) sin νπ 50 (sin 0)1/2 \λ, where use has been made of the relation

v £ fcaSrt

The height function g(z0) occurring in the expression for Er can be simplified at low heights. For example, if |fcCnz0| < 1 which is the usual case

Thus, the vertical electric field of a horizontal dipole is well approximated by

Er ^ E0 ^ \J^-\ 1/2 V ^ " ^ £ ÔSy2 e-i2nSn(äI» (9β37) iV Lsinrf/aJ (/z/2) „t^

where E0 = i(i|/A)J d5(e-i27td/A)/d. (9.38)

It is of interest to compare this with Eq. (6.24) for the vertical electric field of a vertical dipole with the same moment. It is seen for a given mode

£j/° (for horizontal dipole) cos φ I S2n\1/2

E^n) (for vertical dipole) c o s a / Si\/2

Since Sn ^ 1, it is seen that the ratio does not depend critically on mode number n, thus

Er (for horizontal dipole) ^ cos φ I 1 \1/z

Er (for vertical dipole) S Ng \ Nj / ( " }

In most cases |Λ | > 1 so the ratio is of the order of l/Ng which is small. In particular, at very low frequencies

cos φ ^ /εω\1/2

e'*74 cos φ (9.41)

which indicates that the ratio varies as the square root of the frequency. This is in disagreement with Schumann (1954b) who finds that dependence is the inverse first power of frequency. In the direction φ = 0 of course the ratio derived here turns out to be nothing more than the "wave-tilt" for a vertically polarized plane wave at grazing incidence on a flat earth. Thus Schumann's results for the horizontal dipole would seem to be in error.

The horizontal dipole also, of course, radiates horizontal polarization. The simplified expression for Hr can be written by employing the single term asymptotic representation described above. Thus

Ä E0 sin φ(άΙλ)* Γ d/a 1 Δ ί[(2π<ί/;)-(π/4)]

Γ~~ η (Νλ) [sin d/a]

x Σ «ί. e-ikaes~S%fl(z0)fhm(z) (9.42)

10. HIGHER APPROXIMATIONS TO THE CURVED EARTH THEORY

In the previous sections the mode series for a concentric spherical earth-air-ionosphere system was developed. In order to simplify the discussion and lead to results suitable for immediate use, rather crude approximations were introduced. In this section the problem is reformulated in a more rigorous fashion and higher order approximations for the various spherical wave functions are introduced. This analysis is really an extension of the work of Watson, Rydbeck and Bremmer. The final results indicate the range of validity of the lower order approximations used in the earlier sections. The formulas are in a form which is suitable for numerical computation.

The earth is represented by a homogeneous sphere of radius a and is surrounded by a concentric homogeneous sharply-bounded ionosphere of radius c. The source is a vertical electric dipole of strength / as and is located at r = b. The electrical constants of the air space are denoted ε and μ and subscripts g and / are added to these when reference is made to the ground and the ionosphere, respectively. (See Fig. 5 for example.)

The fields can be expressed in terms of a Hertz vector which has only a radial component U, and thus, for the region a < r < a + h,

*r=(k2+î?Yrv) H'=° Ee = -£rArU) roroO

Εφ = 0

tf» = o

• dU

(ΙΟ.ί)

where k = (εμ)1/2ω.| A subscript g and i are also added to the field quantities when reference is made to the regions r < a and r > c, respectively. Furthermore, kg = (εθμβ)ί/2ω and k{ = (ε,μ^ω are the respective wave numbers for these 2 regions.

The Hertz functions satisfy the inhomogeneous wave equation

<y> + e)u = c^-2my (10.2)

for a < r < a + h, where the <5's are unit impulse functions. The factor 2nr2 sin Θ is the Jacobian of the transformation from rectangular to spherical coordinates. The constant C is to be chosen so that C/has the proper singularity at the dipole, that is

ç-ikR -bU->- -Ids forR->0

Απιω&κ where R = [r2 + b2 - 2br cos 0]i/2, and therefore C = (//ωε)/ as.

The field in the region a < r < a Λ- h is now written as the sum of the 2 parts Ue + Us9 where Ue has the proper dipole singularity at R = 0, and i/s is finite at the point. As Us is a solution of the homogeneous wave equation, it can be written in the form

ikC °° tfs = I - Σ (2« + l)D4e/42)(fcr) + ß4;e(/cr)]P4(cos 0) (10.3)

where ye(A:r) and h(q

2)(kr) are spherical Hankel functions of the first and fourth kind, respectively, and Pq(cos Θ) is the Legendre function. The summation is over positive integral values of q. The corresponding expression for Ue is given by

4π etO * ΙΛ« (fcr)/fl(fc&); for r > 6 Since there are no singularities other than the source dipole, the Hertz functions Ug and L are solutions of the homogeneous wave equations

(V2 + k2g)(Ug) = 0 for 0 < r < a (10.5) and

(V2 + kf)(Ui) = 0 for r ^ c (10.6)

Noting that Ug is to be finite at r = 0, the solution must be of the form

ikC °° U. = ~r Σ (2« + DJVcos 0)<y4(V) (10.7)

t The function U = — /τ;0 in terms of scalar function ψ used previously for the potential.

where aq is a coefficient which is independent of r and 0. Furthermore, since Ui is to give rise to an outgoing wave at r = oo, the solution is of the form

ikC °° Ui = — Σ (24 + l)^(cos eWJifXk?) (10.8)

471 q = o

where 6e is a coefficient. The 4 unknown coefficients Aq, Bq, aq and £€ can be found from the boun

dary conditions at r = a and c. These require the continuity of the tangential field components. In order to facilitate the solution and to readily permit later generalizations, the 4 boundary conditions as stated above can be replaced by 2 impedance type boundary conditions. For the #'th terms of the series these read

E(eq) =-Z(gq)H^q) at r = a (10.9)

and

where E(

eq) = Z[q)H^ at r = c (10.10)

d

* > . ' d r WAr)l ieco rjq(kgr)

and

ZP = - 7 1 dr i[rt?W)]

ίεω rh^Xkif)

Replacing kr by x, Eqs. (9) and (10) may be rewritten

1 s)

- — (xU) = i(Zgq)ln)U for x = ka (10.11)

and i /) - — (xU) = - i(Z\q)h)V for x = fcc (10.12)

Applying these to Eq. (3) enable Aq and Bq to be obtained explicitly in terms of known quantities. Using these results the following exact solution for a ίζ r < è i s obtained:

ikC °° F tf = -7 - Σ (2β + l ^ W ^ f c r W c o s 0) -S (10.13)

47Γ q = 0 L/q

where

r, „.L^ffiffi ( 1 0 15)

(4) = _ ln'Cfcaft^fcq)] - f Z < ^ 9 ln'[/ca/i<2)(fca)] - iZ<«>/f/ (10.16)

ln'[/cc/42>(fcc)] + iZJ*/i, ' l n ' E f c c ^ ^ c ) ] + iZfO/rç U '

The symbol In' denotes logarithmic differentiation, for example

ln'CfeefcJ^fce)] = dx - lxh?\x)]

χΗγΧχ) (10.18)

\x = ka

The above result, although rigorous, is not of practical value for v.l.f. propagation calculations because of poor convergence of the series solutions. In fact, something of the order of 2ka terms are required to achieve 5 per cent accuracy. At 15 kc, for example, 2ka =" 2000 which is rather a large number. An important observation, however, is that terms of order q beyond 2ka contribute little to the series. Thus the spherical Bessel functions jq(kga) may be replaced by the Debye or second-order approximation since \kga\ > q in the important range of q so long as \kg\ ρ k (i.e. well-conducting ground). Thus

ln'lkgajq(kea)-] s / [ l - ^ ] ' (10.19)

Similarly, for \k(\ > k

lnTAaW«)] = - i [l - f j j 2 (10.20) Since the total field is of the form

U = fj(2q + l)f(q)Pq(cose) (10.21) q = 0

it can be rewritten as a contour integral over q which has poles when q takes integral values. Such a representation is

JCi + q Aq f(q - i)P9-y2icos(n - Θ)] (10.22)

c2 cos qn

where the contour Q + C2 encloses the real axis as illustrated in Fig. 9. Noting that the poles of the integrand are located at q = -J-, ·§·, f, ... etc., it can be readily verified by the theorem of residues that this integral is equivalent to Eq. (21). Now, subject to the validity of the second-order approximations, for the wave functions of order kga and kta mentioned above, the

1 J r

Ci

^ r c2

L X

X X

FIG. 9. The contours in the complex q plane.

function/(# - \) is an even function of q. This means that the part of the contour C± just above the positive real axis can be replaced by C[ which is located just below the negative real axis.f The contour C[ + C2 is now entirely equivalent to L, a straight line running along just below the real axis. Replacing q - \ by v the contour representation for U takes the form

υ—ί (v + i) L sin νπ

/(v)Pv[cosft - 0)] dv (10.23)

It is to be noted that this manipulation of the contours is only strictly justified when/fa - \) is an even function of q. This is well justified when \kg\2 and |A:f|2 are both pk2.

The next step in the analysis is to close L by an infinite semi-circle in the negative half-plane. The contribution from this part of the contour vanishes as the radius of the semi-circle approaches infinity because of the exponentially decreasing character of the integrand. The value of the integral for Ualong the contour L is now equal to -Ini x the sum of the residues of the integrand evaluated at the poles of f(v) located in the lower half-plane of v. It then follows that

U = _ ike Σ ( - ^ h?Xkb)h<lXkr) PvLcos(n - 0)] v sin νπ Dv (10.24)

t See Fig. 9.

where D'v = dDJdv. All quantities in the summand are to be evaluated at the poles of/(v) which are the roots of the equation

Dv = 0

This equation is precisely the same as the one discussed earlier (i.e. Eq. (5.5)). At that time the relevant spherical wave functions of order ka and kc were simplified by the use of the Debye or second-order approximation. The Hankel or third-order approximation will now be employed. It may be written [Sommerfeld, 1949]

<*> Γ π χ 1 1 / 2 x Γ ( v + ï ) 2 V A <*> (1)

xWXx) s [ y j ^ [l - l - ^ - J tfi»exp[± K-à* - pW?\x) (10.25)

where

(10.26) ( i )

and H\^\p) is the Hankel function of order J of argument p given by

'--T%ÌÌH (ια27)

For Re (v + £) ^ x or for |p| > 1, the above reduces to

xh?\x) s S?\x) (10.28)

which is the Debye approximation used in Section 5. The third order representation for the logarithmic derivative is

taT5Ä)[| a e*«»Γΐ - * ± * \ * ί | ω «0.29, L ' ' HgW

while the corresponding second order approximation, valid for \p\ > 1 is simply

2Ί Vi

ln'[xfc(v2)(x)] s ± i h _ V V (10·30)

For convenience in what follows, it is desirable to introduce 2 new spherical reflection coefiicients rg and rf which are connected to Rg and i?f in the following manner

! _ ^ ln'lkahl2Xka)l „ In'Çfca/^fai)]

re=- ,,,,.-,_ w.w,^- , *ο = 7Ί ( 1 0 · 3 1 ) Xn'lkaWXka)-] ' 1 iA„

and

1 -

1 +

ln'[fca/i<2)(fca)]

iAj

where

and

h'IkchïXkc)·} „ T ln'[fcc/t<2>(fcc)] rf - " ln'[fec^2>(fec)] *' " »A, U U · 3 ^

ln'Cfccft^fcc)]

*·ϊ['-(£)Τ These new reflection coefficients may be expressed to a high order of approximation by using the Hankel or third-order approximation for the spherical Bessel functions of argument ka or kc. Thus

TO

and

(10.35)

where

and

1 + {1 - [(v + i)/fc«]2>* e X P \ ' 6/ H i j V J

t Δ, / . πΧΗ^ρ . ) {i-Kv + i)/**]2}* P I V H ^ ( P C )

1 + {1 - [(v + mcl2}* P l 6J H%(pc)

i%

(10.36)

-HMgSH v + i T (kc)2 "I*

'■—Hirn?"1] ( 1 0 · 3 8 )

To this same approximation _ - c-i4n/3 H % } ( ^ ) ^ U W n Qn 39)

and

ri = e H^ipc)Efipc)Ri ( 1 ° - 4 0 )

When \pa\ and |pe| > 1

r ^ R ^±z[t±m^}l=A (1041) and

ri = Ri={i-i(v + mcVY/> + Ai ( 1 0 · 4 2 )

On writing v + \ = fcaS = fccS", these latter forms are readily identified as the Fresnel reflection coefficients

9 - β - N 2 (I - s2)* + ov2 - s2)* u ' and

r w Ä W «w^w (1044) 1 = ' = Nf [1 - (S')2]'/2 + [Nf - (S')2]* U ;

Attention is turned specifically to the determination of the roots of the equation

Dv = 0 (10.45) This may be written

^ ' ^ ( i k a ^ d k c ) - 6 ( i a 4 6 )

where i^ and i?f are defined by Eqs. (31) and (32) and n may take integral values. Employing the third-order approximations, this may be rewritten, for Re(v + \) < ka

where

and

v'!llf^-, i<"-"JeU"-->-s-' !" (ΐο·47)

r u_i^m\x (10.48) J v+i/2 L X J

r ._r uji±m\x (10.49) J v + i/2 L X J

while, for kc > Re(v + i) > ka,

H%Xpc)H%XPa) V ' H%(Pc)H%(pa) ( }

In the above formulas the (spherical) reflection coefficients rg and rt are defined by Eqs. (35) and (36). When \pa\ and \pc\ > 1 or if Re(v + \) <ζ ka the relevant equation for the modes is simply

rgriQ-i2^-ya) = Q-ilnn (10.51)

where rg and rf are defined by Eqs. (41) and (42) which are the Fresnel form of the reflection coefficients. Equation (51) is identical to Eq. (5.10) which was discussed previously.

11. INFLUENCE OF STRATIFICATION AT THE LOWER EDGE OF THE IONOSPHERE

Attention in previous sections has been largely confined to a sharply-bounded homogeneous ionosphere. In view of the general uncertainty about the electrical properties of the lower edge of the ionosphere, a more elaborate model might hardly seem worthwhile. Furthermore, despite the geometrical simplicity of the above models, the computation of the modes is very involved in the general case. Despite these disparaging remarks, the inhomogeneity of the lower ionosphere may be considered in some cases without greatly increasing the complexity. Some of these generalizations are discussed here for what they are worth.

The theoretical treatment given in Section 3 for a vertical electric dipole located in the air space between a flat ground and a plane interface of a homogeneous ionosphere may be easily generalized to a stratified ionosphere. The essential modification is to replace the ionosphere reflection coefficient Ri(C) by a more elaborate form which is denoted ^ ( C ) . For example, a 2 layer ionosphere is chosen. The lower edge is at z = h and from there to z = h + s, the refractive index (assumed constant and isotropie) is JV ; at this point the refractive index (also assumed constant and isotropie) is N2 and remains at this value thereafter. It is not at all difficult to show that, for vertical polarization [Wait, 1958c],

N\C - (N\ - S2)i/2Q Ri(Q = NlC + iNl-S^Q ( 1 U )

where _ Nl(Nl - S2)* + Nl(Nl - S2)* tanh[ifa(JVf - S2)*l

Nl(Nl - S2)* + Nl(Nz2 - Ξψ tanh[ifcs(^î - S2)*] l " ;

Here it may be observed that if \kNts\ <ξ 1 the reflection coefficient becomes

N\C - (N\ - S2)Vl

Ri(c) = N2c + (Ni-sy> (1L3)

whereas if \kNxs\ > 1, it becomes

KAC)-NlC + (Nl-S2)^ (UA)

These 2 limiting cases correspond respectively, to the conditions of an electrically thin and an electrically thick stratum. In the former case the effective reflection level is at z = h + s and in the latter case it is at z = h.

The formula for Kf(C) may easily be generalized to any number of layers. For example, in the case of discrete layers or strata, 0 < z < h corresponds to the air, h < z < h + s± corresponds to a stratum with index Nl9

7

1 s, f

1 s2

t t h

1

N2

N,

FIG. 10. Stratified model for ionosphere.

A 4- s± < z < h + st + s2 corresponds to a stratum with index N2, and so on.f With this generalization, g, in Eq. (1), is to be replaced by

n _ N\(N22 - Sa)*Qa + N2

2(Nl - Ξψ tanhpfa^y; - S2)1^ y i Nl(Nl - S2f* + Nl{Nl - S2)1/2ß2 tanh[ifeSl(iVÎ - Ξψ] K ' }

η Ν2(Ν2 - Sa)*e3 + ΝΙ(ΝΙ - Ξψ tanh[ffa2(NJ - S2)*! " NÌ(N2 - S2)1/2 + N2(N2 - S2)1/2ß3 tanh[i/c52(JV2 - S2)1^ l * ;

and so on. g3, g4, Q5,... are obtained by cyclic permutation of indices. It should be noted, however, for M discrete strata that Qm = 1 since effectively ^M = oo. The resultant Hertz vector for the air space 0 < z < h is then formally given by Eq. (3.13) with the more general meaning now attached to the ionosphere reflection coefficient. In the general case, the rigorous evaluation of the integrals would be extremely involved. However, using arguments similar to these for the homogeneous ionosphere, the field may be approximated as a sum of residues evaluated at the poles of the integrand. Thus the contributions from the branch points are again neglected since for finitely conducting layers they correspond to heavily damped waves. Therefore, the residue series formula given by Eq. (3.15) is also applicable if the

t See Fig. 10. N

reflection coefficient Ri(Cn) is replaced by Ri(Cn). The modal equation now reads

Ki(CM)^(Cn)exp(-2ife/ICw) = exp(- ilnn) (11.7) where n takes integral values.

A numerical treatment and an application for the special case of a two-layer ionosphere has been carried out and reported in the literature [Wait, 1958a]. There is no intrinsic difficulty in extending such calculations to an indefinitely large number of such layers each with infinitesimal thickness. For finitely conducting strata such a process converges and leads to an adequate representation for a continuous refractive index profile.

A great simplification to the formulas for a stratified ionosphere is effected if the refractive indices for all layers are large. For example, if liV , \N2\ ... \NM\ > 1, then

^ NXQ2 + N2 tanhÇ/fc iVP ^ 1 - N2 + N&2 tanhiiks^) l ' }

^ N2Q3 + N3 tanh(ifcs2iV2) ^2 " N3 + N2Q3 tanh(ifcs2iV2) l ' '

and so on. Thus to this approximation, Q1 does not depend on the angle of incidence or the factor C. In this case, the modal equation simplifies to

khCn^nn + iA/Cn (11.10) where

Regarding A/Cn as a small quantity, this can be solved to give

*- ["©Τ- 'Μ'-©Γ <"·,2> where s0 = 1, εη = 2 (n Φ 0). This is valid if \Akh\ < 1 and |Δ| < kh[\ -(πη/kh)2]. Thus, at low frequencies and for highly conducting layers, the propagation factor Sn is expressible in a relatively simple form.

The special case of a 2 layer ionosphere has been considered in some detail [Wait, 1958a]. Such a model was sufficient to explain the variation of the observed attenuation rate for frequencies of the order of 500 c/s-15 kc/s.

Exponential Profiles At the extremely low frequencies it was seen that for a stratified model of

the ionosphere that the factor Q does not depend on C or S. In fact, it is not difficult to show that the surface impedance Z at z = h looking outwards is given by

Z = loQJNi where η0 s 120π

From this viewpoint it becomes quite easy to write down expressions for Qx for other profiles. For example, if the refractive index varies in an exponential fashion relatively simple formulas are obtained. Two cases which could be considered are

N(z) = 1.0 for 0 < z < f t

N(z) = JVexp[+(z--/i)//] for z>h

where the (—) sign corresponds to a refractive index descreasing with height and the (+) sign corresponds to a refractive index increasing with height. In the above, /is a scale factor; for example, within a distance / above the lower edge of the ionosphere, the refractive index has changed from N to N/e or Ne for the 2 respective cases.

Within the layers the ionosphere propagation is vertically upwards and thus a component E of the electric field satisfies

^4 + k2N2(z)E = 0 for z > ft oz

As indicated in Chapter III, solutions of this equation for the exponential form of N\z) are

const x I0(iNU-(z-h)/l) m n i " const x K0(iNl e+<»-*>") ^ }

for the 2 respective cases, where I0 and K0 are modified Bessel functions. The transverse component of the magnetic field is then found from Maxwell's equations, e.g. ίμωΗ = dE/dz. The surface impedance Z at the lower edge of this model of the ionosphere is then defined by

Z = ElH-]2=h

For the case when the refractive index decreases with height

and when the refractive index increases with height,

For low frequencies, iNl s ^/(ί)χ with x = |N|/. The arguments of modified Bessel functions are thus proportional to ji. Numerical values are shown in the Table for certain real values of x. In both these cases, it may be observed that as the scale length / approaches infinity, Z becomes η0/Ν as it should.


The simplicity of the above formulas for the exponential profiles is due to the inherent assumption that the refractive index N(z) is large compared to unity for z > h. When this condition is violated, such as it would be at frequencies above several kilocycles, the solution is not expressible in closed form, except for horizontal polarization [Wait, 1952] which is really only of academic interest at v.l.f.t Nevertheless for calculation of attenuation rates at extremely-low-frequencies (less than 1 kc, say), the exponential models find direct application. For example, if Q = 1 the homogeneous ionosphere is regained and the attenuation rate (—k Im S0) is proportional to ω1/2; on the other hand, for an exponential profile with N(z) increasing upwards, the attenuation rate increases with frequency more rapidly as suggested by experimental data. In fact, it is quite easy to see, for both decreasing and increasing exponential profiles, that

Attenuation Rate for Exponential Model • = V(2)|ß| s t o ß - a r g ß ) Attenuation Rate for Homogeneous Model

The right-hand side of the above equation is a function of the quantity x = INI! £ (ωΓ/ω)1/2/ where ωΓ = σ/ε0 is the conductivity at the lower edge of the ionosphere and /is the vertical distance in whichiV changes by afactor 2.718.

These results have been used to interpret experimental data at e.l.f. (extremely low frequencies) [Wait, 1960a, 1960b].

Table 1. Selected numerical values of the factor Q

X

0.2 0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 7.0 8.0 9.0

10.0

0.1 0.3 0.7

wo» u wo» lei 1

10.000 4.003 2.026 1.417 1.180 1.100 1.083 1.084 1.073 1.060 1.051

1 1.045 1.040 1.035

arg g

-44° 43' -43° 13' -37° 55' -29° 52' -20° 59' -13° 44' - 9 ° 09' - 5 ° 42' - 4 ° 37' - 3 ° 49' - 3 ° 19' - 2 ° 47' - 2 ° 26' - 2 ° 11'

J W O » * KiWiOx)

ÔT~ 0.3854 0.5880 0.7344 0.8047 0.8459 0.8728 0.8919 0.9169 0.9325 0.9433 0.9490 0.9497 0.9618 0.9654

0.25518 0.4719 • 0.6622

argß 23° 58' 17° 16' 11° 59' 9° 15' 7° 28' 6° 20' 5° 28' 4° 18' 3° 33' 3° 02' 2° 37' 2° 16' 2° 06' 1°54'

27° 22' 21° 08' 14° 38'

t Some extensions of the solution have recently been discussed by Galejs [1961], and Wait and Carter [I960].


12. AVERAGE DECAY LAWS

Although representation of field strength in terms of waveguide modes has been quite useful in explaining observed phenomena at great ranges, the convergence of the mode sum becomes very poor at short distances from the source. Recourse can be made to a ray or hop theory but this is only convenient at the fairly short ranges. At v.l.f. there is a distance range from about 400 to 4000 km where both ray theory and mode theory predict that the field strength is an undulating function of distance. The precise location of the maxima and minima in the interference patterns depends critically on the effective reflection height and to some extent on the electrical characteristics of the ionosphere and the ground. In many applications one is not too interested in the fine structure of the field strength vs. distance, but rather one is more concerned with the average rate of decay of the field.

In this section it is shown that many of the broad characteristics of v.l.f. fields, particularly in the frequency range from 10 to 25 kc/s, are compatible with a simple formula relating average field strength, distance, and frequency. The simplicity is achieved by making a number of reasonable approximations which can be justified on the basis of the previous work using more elaborate methods.

As before, the space between the ground and the ionosphere is represented by a parallel plate waveguide of width h and whose effective surface impedances are Zg and Zf. The source is taken to be a vertical electric dipole located on the lower surface (i.e. the ground). From Eq. (3.28) the vertical field E at distance d and height z can be written

E^E0W (12.1) where E0 is the field of the dipole if the ground were flat and perfectly conducting, and if the ionosphere were not present (i.e. h -> oo).| The complex function W is thus a correction factor which contains the essence of the propagation problem. Using the results of Section 3, one obtains

W = m^- e ' ·"2^)-^4)] g δβ% Q-^nä/λ cos( fcCnZ) (12.2) («M) n = 0

where 1

^ = r .g(^ig/äc] ( 1 2 · 3 )

Ri = ^ χ , Δ, = Ζ,/IJ, η = 120π

t EQ can be loosely described as the "inverse-distance field."

and Cn is a solution of

RiRg Qxpl-i4nhC/X2 = exp(-2nin) (12.4) and finally

S„ = ( l - C 2 ) *

Here the influence of earth curvature and the terrestrial magnetic field are neglected since, for present purposes, the parallel plate waveguide with homogeneous isotropie walls is adequate for this discussion. Thus, we may write

A = G,/2 e1'*'4 (12.5)

[*-r Δ{= i f— (12.6)

JL·

where G and L are constants independent of C. In terms of the conductivity ag and dielectric constant eg of the ground,

ECU G = where ε = 8.854 x IO-12

σβ + ΐεβω

In most cases εβω <ζ σβ, so that,

is regarded as a real quantity in what follows. In terms of the (angular) plasma frequency ω0 and the collisional frequency v of the ionosphere,

2

L = — where ωΓ = — (12.7) COr V

The square of the amplitude of the field in the waveguide can now be written

\E2\ = \E20\\W2\ (12.8)

where

yjHA) n = o m = 0 x cos(fcCnz) cos(feC*z) (12.9)

The double summation is not particularly useful in this form. However, the average value of \W2\ can be represented in a very simple form if approximations are made which are valid for obHque incidence modes. A similar

approach has been used by Brekhovskikh [1949] in connection with acoustic propagation in the ocean. Specifically, if

CJAi<l and CJAg> 1 for all important modes, it follows from Eq. (4.19) that

where _n(n-j) ( n - f l

Ln kh - (2Λ/Α) ( 1 2 , 1 1 )

The restrictions stated above may now be replaced by

(C„)2«;~L and (Cn)2>G

Since L is of the order of 0.5 and G <ξ 1 in the frequency range of 10-25 kc/s, these conditions are well satisfied for the important modes.

The double summation for \W2\ may thus be approximated by

where δη and <5m have been replaced by unity. In this case the modes of lowest attenuation are of order one following earlier adopted conventions.

We now consider the value of the power intensity or (field amplitude)2

averaged over the width of the waveguide. This average is obtained from \E2\ = | ^ | | ÎF 2 | where

|ïf2| = i | \w2\dh (12.13) " Jo

In view of the relation

f* Γπ(η-*Κ1 \n(m - ±)z\ A (1 for n = m , „ 1ΑΛ

J o C0SL Λ JC0SL Λ J d Z = lO forn Φ m (12*14)

W^ = ^l*-ikiSn-Sn*)d (12.15)

it readily follows that

2(hß)2 „ f x It may be noted

S„-S; = 2i Im S„ (12.16) and from Eq. (10),

Im * » - vès M^ L + j z ) + v° ] <1217>

Thus

'■I'^i·'!·*-^ (1218)

where

and

a = ^ n ( V L + ^;) (12·19)

Pg = expl-y/(2G)(dlhy] (12.20)

The summation indicated above can be written explicitly in terms of the Jacobi theta function 02(v, κ) as defined by Jahnke, Emde, and Lösch [I960]. Thus

When ad is small a fairly large number of terms in the summation in Eq. (18) are required and for a first approximation, the sum may be replaced by an integral. Thus

This is valid for ocd < 2 (see appendix). Making use of Eq. (19), this can be written in the compact form

exp[-V(2G>P] (12·23)

Since

and m=N2I w>\

const En V = 01 ~~dr~

it is seen that distance dependence is described by

\E2\ oc - ^ exp[-V(2G)d/fc] oc for G = 0 (i.e. σ, s oo) (12.24)

To test the validity of this simple law | W2\ given by Eq. (23) and | W2\ given by Eq. (9) and plotted as a function of d/λ for the conditions, h/λ = 5, L = 0.5, G = 0. It is seen that over a range of d/λ from about 20 to 200, the straight line, which corresponds to \W2\, is a reasonable average of the undulating curve for \W2\. It may also be demonstrated [Wait, 1957b] that if hjX is changed by 10 per cent or so, the computed mode sum for \W2\ has maxima


and minima that are shifted appreciably from the curve shown in Fig. 11. However, the same straight line is still a good measure of the averaged intensity for the distance range from 20 to 200 wavelengths.

!£

15

10

5

n

-5

-10

-15

-

1 1

|w* |

I I !

- | w » |

1 1 1 !

G = 0

hA = 5 L = 0. 5

I . I . 10 20 30 40 50 100

dA 200 400

FIG. 11. A comparison of the intensity and the averaged intensity for a concrete case.

When ad > 1, the summation in Eq. (18) reduces essentially to one term. Thus,

W2\^ d\X 2(hixy

, Q-ad/4 e-V(2G)(d/Ä) (12.25)

The average intensity is now exponentially damped. This corresponds to the region where only one waveguide mode is predominant. This particular case has been discussed at length in the literature [Wait, 1957b; Budden, 1952]. It should be mentioned, however, that earth curvature increases the effective value of the attenuation of the dominant mode [Wait and Spies, 1960], as will be discussed in a later chapter. However, the flat-earth approximation is justified for the higher modes and thus averaged intensity in the intermediate distance range is not appreciably influenced by earth curvature.

Equation (23) for \W \ allows for a very simple physical discussion of the averaged field intensity in the intermediate distance region. In the first place, it can be seen that \E2\ varies as d~3/2 when the ground losses are negligible (such as over sea water). The ground conductivity results in additional attenuation per unit distance given by (6A4/h)y/G x I03dh per 1000 km, when h is expressed in kilometers. For example, if h = 80 km,

/ = 16 kc, and σβ = 2 mmho/m, this amounts to 1.6 dB/1000 km. This agrees quite well with the results obtained from a more rigorous numerical treatment of the modal equation [Wait, 1957b; Howe and Wait, 1957]. It is interesting to note that this attenuation factor is proportional to/1/2 and σβ"ν\


Again, it should be stressed that this is only true for the intermediate distance region. At very great distances (i.e. d > 3000 km) the influence of earth's curvature on the dominant mode leads to a more complicated dependence of ground losses on frequency and conductivity [Wait and Spies, I960]. This is discussed also in a later chapter.

Since the averaged field intensity is proportional to the factor

we see that a broad maximum occurs for a frequency ω = ωΓ. Since œr is of the order of 105 this corresponds to a frequency of the order of 15 kc/s. Such a maximum is observed when the full mode sum is plotted as a function of frequency. Also such a maximum has been deduced by Watt and Maxwell [1957] on the basis of experimental data. It should be mentioned that the frequency dependence is a function of the ionospheric model. For a stratified or a "diffuse" model the factor y/L + 1/y/L is replaced by a more complicated function. This is also true when the earth's magnetic field is included in the analysis [Wait and Spies, I960],

10 r 1 1—r—i—i—| 1 1 1 1 1 i »—i ι J

1 0 I i i i i i I i i i i i i i i i I

0.4 0.6 1.0 2.0 4.0 6.0 10.0

ad

FIG. 12. Average Decay Laws.

13. APPENDIX

From Eqs. (8) and (21) it is seen that the averaged field intensity behaves in the manner

\E2\ocCl where

°-sM0»ir) 9 oo

__±. y e-(»-%)2«i ad M=i

The function Ω is plotted in Fig. 12 as a function of ccd. On the same graph the integral approximation

2 f00 π1/ζ

ad J0 (ocd)/2

is also shown for comparison. As can be seen, if cud < 2, the summation is well approximated by the 3/2's power law. In the example shown in Fig. 11 this corresponds to d/λ < 300.

REFERENCES

AL'PERT, IA. L. (1955) Low frequency electromagnetic wave propagation over the surface of the earth, Monograph published by Academy of Science, U.S.S.R.

AL'PERT, IA. L. (1956) The field of long and very long radio waves over the earth under actual conditions, Radiotekj. i Elektron, 1, 281.

BREKHOVSKIKH, L. M. (1949) Sound propagation in an underwater sound channel, Dok. Akad. Nauk. S.S.S.R., 69, 157-167.

BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier, Amsterdam, Netherlands. BUDDEN, K. G. (1952) The propagation of a radio atmospheric, II, Phil. Mag. 43, 1179. BUDDEN, K. G. (1953) The propagation of very-low-frequency radio waves to great distances,

Phil. Mag., 44, 504. BUDDEN, K. G. (1957) The "waveguide mode" theory of the propagation of very-low-

frequency radio waves, Proc. I.R.E., 45, 772. CHAPMAN, F. W., and MACARIO, R. C. V. (1956) Propagation of audio frequency radio

waves to great distances, Nature, 177, 930. FRIEDMAN, B. (1960) Low frequency propagation in the ionosphere, in Electromagnetic Wave

Propagation, Academic Press, New York, N.Y. GALEJS, J. (1961) e.l.f. waves in the presence of exponential ionospheric conductivity

profiles, Trans. I.R.E., AP-9, No. 6, 554-560. HOWE, H. H., and WAIT, J. R. (1957) Mode calculations for v.l.f. ionospheric propagation,

v.l.f. Symposium Paper 36, Boulder, Colorado. JAHNKE, E., EMDE, F., and LOSCH, F. (1960) Tables of Higher Functions, McGraw-Hill,

New York, N.Y. KADEN, H. (1957) Die reflexions und Schirmwirkung metallischer hüllen in einer ebenen

elektromagnetischen welle, Sonderdruck aus Archiv der Elektrischen Übertragung, 403.


KARBOWIAK, A. E. (1959) Radiation and guided waves, Trans. I.R.E., AP-7 (Special Supplement), 191-200.

LIEBERMAN, L. (1957) Anomalous propagation below 500 c/s, Proc. Symposium on Propagation of Radio Waves, 3, paper 25, Boulder, Colorado.

MAGNUS, F., and OBERHETTTNGER, F. (1949) Special Functions of Mathematical Physics, pp. 57, 71, 72, Chelsea, New York, N.Y.

NAMBA, S. (1933) General theory of the propagation in the upper atmosphere, Proc. I.R.E., 21, 238.

RYDBECK, O. E. H. (1944) On the propagation of radio waves, Trans. Chalmers University, 34.

SCHUMANN, W. O. (1952) Über die ausbreitung sehr langer elektrischer wellen um die erde und die signale des blitze, / / Nuovo cimento, IX.

SCHUMANN, W. O. (1954a) Über die oberfelder bei der Ausbreitung langer, elektrischer wellen im system erde-luft-ionosphare und 2 anwendungen (horizontaler und senkrechter dipoi), A. Angew. Phys., 6, 34 .

SCHUMANN, W. O. (1954b) Über die Strahlung langer wellen des horizontalen dipois in dem lufthohlraum zwischen erde und ionosphare, Z. Angew. Phys., 6, 225.

SCHUMANN, W. O. (1957) Über elektrische eigenschwingungen des hohlraumes erde-luft-ionosphare arregt durch blitzentladungen, Z. Angew. Phys., 9, 373.

SOMMERFELD, A. N. (1926) Über die ausbreitung der wellen in der drachtlosen télégraphie, Ann. Physik, 81, 1135.

SOMMERFELD, A. N. (1949) Partial Differential Equations, Academic Press, New York, N.Y. SPIES, K. P., and WAIT, J. R. (1961) Mode calculations for v.l.f. propagation in the earth-

ionosphere waveguide, Nat. Bur. Çtand. Technical Note No. 114. TAYLOR, W. L., and LANGE, L. J. (1959) Some characteristics of v.l.f. propagation using

atmospheric waveforms, Proc. Second Conf. on Recent Advances in Atmospheric Electricity; p. 609, Pergamon Press, New York, N.Y.

WAIT, J. R. (1952) Oblique reflection of radio waves from an inhomogeneous region, / . Appi. Phys., 23, 1403.

WAIT, J. R. (1956) Low frequency radiation from a horizontal antenna over a spherical earth, Canad. J. Phys., 34, 586.

WAIT, J. R. (1957a) On the mode theory of v.l.f. ionospheric propagation, Rev. Geofis. Pura e Appi, 37, 193.

WAIT, J. R. (1957b) The mode theory of v.l.f. ionospheric propagation for finite ground conductivity, Proc. I.R.E., 45, 760.

WAIT, J. R. (1958a) An extension to the mode theory of v.l.f. ionospheric region, / . Geophys. Res., 63, 125.

WAIT, J. R. (1958b) Propagation of very-low-frequency pulses to great distances, / . Res. Nat. Bur. Stand., 61, 187.

WAIT, J. R. (1958c) Transmission and reflection of electromagnetic waves in the presence of stratified media, / . Res. Nat. Bur. Stand., 61, 205.

WAIT, J. R. (1960a) On the theory of the slow-tail portion of atmospheric waveforms, / . Geophys. Res. 65, No. 7, 1939-1946.

WAIT, J. R. (1960b) Mode theory and the propagation of extremely low frequency radio waves, / . Res. Nat. Bur. Stand., 64D, No. 4, 387-404.

WAIT, J. R., and CARTER, N. F. (1960) Field strength calculations for e.l.f. radio waves, Nat. Bur. Stand. Technical Note No. 52.

WAIT, J. R., and HOWE, H. H. (1957) The waveguide mode theory of v.l.f. ionospheric propagation, Proc. I.R.E., 45.

WAIT, J. R., and PERRY, L. B. (1957) Calculations of ionospheric reflection coefficients at very low radio frequencies, / . Geophys. Res. 62, 43.

WAIT, J. R., and SPIES, K. P. (I960) Influence of earth curvature and the terrestrial magnetic field on v.l.f. propagation, / . Geophys. Res., 65, No. 8, 2325-2331.

WATSON, G. N. (1919) The transmission of electric waves round the earth, Proc. Roy. Soc, 95, 546.

WATT, A. D., and MAXWELL, E. L. (1957) Characteristics of atmospheric noise from 1 to 100 kc/s, Proc. I.R.E., 45, 787-795.


Additional References AL'PERT, IA. L. (1957) On the speed of propagation of electromagnetic waves at audio

frequencies, / . Exp. Theoret. Phys. (U.S.S.R.)., 33, 1305. AL'PERT, IA. L. (1956) Investigation of the propagation of long and very long radio waves

by the method of analyzing the shape of atmospherics, Radiotekh. i Elektron, 1, 293. AL'PERT, IA. L., and BORODINA, S. V. (1959) On the velocity of propagation of audio fre

quency electromagnetic waves, Radiotekh. i Elektron, 4, 195. AUSTIN, L. W. (1926) Preliminary note on proposed changes in the constants of the Austin-

Cohen transmission formula, Proc. I.R.E., 14, 377. BARRON, D. W. (1959) The "waveguide mode" theory of radio wave propagation when the

ionosphere is not sharply bounded, Phil. Mag., 50,1068. (This is a continuation of early unpublished work of Budden.)

BREMMER, H. (1958) Propagation of electromagnetic waves, Handbuch der Physik, 16, 423. BREMMER, H. (1959) Mode expansion in the low frequency range for propagation through

a curved stratified atmosphere, / . Res. Nat. Bur. Stand., 63D, 75. BUDDEN, K. G. (1953, 1954) A reciprocity theorem on the propagation of radio waves via

the ionosphere, Phil. Mag., 44, 604. ECKERSLEY, T. L. (1932) Studies in radio transmission, / . Inst. Elee. Engrs., 71, 405. FRIEDMAN, B. F. (1951) Propagation in a nonhomogeneous atmosphere, Theory of Electro

magnetic Waves, Academic Press, New York, N.Y. KENDRICK, G. W. (1928) Radio transmission formulae, Phys. Rev. 31, 1040. POEVERLEIN, H. (1959) Lang-und Langstwellausbreitung, Fortschr. der Hochfrequenztech.,

4, 47. (Many additional references are found in this excellent review article.) SCHMELOVSKY, K. H. (1958) Problem der ausbreitung in tropospherischen und ionosphischen

Wellenleiter, Abhanglungen des Meteorologischen und Hydrologischen Dienstes der Deutschen Demoktratischen Republik, 7.

WAIT, J. R. (1957) The attenuation versus frequency characteristics of v.l.f. radio waves, Proc. I.R.E., 45, 768.

WAIT, J. R., and HOWE, H. H. (1956) Amplitude and phase curves for groundwave propagation in the band 200 c/s to 500 kc, Nat. Bur. Stand., Circular No. 574.

WAIT, J. R. (1956) Radiation from a vertical antenna over a curved stratified ground, / . Res. Nat. Bur. Stand., 56, 237.

WAIT, J. R. (1954) Note on the theory of radio propagation over an ice covered sea, Def. Research Tele. Est., Radio Physics Lab., Project Rept. 18-0-7. (Gives solution for dielectric slab over a conductive base.)

WAIT, J. R., and MURPHY, A. (1956) Multiple reflections between the earth and the ionosphere in v.l.f. propagation, Geofis. Pura Appi, 35, 61.

WATT, A. D. (I960) e.l.f. electric fields from thunderstorms,/. Res. Nat. Bur. Stand. 64D, (Gives details on the lightning discharge source magnitude and spectral distribution).

Chapter VII

CHARACTERISTICS OF THE MODES FOR V.L.F. PROPAGATION

Abstract—An attempt is made to present a concise derivation of the mode theory at v.l.f. propagation. Taking note of the fact that the important modes for long distance propagation are near grazing, suitable approximate forms are introduced at the outset rather than at the end of the analysis. The derived formulas are used to obtain numerical results for the attenuation, phase velocity and excitation of the dominant modes in v.l.f. radio propagation. The physical and practical significance of these results is described.

1. INTRODUCTION

In this chapter some features of propagation between concentric spherical surfaces are illustrated. Although the general theory has already been given, the derivations of the basic formulas used here are outlined very briefly. The approach [Wait, 1961] is somewhat different in that certain approximations are introduced into the analysis at the outset rather than at the end. In most of the numerical results presented, the parameters and constants are chosen so that application may be made directly to v.l.f. propagation in the earth-ionosphere waveguide.

2. THE GROUND WAVE

We will start by assuming that the source is a vertical electric dipole located on the surface of a smooth spherical earth of radius a, conductivity σ and dielectric constant ε. Spherical coordinates (r, 0, φ) are chosen with the dipole located at r = a and 0 = 0. For harmonic time dependence the radial electric field component is written in the form

^Φΐ^Ψ v° (1)

apart from a constant factor. In the case of an airless earth, in which the ionosphere is neglected, it was shown in Chapter V that V0 may be written in the form [van der Pol and Bremmer, 1937; Fock, 1945]

VoSTinx)**-*» Σ ^ , W - ^ ê > (2) s=i,2,... (ts-q ) Wi(y

196

Characteristics of the Modes for V.L.F. Propagation 197

where

and Δ = / ie0œ \1/2Γ _ ιε0ω Ί1/*

\σ + ιεω/ |_ σ + ι ε ω ] The coefficients is are solutions of the equation

wi(0 - «Wi(0 = 0 (3) where ^ ( i ) is an Airy integral and the prime indicates a derivative with respect to t. In terms of Hankel functions of order one third,

Wl(0 = expC-2Wi73)(-Wi/3)%flg>[ÖX-0%] (4) The above formula for V0 is usually called the residue series representation

for the ground wave field [Bremmer, 1949]. It is valid when ka > 1 and r — a <ζ a. It is also required that the values of |fs| for the important modes are not large compared with unity. Such approximations as these are certainly valid for v.l.f. waves

For the purposes of the subsequent analysis V0 is now written as a contour integral in the manner

V0 = e ^ x w 4 Q ^ àt (5) J Wi(i) - qwt(t)

The contour encloses the poles at t = ts in a clockwise sense. The equivalence of Eqs. (2) and (5) is easily verified on noting that

wïfo) - «witt) = ('. - « V i t o since

wî(i) = ίνν^Ο and

Witt) = «w^O

3. THE SKY WAVES

We will now enclose the earth by a concentric reflecting shell located at r = a + h as indicated in Fig. 1. The electrical properties of this layer will not be specified at the moment, but it is assumed that an upgoing wave will be converted to a downgoing wave. Thus, after one reflection

e'^SviO - y) is converted to

A(t)e~ixtw2(t-y) where

w2(i) = exp(2™/3)(-nf/3)*Hij>[(D(-/)%] (6)


and A(t) is unknown function of /. The boundary condition at r = a + h is written in the form

|"dw(i - y)

where

and

Ay - qtw(t - J O | = o (7)

2\ ' /J

m

\kaj

The quantity A( involves the properties of the layer beyond r = a + h; for the moment it is not given explicitly. Formally, Af = Ζ/η0 where Z is the radial surface impedance at r = a + A. Most generally Af (or Z) is a function of f but usually it may be taken as a constant [Wait, I960]. On identifying w(t - y) with wt(t - y) + i4(0w2(* - j>), it readily follows from Eq. (7) that

A(t) wi(i - >>o) + 4iW2(* - J O ) J

(8)

iSS^ur?!·) l i 0 ^

FIG. 1. A sketch of the model employed. The surface of the ground and the lower edge of the reflecting layer are

concentric spherical surfaces.

The downcoming wave characterized by the function w2(t - y) is now reflected at the ground and thus it generates a new upgoing wave of the form

The boundary condition at r = a may be written

Then, on identifying w(t — y) with the sum of the downcoming and the upgoing wave, it follows that

The process may be repeated any number of times. The resultant field can thus be written as a sum

Q-ika0

Er = a(9 sinfl)* ( U )

where V= Σ Vj (12)

j = 0,l,2...

and

for j even, while Vj = ψΫ&^ι(ί ? lA{i)B(t)y12 ài (13) e

V' - ''""(ff 9 ^ 1 «»"*""« ""'"2<" (14) for y odd. Formally these are geometrical progressions, so they can be summed. On interchanging the order of integration and summation, this leads to the integral representation

W J MM - qWl(tmi - A(t)B(m

= *4*(*Ϋ& e""[^-y)+^(0w2Q-^)] W J wi(0-9w1(i) + /l(i)[wi(0-ew2(i)] l '

Now the contour is to enclose the complex poles which occur at t = t„ where /„ is a solution of

1 - A(t)B(t) = 0 (16)

The residue series representation for the total field is thus given by

F--*«*)*·-«* Σ C~f*'"[Wt(<"-^+rf>*(V)] (17)

n = 0,1.2... [wKO-ew^JD^iiCiiBCo] As in the residue series formula for the ground wave field, Eq. (17) is valid

for kai> \ and h < a. It can be shown that this equation is equivalent to Eq. (11.24) of the previous chapter, derived by a more traditional method.


The interpretation of the preceding results is now discussed and certain simplifications made to facilitate computation. If (—t) > 1, it follows that

B(ù - \W'2(t) ~ qW2(t)] K ' Lwi(0 - qWl(t)l

„ Γ-«-0*-«Ί -» [££ ' . . / ] e 'w a»p^-^ (18)

On making the substitution ( - t)Vi = (ka/2)i/3C and noting that

iq = (fca/2)'/3A it follows that

B(t) S ^ exp|>/2] exp[i 5 ( y ) c 3 ] (19)

If C is identified as the cosine of an angle of incidence, (C — A)/(C + Δ) can be recognized as a Fresnel reflection coefficient. Similarly, if (y0 — t) > 1

A(t) s l ^ - 1 exp[-/π/2] exp[- i 1 ( y ) (C)3] (20)

where (ka/2)l/3C = (j>0 — i)y2. C" may be identified as the cosine of the angle of incidence at the ionosphere. It is also noted that

C=(c2+^y2 (21) The factor (C — ^d/(C + Δ,·) is, of course, a Fresnel reflection coefficient referred to the bottom of the layer at r = a + A.

The modal equation (16) which determines the coefficients tn may thus be written in the approximate form

RgR i exp(- il) = 1 = e~i2nn (22)

„ C - Δ Λ η C ' - Δ , where

and f = ^ [ ( C ' ) 3 - C 3 ]

if A/a ^ C2, then JRi = (C - A,)/(C + Δ,) and 1 s 2&AC which corresponds to the modal equation for the flat-earth case.


For numerical work it is desirable to express the function

A(t)B(t) ] explicitly in terms of Airy functions.

First, it is noted that Γ3 w'2(t) - qw2(m = 2i(tn - q2) [dt wi(0 - «WiioLfc = [wKO - qw^tj]2 (23)

where use has been made of the relations (1) wl(t) = twx(t) and (2) w1{f)W2{f) - H>i(i)w2(0 = —2/which are valid for any value of t.

Similarly,

rdw'1(t-y0)+qiw1(t-y0y\ - 2 ι ( * „ - y 0 - g?) |_δί w2(i - y0) + 4iW2(* - yo)\t=tn [wi(i„ - )>o) + «iH>2(i» - )>o]2

Thus

$w 2 (0]

2i(i„ - y0 - « f r i t t i ) - qWiiQl (25) [wi(in) - iWiiOlCwi^ - y0) + ^νν2(ίη - y 0 ) ] 2

Therefore, the complete residue series may be written

Κ β ( π χ ) * β - | κ / 4 Y e ' ^ Î w ^ r . - y) + A{tn)w2{tn - y)] »-0.1.2 Γ *„ - g2 (r. - y0 - qf)[w'2(Q - gw2(tw)]1

L w2(in) - qw2(tn) [w2(f„ - y0) + QïWiitn - >>o)]2 J (26)

4. THE ROOTS OF THE MODAL EQUATION!

The modal equation which determines the roots tn is given by Eq. (16). It may be written

Γνν2(0 - qw2(m [w[(t - y0) + q^x{t - y0)l = e_i2nn

Lwi(0 - gwi(0J Lw2(i - y0) + «iw2(i - yo)\ where iv^i) and w2(t) are Airy integrals. For most applications this rather formidable looking equation can be simplified since y0 is reasonably large

t Some of the remaining material in this chapter is based on the work of K. P. Spies, the author's colleague.

compared with unity. Thus, on the assumption that (y0 — i) > 1, Eq. (27) may be transformed to

RgRiFgQ-ii = Q-i2nn (28) where,

and

Now,

, as before

however, an

h

R„

Ri

C

= f fca[(C')3 - C3] C-A

~ C + A C'-Ai

C +Δ;

= (fea/2)'/3C

= (C2 + 2hla)'A

additional factor Fg appears which is defined by

F „ =

w'2(t) - qw2(t) w'i(Q - qwj.it)

>«*-*Α=Ε$ΞΪ\ (29)

The denominator of this expression is the asymptotic approximation (for large negative t) of the numerator. Thus, if

(fefl/2),/3C > 1

Fg can be replaced by unity and Eq. (28) reduces to Eq. (22). As we will see, for most cases of practical interest in v.l.f. propagation the preceding inequality is seriously violated and consequently Fg cannot be replaced by unity.

In Eq. (28), if we make the substitution

R. = -ea i C / = -exp L (c2 + ^ \ Ί (30)

it follows that the mode equation may be conveniently written

(31)

Strictly speaking, αχ is a function of C. However, if |C'/A(| < 1, it is seen that

http://qwj.it


For example, in the case of an isotropie ionosphere of refractive index Nt

-2Nf ^ -2Nf "* "" [JV? - 1 + (C')2],/2 S (Nf - 1),/2

where ωΓ is a constant. In other cases, such as for stratified ionosphere ax may also be regarded as a constant for the modes of low attenuation.

Before attempting to solve Eq. (27) for the general case it is first desirable to consider its limiting form for perfectly reflecting boundaries. Thus we choose q = 0 or Rg = 1, corresponding to perfectly conducting ground, and at = 0 or Rt = - 1 , corresponding to perfectly reflecting ionosphere. Then, Eq. (31) can be writtenf

This equation may be solved readily by graphical methods. The values of t which satisfy this equation are either real or purely imaginary. Only the roots corresponding to the real solutions are of practical interest since they are unattenuated. For this case, the phase velocity, v, expressed as a ratio to the velocity of light c, is given by

v 1 C2

c o 2

Curves of (v/c — 1) as a function of frequency in the v.l.f. range are shown in Figs. 2a and 2b for n = 1 and 2, respectively, and for h varying from 60 to 100 km. These results are based on Eqs. (32) and (33). It is interesting to note that v may actually be equal to c in certain instances. Actually, this happens when / = 0 is a solution of the modal equation. Returning now to the general mode Eq. (31) and making the substitution z = C2, it follows that

F(z) = 0 (34) where

^ , „ / 2hY/2 . / 2hV2 F(z) = ikalz + — I + IOLAZ + — I

and where t= ~{ka\2f*z (36)

t The Inverse tangents are chosen such that tan*11/(0)/«'(0) = — π/6.


0.010

8 12 16 20 24 28 FREQUENCY, kc /s

FIG. 2a. Phase velocity of the first or dominant mode for a perfectly conducting earth and a perfectly reflecting ionosphere.


U.UIÖ

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

0.002

0.004

\ \ \

\ \

--

1 i i

A \ \ \ [

1 \ \ \ \ \ \^K

\ \ \ \ \ \ \> \ \ \ \ \ \ A N \ \ \ \ \ \ ^ \ \ \ \ ψ \ \ \ \ τ \

N #A

1 1 1 1 1

crg=oo ;α,^Ό]

\E3

\\\\ Y&

_ J L 1 8 12 16 20 24 28

FREQUENCY, kc /s

FIG. 2b. Phase velocity of the second mode for a perfectly conducting earth and a perfectly reflecting ionosphere.


Now, following Newton's method, if z0 is an approximate root of Eq. (34), a next approximation zt may be found from

zt = z0- Az (37) where

Δ ζ = F V T Ì ( 3 8 )

Making use of the Wronskian relation

w[(t)w2(t) - W l ( i M ( 0 = li (39) one finds that

ikaC» + falC + i l o g [ W y ° ; - g t V f o ) l - (4„ - 1) Î . *Wi(to) - qwi(to)\ J2 Δζ = 1 7Γ7Τ% v _ „ Ι Λ (40) kaC +

' ! « - ' ♦ ( * ) fca\% 2(ί0 - s2)

[wi(io) - «*ι('ο)]Ι>2(ίο) - qw2(t0)2 where

\ ° + w an *°= "" I T ) Z°* Provided the process converges, Newton's method may be applied successively to obtain higher order approximations. As a zero-order or starting solution t0 for Eq. (34), it is convenient to choose the solution of Eq. (32). Usually only one or two iterations are sufficient. A more detailed discussion of this procedure is given elsewhere [Spies and Wait, 1961].

Using the method, briefly described above, values of phase velocity and attenuation of the first 2 modes have been calculated as a function of frequency from 8 to 30 kc/s. The phase velocities, calculated from the relation

c Re S v }

are shown in Figs. 3a and 3b, for n = 1 and 2, respectively, when œr = 2 x IO5

and ag = oo. The corresponding attenuation curves are calculated from: Attenuation in decibels per 1000 km of path length = — Im 5'(2π/^)8.68 χ

IO3 where λ is the wavelength in kilometers. The curves in Figs. 3a, 3b, 4a and 4b correspond to a perfectly conducting

earth and a finitely conducting and isotropie ionosphere. This would be a suitable idealization of v.l.f. propagation over sea water for an undisturbed ionosphere. The value of ωΓ = 2 x 105 is typical of daytime conditions. The influence of changing this parameter is mentioned below. Also, in a later chapter, the effect of anisotropy in the ionosphere is considered. It is important to remember that in the present chapter the influence of the terrestrial magnetic field is not considered.


8 I? 16 20 24 28 FREQUENCY, k c / s

FIG. 3a. Phase velocity of the first mode for a perfectly conducting earth and an imperfectly reflecting ionosphere.


h \\ \ \ \ \ \ \ \

Γ \ \ \ \ \ \ \

1* \ \ \ \

Γ \ \ \ \ Γ \ W Γ W h \

L h

L_J 1 1 L_J U

σ-g =oo ; ωΓ= 2 x I05|

[â3

\ \ \ $ \

\ \ Ν τ \ 1 Λ\ \Λν \ Y \ \ 1 Vw\V| V ^ s \ j N ^ ^ O ^ v J

^^o _J i i i—1

16 20 24 28 FREQUENCY, kc /s

FIG. 3b. Phase velocity of the second mode for a perfectly conducting earth and an imperfectly reflecting ionosphere.


16 20 24 FREQUENCY, kc /s

FIG. 4a. Attenuation rate of the first mode for a perfectly conducting earth and an imperfectly reflecting ionosphere.

Γ i \ \ \ \

Γ \ \

[ \ h

u

l·

1 1 1

σ ς = 0 0 ; ωΓ = 2 χ ΐ 0 5

IHZU

\ \ \ \ \ \ V\ \\\\\\W \ \ \ \ \ \ Ψ \ WWW τ" \ \ \ \ \ \ \ v·\ \ 1 \ \ \ \ v \ \ \ \ \ \ JÊP\ X \ i

\ \ m \ \ \ % < ^ ^ ^ ^

I I I 1 1 1 1 1

16 20 24 FREQUENCY, kc /s

28

FIG. 4b. Attenuation rate of the second mode for a perfectly conducting earth and an imperfectly reflecting ionosphere.

υ.υιυ

0.008

0.006

0.004

0.002

0.000

-0.002

-0.004

-0.006

-0.008

-n nifi

-

_ \ *

1

h = 70km-,GUr = 2 X I 0 5

FU

J<Jq = CO MILLIMHOS /METER|

/ JTÔ]

1 ! - - 1 1 8 12 16 20 24 28

FREQUENCY, kc/S

FIG. 5a. Phase velocity of the first mode for an imperfectly conducting earth and an imperfectly reflecting ionosphere.


16 20 24 FREQUENCY, kc /S

FIG. 5b. Attenuation rate of the first mode for an imperfectly conducting earth and an imperfectly reflecting ionosphere.

The influence of finite ground conductivity is shown in Figs. 5a and 5b for h = 70 km, cor = 2 x 105, and n = 1. Here the result for phase velocity and attenuation were calculated by the above described method. For the case Gg = 20 mmho/m, the initial solution z0 was chosen to be the solution for ωΓ = 2 x IO5 and σ9 = oo. This resulting solution was then used as the starting solution for ag = 10 which, in turn, was used as the starting solution for Gg = 5, and so on. In every case, only one application of Newton's method was adequate to produce convergence, although a second application was carried out as a check. Because of its practical importance, the calculation was also carried out for ag = 4000 mmho/m corresponding to the conductivity of sea water. The results were indistinguishable from ag = oo on Figs. 5a and 5b.

At least for the n = 1 mode, it is apparent that the finite ground conductivity has a marked effect on the attenuation, particularly at frequencies around 18 kc/s. For higher and lower frequencies, the effect is diminished somewhat. The phase velocity curves are only slightly influenced by finite ground conductivity.

5. COMMENTS ON A MORE ACCURATE FORM OF THE MODE EQUATION

In the present development of the mode theory, it is assumed, at the outset, that h/a is small compared with unity. In the previous chapter this restriction was not made. Although h/a is of the order of 10~2, it is desirable to obtain some idea of the consequences of using the simplified form of the mode equation. In particular, one might expect errors in the computed value of (v/c) — 1 since it is only of the order of 10"2.

The full modal equation in terms of Hankel functions of orders 1/3 and 2/3 was given in the previous chapter. These are related to Airy functions and their derivatives by

Wl(i) = e - ^ / 3 ^ y / 2 H < j > [ K - i ) % ] (42a)

w2(i) = e + ^ ^ y V ^ C K - O * ] (42b)

H>1(0 = e-'«'30) V/)flg>H<-0%] (42c)

W'2(t) = e ^d jVof fg ïK-O*] <4 2 d>

where


On making use of these identities, Eq. (10.47) of the previous chapter becomes

K ( Ü - 4w2(*Jl Kite) + 4iwi(01

<!> = 2(7c-ya)-2(pc-pa), (Re(S)<l)

= 2yc-2pc, ( l < R e ( S ) < l + £ )

feaSr(kc)2 I * fte Γ (fcaS)2"]*

and P„ = K-<«)% ί. = "(ÌP«)% 4 = «S"*

Pc = l(-ic)%, ie = "(iPc)% 4, = qtS~%

c = ft + a

Equation (43) is not restricted by the condition h/a <ξ 1. By carrying out the above integrations and a certain amount of algebra, this mode equation may be written (if z = C2)

ι

where

and

. l o g [ ^ ) - f f>1 + | logoff! + Φ - 2 η * - 0 (44)

fâ% z _ _ / M % z + (2/t/a) + (h2/a2) '■" U j (1-2)%' ' « " \2} ( l-z)%

+Va-)[--va-)-coS-.(^g)]}

7("?*$-*-*-"(£B3)] = 2fca

2ka(Z+(2fo/a)+(/,2/a2)J/Q • -5 : (Re (z) < 0).

j 1 — z


The (complex) inverse cosines above are made unique by requiring them to vanish as their arguments approach unity. Since the arguments are in fact close to one, an infinite series representation convenient for this range is obtained by considering cos_1(l — u) where \u\ is small. Now

π - π f1"" di

cos-(1_M)=__sin-(1_M) = _ _ j o _ _ _ so

cos" 1(1 M)_2 Uova-*2) .Lvd-d".Lva- ' 2

) Introducing a new variable of integration by means of the relations = 1 — t, one gets

cos 1 ( 1 - M ) = - - T - ^--τζ \ w U - T dw

Expanding the right-hand integrand into an infinite series (using the binomial theorem) and integrating term-by-term gives the desired result:

in Λ m J , , * , (1X3) 2 ■ aX3X5) 3 ^ I ( 1 -") = ^(2"Η1+Π(3Χ^"+2-!(5χ2Τ" +3!(7Χ2Τ" H (45)

This result is now used to obtain infinite series for

c o s - y ( l - z ) and c o s " 1 ^ 1 ~?) v \l+hla)

Expanding ,/(l — z) into an infinite series (again using the binomial theorem), one has

c o s - V ( 1 - z ) = c o s _ 1 ( l - « 1 ) ( 4 6 a )

where u - 1

z+ 1 Z2 , (1X3) 7 3 , (1X3X5) 2 4 , afths

Now V ( l - z ) / ft/α \ 1 [ft ft 1

if^H^1-^ (47a)

Μ' = Η Γ 6 + Μι) (47b)

Thus

cos

where


and

When the ground is perfectly reflecting, q = 0; when the ionosphere is perfectly reflecting, qt = oo. The mode Eq. (44) then becomes

Again, real solutions of (48) are desired since the modes propagate without attenuation when the boundaries are perfectly reflecting. For real arguments,

where the inverse tangents are continuous functions of / such that

_.[V(0)1 π Λ _ ,1X0)1 π tan x\-i±\ = —r and tan ' U r = +7

Lu'(0)J 6 L«(0)J 6

The mode equation for perfectly reflecting boundaries can now be written as

t a n - 4 ^ 1 - t a n - r a + i O - „ n = 0 (51) lu(tc)] Lw(OJ

where ta, tc and Φ are defined after Eq. (44). The solutions of the mode equation (32) should be a fair approximation to

the solutions of the mode equation (48) above. With these solutions as a guide, a single application of the "method of false position" has been used to obtain a further approximation for n = 1 when h = 60, 100 km. These results are shown in Fig. 6. As expected, they differ but little from those obtained by solving the mode equation (32).

Newton's method may also be used to solve the general mode equation (44). It is rewritten

F(z) = 0 (52) where

KO«) - alitai] F(z) = i log ^ ^ f f j] + " o , [ ^ + f f^l +Φ(ζ)-2η, (53) and ta, tc, Φ(ζ) are defined following (44). Then according to Newton's method, if z0 is an approximate root of (53), a next approximation ζγ is given by

z1 = z 0 — Δζ where

F(z0)

Characteristics of the Modes for V.L.F. Propagation

0.010

0.008

0.006

0.004

0.002

o 0.000 >

-0.002

-0.004

-0.006

|q = 0 ; q,· s~cöj

-0.010

jSOLID CURVE: RESULTS USING MORE ACCURATE FORM (51) OF MODE EQUATION

DASHED CURVE: RESULTS USING APPROXIMATE FORM (32) OF MODE EQUATION

V ^

V

J(Po/

I I I I 1

8 12 16 20 24 28 FREQUENCY, kc /s

FIG. 6. Comparison of 2 methods of calculating phase velocity.


Differentiating (53) with respect to z and using the relation

wi(0 w2(i) - W l(0 w'2{t) = li (54) gives*

„, Λ 2(fca/2)% ( (t. - 42)(1 - iz) * ( * ) = ,- —, 134 (1 - z)(l - z)% l [wl(0 - qw^mw'ziQ - qw2{tj]

2 / 2h h2"

. , , x . ^ — , . ^ ,,. v .—T—TTT^I dz (55)

where dO

dI~'vlV(

[wi(ie) + 4iW Milite) + i4w2(ie)]J

- l°(vö^j h" ' (τ^τ) -cos"V(1 - z ) ] ] (56a)

1 / 2Λ /i2\3/2

6. THE HEIGHT-GAIN FUNCTIONS

It is seen from Eq. (17) that the height dependence of the field is described by the function

f(t„ y) = w ^ -y) + A{tn) w2(tn - y)À (57)

Making use of the modal equation (16) and the definition for B(t„)9 it is seen that

f(tH, y) = Wi(iB - y)

At y = 0, this reduces to li

yv'iOn) - With ■à] „)J

w2(i„ - y) (58)

/ ( ' „ , 0 ) = — — — (59) w2(t„) - qw2{t„)

where use has been made of Eq. (54). By definition the height-gain function G„{y) for the /j'th mode is

* In writing Eqns. (56) it is assumed that q and qi are independent of z.

This function describes the behavior of the field as a function of height for a given mode. It may be expanded about y = 0 in the manner

co m r d m -i

Then, on making use of the boundary condition

-£/('„, JO] +«/(*., 0)=0 (62) ay y=o

and the differential equation

d2f(t,y) ay2

one finds that

-(t-y)f(t,y) = 0 (63)

Οηω^Σ^-Λ-ψΒ», (64) m m ·

where the coefficients are given by

B0 = 1 , Bl = q, B2 = tn

B3 = 1 + ί Β ί , Β4 = ίπ2+2ς[

£5 =4i„ + iw2tf, Β6 = (ί»+4) + 6ί(Ιί

*7 = 9ί2+(ίη3 + 10)<ζ £8 = (£ + 280 + 12**«

2*9 = (16^ + 28) + (tf + 520« *1 0 = (i„5 + 100*2 ) + (20i3 + SO)q * n = (25tt + 280O + (fn5 + 160i2)«

and so on. Remembering that y = (2/ka)i/3kz and q = -i(ka/2)i/3A the expansion can

be written in the form

Gn(y) = i + iAkz + 1-^j%tK(kz)2

" i f 1 - ' (y ) / 3 A i " l ( f c z ) 3 + t e r m s in ( / c z ) 4' ( k z ) 5'e t c · (65)

Another useful form is obtained by noting that

(ka\<

and therefore

Gn(y) = 1 + iAkz - ^ψ- - I[JL + i^\{kz? + ... (66) Since

and

Kef <° one finds, for small departures from unity, that

Δ^|Δ | β ^ = |Δ|(!±ί)

ι^ι^-^-Γί-Ή2

For propagation over sea water the second term is usually negligible. In this case, it is rather interesting to note that if the phase velocity vn is greater than c the field initially decreases with height z. On the other hand, if the phase velocity is less than c the field initially increases with height.

The change of |(/„Ο0| with height z is not marked. For example, at / = 20 kc/s, h = 80 km, ωΓ = 2 x 105, and ag = 4 mho/m the value of {vjc — 1) for n = 1 is found from Fig. 3a as —3.3 x 10"3. At a height z of 10 km we find that

\Gn\ - 1 s 3.4 x 10"2

or approximately 3£ per cent which is hardly measurable. However, for higher heights the effect becomes noticeable. For example, at z = 20 km

K | - 1 = 0.14

which is certainly measurable. The height-gain function would continue to increase for still greater heights, however, for calculation it would be necessary to use the complete expansion by Eq. (64).

It is also of interest to note that when \Gn\ is near unity that

phase of Gn ^ !—L-— a„kz2

where a„ = k Im C2/2 is the attenuation (in nepers per unit length) for the rt'th mode. Thus, to a first order, the phase of the height-gain function is directly related to the attenuation of the mode.

It is possible that a study of the height-gain function could lead to a direct measurement of phase velocity and attenuation for a given mode. In such a scheme it would be necessary to probe the field to fairly great heights.

7. THE EXCITATION OF V.L.F. MODES

An important consideration in any application of mode theory is the degree to which the modes are excited. To examine this question we shall rewrite Eq. (26) in the following form

A(nx\Vz

V = ±2- e-""4 Σ «Φ[- txQGJWi. (67) yo n

where

K = ^\(tn~q)~l^tn-yo) + qi^tn-y0)^ ) (68)

and use has been made of Eq. (54). A more familiar form is obtained by replacing x9 tn and y0 by their usual equivalents. Thus

V = 2W (69) where

w = (im e"m ?exp [ikd Ψ\Qn{y)K (70)

and d = αθ is the great-circle distance. The similarity of Eq. (70) with Eq. (3.28) of the preceding chapter is worthy of note. Here it has been assumed that the source dipole is on the surface. Actually, the more general form of W for the spherical problem is simply obtained by replacing G„(y) by G„(y) Gn(y') where

and z0 is the height, above ground, of the source dipole. Thus, in general

w=m- e " / 4 .1 e x p r Y H G ^ > A » <71> The radiation field Ez is then obtained from

E2 = E0W (72) where

E0 = ι(ηΙΧ)(Ι ds/d) exp(- ilnd/X) is a standard reference field for the electric dipole of moment Ids.

The excitation factor is defined explicitly by Eq. (68). To discuss its behavior, it is assumed that q = 0 and ql = oo, corresponding to a perfectly conducting ground and a perfectly reflecting ionosphere. In this case,


8 10 12 14 16 18 20 22 24 26 28 30

FREQUENCY, k c / s

FIG. 7. The excitation factor of the first mode expressed in decibels.

Now, if - tn = (kaß)V3Cl > 1, the Airy functions can be replaced by the first term of their asymptotic expansions. Thus, in an asymptotic sense

yo_ 2tn

Then, if

1

-HI (74)

y0 _ 2h t„ üLsn

— = — <!

A ~ 1

The latter limiting condition is equivalent to assuming that the earth is flat. The value of Λ„ is a measure of the relative efficiency of launching a mode into the earth-ionosphere waveguide. For a flat earth with a finitely conducting ground and ionosphere, it is related to the coefficient <5„ of the previous chapter b y

An = <U?(0) = <5M-T2^ = ^ (75) 'CM2-A2

+ 4

-10

n 1 1 1 1 r "i 1 r

h= 70km

JL. \ _J l L 8 10 12 14 16 18 20 22 24 26 28 30

F R E Q U E N C Y , k c / s

FIG. 8. The excitation factor of the second mode expressed in decibels.


where

δ e* ίΔ ΐΔ;

+ kh(Cî - Δ2) + kh(C2n - Δ?)

For v.l.f. waves in the frequency range of 8-30 kc/s the latter form is of no practical use since earth curvature cannot be neglected for low order modes.

The excitation coefficient Λπ (expressed in decibels) for the curved earth (with q = 0, q{ = oo) is shown in Figs. 7 and 8 for n = 1 and 2, respectively, with heights h ranging from 60 to 100 km. In this particular case, since tn is real, Λη is also real. For heights of the order of 70 km and frequencies less than 16 kc/s, the excitation factors do not depart significantly from unity. However, for higher frequencies or greater heights, the excitation of the first mode becomes relatively weak.

8. DISCUSSION OF THE EARTH DETACHED MODE

The v.l.f. mode of lowest attenuation may not be easily launched when the phase velocity vt is less than c since Λ may then be quite small. When this situation prevails, it is also found that the height-gain functions tend to increase with height as indicated by Eq. (66) et seq. Also, under these conditions, the phase velocity of the first mode does not depend significantly on ground conductivity as evidenced by the curves in Fig. 5a. A mode with these features can be imagined as becoming "detached" from the lower boundary of the spherical earth-ionosphere waveguide.

The gross features of the "earth detached mode" are illustrated by noting that the full mode equation (27) can be approximated by

"iC-JO) -i2nn w2(t - y0)

S e-12™ (76)

provided Ms a positive quantity somewhat greater than unity and |^f| is very large compared with unity. Then, if in addition, t — y is sufficiently negative, the above becomes simply

e x p [ - ίπ/2] e x p [ - i$(y0 - i)3/2] = e x p [ - ilnri] (77)

This is equivalent to

2ka Γ 2 2 f t p π „

or 2k j (c2 + — ) Λ dz + ϊ = 2π« (79)


where i = -aC2/2. The latter form is related to the "phase integrals" discussed in Chapter IV. The condition states that electric length of a "to-and-fro" path from h to è is Inn — π/2 rad. The π/2 can be associated with a — π/2 phase change at the "reflection level" z = Î and a π phase change at z = h. If £ is somewhat greater than zero, corresponding to C2 being negative, the "reflection level" is above the ground i = 0. The waveguide in this case can be imagined as a channel extending from z = Ì to h and thus the term "earth detached mode" is indeed appropriate.

The phenomenon described above is very similar to the whispering gallery effect for sound waves analyzed in considerable detail by Lord Rayleigh (1910,1914). A striking example is the dome of St. Paul's cathedral. An audible sound will travel around the inside of the dome with exceptionally low attenuation.

REFERENCES

BREMMER, H. (1949) Terrestrial Radio Waves Elsevier, New York, N.Y. and Amsterdam. FOCK, V. A. (1945) Diffraction of radio waves around the earth's surface, / . Phys. U.S.S.R.,

9, 256. RAYLEIGH, LORD (1910) The problem of the whispering gallery, Phil. Mag. 20, 1001-1004. RAYLEIGH, LORD (1914) Further applications of Bessel functions of high order to the

whispering gallery and allied problems, Phil. Mag. 27, 100-109 . SPIES, K. P., and WAIT, J. R. (1961) Mode calculations for v.l.f. propagation in the earth-

ionosphere waveguide, Nat. Bur. Stand. Technical Note No. 114. (Available from Office of Technical Services, U.S. Dept. of Commerce, Washington 25, D.C., price $1.50).

VAN DER POL, B., and BREMMER, H. (1937) The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, Phil. Mag., 7, 24, 141.

WAIT, J. R. (1960) Terrestrial propagation of v.l.f. radio waves, a theoretical investigation, / . Res. Nat. Bur. Stand., 64D, 153. (Note the inequality at the middle of p. 172 should read {kaßY^O 1.)

WAIT, J. R. (1961) A new approach to the mode theory of v.l.f. propagation, / . Res. Nat. Bur. Stand., 65D, No. 1, 37-46.

Added in Proof BUDDEN, K. G. (1962) The influence of the earth's magnetic field on radio propagation by

wave-guide modes, Proc. Roy. Soc. A 265, 538. BUDDEN, K. G., and MARTIN, H. G. (1962) The ionosphere as a whispering gallery, Proc.

Roy. Soc. A. 265, 554. WAIT, J. R. (1961) A diffraction theory for l.f. sky-wave propagation, / . Geophy. Res., 66,

No. 6, 1713-1724.

Chapter VIII

PROPAGATION IN STRATIFIED MAGNETO-PLASMA MEDIA

Abstract—It is the purpose of the present chapter to consider the interaction of electromagnetic waves and plasma for a special class of two-dimensional problems [Wait, 1961]. The geometry is chosen initially so the wave propagation is essentially transverse to the magnetic field. Such a restriction permits a straightforward and exact solution of a relevant boundary value problem to be obtained. When the magnetic field is not transverse to the direction of propagation the formulation is a great deal more complicated; however, approximate and useful results for such cases are considered in a relatively simple fashion. Finally, the results are used to demonstrate the importance of the terrestrial field in v.l.f. propagation.

1. INTRODUCTION

A neutral mixture of positive (and negative) ions and electrons is often described as a plasma. For example, flames, gaseous discharges, strong shock waves, and the ionosphere are various kinds of plasma. The behavior of electromagnetic waves within and in the vicinity of plasma is of great current interest.

Mainly, because of the presence of the free electrons, the plasma is a dielectric. The collisions of electrons with the molecules and ions cause dissipation of energy and thus the dielectric is lossy. For this reason, the iono-osphere can often be represented as an isotropie conductor as we have done in previous chapters. However, in the presence of a steady magnetic field, the plasma becomes anisotropie so that the dielectric constant is of tensor form and thus, in general, propagation is not reciprocal as previously assumed.

An excellent introduction to the theory of propagation of electromagnetic waves in plasma has been given recently by Whitmer [1959]. A more comprehensive treatment is found in a recent monograph by Ratcliffe [1959]. The dielectric behavior and the molecular properties of a plasma have been dealt with by a number of authors [Chapman and Cowling, 1939; Cowling, 1945; Compton and Langmuir, 1930; and Cravath, 1930]. In particular, Spitzer [1956], has given a very thorough discussion for fully ionized gases.

226

Propagation in Stratified Magneto-Plasma Media 227

2. THE DIELECTRIC PROPERTIES OF A PLASMA

Since the problems to be discussed in the following sections deal with two-dimensional geometry, it is desirable to take the z axis of the Cartesian or the cylindrical coordinate system in the direction of the applied magnetic field H0. In this case the dielectric displacement D is related to the electric field E by the relation

D = W (1)

where (ε) is the tensor dielectric constant. For an implied time factor exp(/co/), it has the form

(ε' -iq 0

iq a' 0 | . (2)

0 0 The quantities ε', ε" and q are functions of the density of the electrons and the ions and the frequency of collisions between them. They also depend, of course, on the strength of the applied magnetic field H0 (see Appendix A).

The case usually considered is when the electromagnetic forces only influence the electrons. Furthermore, the motion of the ions is commonly neglected. For this situation, the properties of the plasma can be approximately described in a macroscopic sense in terms of the following quantities :

ω0, the (electron) plasma frequency ωΓ, the (electron) gyro frequency v, the effective collision frequency (for electrons).

The elements of the dielectric tensor are then given explicitly by

a' i(v + iû))a>o/co ε0 ω\ + (v + ίω)

q _ —ωτωΙΙω a0 ω\ + (v + ίω)2

= î - ;ν;τ 7 ^2 (3)

(4)

! ! _ ! _ * ! * (5) ε0 (ν + ιω)ω

The preceding can be generalized to include the influence of heavy ions by simply adding a summation prefix to the ratios on the right-hand side of Eqs. (3), (4), and (5). Then, in each term, the appropriate value of ω0, ν, and coT must be employed. This approach is valid for a weakly ionized medium and has been employed by Hines [1953]. In the case of a plasma consisting of a neutral mixture of electrons, one type of heavy ions and a relatively large

number of neutral molecules, the elements of the dielectric tensor have the form

ε^ _ i(ve + ΐω)ωΙ/ω _ i(vf + iœ)fiœ20lœ

ε0 ~ G>T + (v* + ico)2 fi2(o\ + (Vf + iœ)2

ωτωΙ/ω β2ωτωΙ/ω œ\ + (ve + ιω)2 fi2œ\ + (vf + iœ)2 (7)

ü! = ! _ ί ωο *#ωο / 8 )

where me mass of electron mt· mass of ion

and ve and vf are the collision frequencies of the electrons and the ions, respectively, with the uncharged molecules. The effect of collisions between the electrons and the ions is neglected. ω0 and ωτ are the (electron) plasma and (electron) gyro frequencies. Since the charge of the ions is equal and opposite to the electrons, it follows that fiœQ and —fiœT are the (ion) plasma and (ion) gyro frequencies, respectively.

It can be seen that if vt <ζ co, the denominators in the second factors for ε' and q behave as fi2œ\ — ω2. Thus, at frequencies near the (ion) gyro frequency, the influence of the ions may be significant even though the mass ratio fi is very small. However, when collisions are not negligible, the effect of the ions is usually quite small. In particular, if v, > ω and ve > ω, this fact can be demonstrated by rewriting the elements of the dielectric tensor in the form

f:=1_^r1+^^i±zü (9)

ε0 ω | + v\l * Ρ?ω\ + vf\

£ _ ! _ , £ * [ ! + 1 ^ 1 (11) ε0 veœ L Vi J

Because the ratio (ω£ + νΙ)/(β2ωγ + vf) is of the order of unity in most cases of practical interest, the square bracket terms may be replaced by unity since fi is of the order of 10"4.

When the plasma becomes strongly ionized, the situation is more complicated. The formalism is given in Appendix A where the relative number of ions and electrons is unrestricted. However, in the case of a fully ionized

plasma, the elements of the dielectric tensor are given by [Ginzburg, 1948; and Fainberg and Gorbatenko, 1959].

Û>O(1 + β)ίω2 - βωτ - ϊων(1 + #)] ε0

9 ε0

ε" ε0

<1 +

ωωτωΙ(β2

1 -

Δ

ω20

ω[ω -

Δ

- ΐ )

(1 + Α) •Μΐ + Α)]

(12)

(13)

(14)

where Δ = ω2ω2

τ(ί - β)2 - [ω2 - /toj - ίων(1 + #)]2. As before, β is the ratio of the masses of the electron and the ion. Also, ω0 and ωτ are the (electron) plasma and (electron) gyro frequencies, respectively. In this case, however, v is the collision frequency between the electrons and the ions which have equal and opposite charge.

It can be immediately seen that if β is set equal to zero, these expressions reduce to the form of Eqs. (3), (4), and (5) which are derived under somewhat different conditions. It can also be seen that Eqs. (12), (13), and (14) have the same form as (6), (7), and (8) if v = ve = vt = 0. However, when collisions are non-negligible there is a fundamental difference between a weakly and a fully ionized plasma. This is a consequence of the coupling between the equations of motion for the electrons and the ions (see Appendix A).

Under the very reasonable approximation that 1 ± β can be replaced by 1, it readily follows that Eqs. (12), (13), and (14) can be written in the form

e^=1_i(v +iconico ε0 ω\ + (v + id)2

q ωτω201ω

ε0 ωί + (ν+ιω)2

E- = l - - ^ - (17) ε0 (ν + ιω)ω

where / ßo)j\

v = v — ΐβωτΐω = vi 1 — i 1. \ νω/

The quantity v could be described as an effective (complex) collision frequency. It should be noted that only the elements ε' and q of the dielectric tensor involve v; the remaining element ε" is not changed.

It is now evident that the motion of the ions, for a fully ionized gas, can be neglected only if

βω\ <ξ νω.


For low frequencies and/or low collision frequencies this condition may well be violated. Also, when v is comparable to ωτ the forms given by Eqs. (15), (16), and (17) may not be valid (see Appendix B).

3. THE FIELD EQUATIONS

Maxwell's equations in a source free region with a (tensor) dielectric constant (ε) and a (scalar) magnetic permeability μ are given by

i(e)coE = cxxnH (18) a n d -ifUDH=cvalE. (19)

It is desirable to write the first of these in the form ΐωΕ = (ε~ι)ο\χήΗ

where (ε"1) is the inverse of the dielectric tensor. It is not difficult to show t h a t IM -IK 0 \

Φ~ι)= \iK M 0 (20)

\ 0 0 ε0/ε'7 where

M = ·—s ? and K = — (e')2-q2 (s'Y-q2

This formula is quite general. The only restriction is that the z axis is to be taken in the direction of the impressed magnetic field. In the following it is assumed that the fields do not vary in the z direction. In terms of Cartesian coordinates, Maxwell's equations are then written

ie0œEx = M ^ + iK^ (21) oy ox

is0coEy = iKd-^-Md-^ (22)

ί 8 ο ω £ ζ = ( β 0 / β « ) [ ^ - ^ ] (23)

-ίμ0ωΗχ = Νψ (24) dy

dx -ίμ0ωΗγ= - N - ^ f (25)

[dEv dEx~\ -ίμοωΗζ = Ν { ^ - ^ \ (26)

where N = μ0/μ.

It is a relatively simple matter to eliminate the transverse component of the fields from the preceding 6 equations. Thus

and

where k = (ε0μ0)ί/2ω = 27r/wavelength. These latter two equations are valid only if the magnetic permeability and the elements of the dielectric tensor are constant for at least a given region.

The fact that Hz and Ez individually satisfy a wave equation means that any solution to our problem can be regarded as the linear combination of 2 partial solutions. In the first of these Ez = 0 and in the second, Hz = 0. Thus without any subsequent loss of generality attention can be restricted to these cases. It should be emphasized that this decomposition into independent partial fields is valid only when the derivatives with respect to z are zero. As we shall see, the solution for Hz = 0 is relatively trivial since the constant magnetic field H0 then has no influence (at least within the limits of magneto-ionic theory).

4. REFLECTION COEFFICIENT FOR A PLANE BOUNDARY BETWEEN FREE SPACE AND PLASMA

With respect to a Cartesian coordinate system (x, y9 z), a homogeneous plasma occupies the space y > 0 while y < 0 corresponds to free space. The constant and uniform magnetic field is parallel to the z axis. A plane wave with harmonic time dependence (i.e. exp(/a>*)) is incident from below as indicated in Fig. 1. The angle of incidence is Θ (measured to the negative y axis) and the wave is polarized such that its magnetic field has only a component in the z direction, denoted Hz

nc. Thus

H'znc = h0 exp(- ifcC>0exp(- ikSx) (29)

where C = cos 0, S = sin 0, and hQ is a constant. Since the reflected field HTei is a solution of the free-space wave equation

and is to have the same dependence with x as the incident wave, it must be of the form

Hf = h0R exp(ifcCj;)exp( - ikSx) (30)

where R is by definition the reflection coefficient. Q

(27)

(28)

'///////////////

if 7

W A / iV

sl ^ co/ ^7 -// Q./

Ί/////////////Λ

V \

FIG. 1. The coordinate system for reflection at a plane interface between a homogeneous plasma (y > 0) and free space (y < 0). The constant magnetic

field Ho is along the direction of the positive z axis (out of the paper).

With similar reasoning, the solution for the plasma (i.e. y > 0) must have the form

H2=f(y)exp(-ikSx) (31)

where f(y) is some function of y. Furthermore, since H2 is to satisfy Eq. (27) it follows that/(y) must satisfy

[$+kÌm-s')}^=° Solutions are of the form

expl±ikl(MNyl - S2)]1/2j>]

(32)

(33)

Since the plasma extends to y = + oo, the solution corresponding to the negative sign in the exponent is pertinent. Therefore, the transmitted wave has the form

Hz = h0T expC-i/cKMiVr1 - S2]1/2y]exp(~ifeSx) (34)

where T is by definition a transmission coefficient. The boundary conditions are that the tangential components of the fields

in the free space and in the plasma are to be continuous at y = 0. Continuity of Hz leads to

1 + R = T (35)

and continuity of EX9 by virtue of Eq. (21), leads to

C(l - R) = TlKMNy1 - S221/2M + iKSß. (36) Thus

where

C - Δ 2C R = cTJ and r = cTÄ <37> Δ = M[(MNyl - S2]I/2 + iKS. (38)

For an electron plasma where the motions of the ions are neglected, it is possible to write Δ in the following form

lc2+ïïJ4rVl^L-L2-^-,>s

(1 + iLf - γ2 ^ '

where L = (v -f- ÌCD)CD/CDQ9 and y = ωτω/ωΙ. We have also set JV = l(i.e. μ = /ί0) although a plasma may be slightly diamagnetic.

The reflection coefficient, essentially in this form, was derived by Barber and Crombie [1959] where the homogeneous electron plasma was to be an idealized representation for the ionosphere. Because of the assumption of a purely transverse magnetic field HQy the horizontal direction of propagation is along the magnetic equator. For propagation from east-to-west, S is positive, while for propagation from west-to-east S is negative, y is then a positive real quantity.

For applications at low and very low radio frequencies, v > ω so that to a good approximation

L s ω/ωΓ where cor = CÜQ/V.

Some numerical results based on Eq. (37) are available [Wait and Carter, I960].

5. REFLECTION FROM A STRATIFIED PLASMA

We shall now undertake to generalize the previous result to a plasma medium which is stratified in layers all parallel to the free-space interface at y = 0. The situation is shown in Fig. 2 where P parallel layers are indicated. The /?'th layer from the bottom is of thickness lp and its electrical properties are described by Mp, Np, and Kp. The index p ranges from 1 to P. It should also be noted that lP = oo.

Again taking the incident wave to be polarized with its magnetic vector parallel to the z axis, it is seen that the field for y < 0 has the form

Hz = ft0[exp(~ ikCy) + R exp(ifcCjO]exp(- ifcSx). (40)


FIG. 2. Reflection from a stratified plasma.

The problem is to find an expression for R which involves the properties of the individual layers. It is possible, of course, to formally extend the results for the semi-infinite case by writing the solution in each layer as a linear combination of the two elementary forms given by (33). The two unknown coefficients for each layer are then found from the two boundary conditions at each plane interface. The resulting 2P linear equations may then be solved in a straightforward but a very tedious manner for any specified but finite value of P. The resultant solution can be found in a more systematic way if the analogy with SchelkunofTs [1943] theory of non-uniform transmission lines is exploited. We use this method here.

The wave impedances for the /?'th layer are defined by

* ; HT and

K: = ==-. H:

(41a)

(41b)

The superscript + signifies that the fields vary with y according to the factor exp [ - ißpy] where

ß. = kl(MpNpyl-S2^


whereas the superscript — signifies that the fields vary with y according to the factor exp[ißpy]. In the present case, the superscript + signifies a wave travelling in the positive y direction (i.e. away from the interface) and the — signifies a wave travelling in the negative y direction.

From Eqs. (41a) and (41b) it readily follows that the wave impedances are

Κ;=η0(Μρβρ+ιΚρΞ) (42) and

Κ;=η0(Μρβρ-ίΚρ8) (43)

where η0 = (μ0/ε0)ί/2 = 120 π. The index p ranges from 1 to P. Because of the quantity Kp9 it is seen that K+ and K~ are not equal as they would be in an isotropie medium.*

The reflection coefficients at the interface between the (P — l)'th and the Pth layer are now defined by

ir— 17 —

K p - i = 7 7 T a n d ?Ρ-ι=τϊ (44)

where the field components are evaluated within the (P — l)'th layer. Thus

R - - K p - i - K f a n d r 1 / g p - i - l / g f r 4 5 .

^-'-κ^+κί and ""-'-iiKF-i + mr ( } The reflection coefficients at the interface between the (P — 2)'thand(P — l)'th layer are then

Kp-2 ~ Zp-i , _ l/i^p-2 — 1/Zp-i Kp-2 + Zp-i l/Kp-2 + 1/Zp-i

where ZP_1 is the impedance seen at the (P - lj'th interface. From analogy with transmission fine theory

where >>_! and ÄP_! are given explicitly by Eq. (44). Now, in general,

l + rpcxp(-i2ßplp) Z"-K" l + RpexV(-i2ßplp) m

so that the process may be continued until p = 1, whence

* No confusion should arise between the symbols Kp and Kp and Kp since the superscripted quantities are used only for the wave impedances.


Finally, for the bottom interface

K0 — Zi

which may be rewritten

κ° = -φττ (50)

C - Δ R-cTi (5,)

where Δ = ΖχΙη0 since K£ = KQ = >/0C. For the special case of a two-layered plasma (i.e. l2 = oo), the explicit

expression for Δ becomes

A nu R . · ^ c\ l + ^ e x p C - 1 2 ^ / 0 . . ^ Δ = ( Μ ι Λ + iKtS) l + RiQxp(_i2ßili) (52)

with

and

_ (M1j81 + i ^ S ) - (Μ2β2 + iK2S) 1 {Μιβ1 - i K ^ + (Μ2^2 + iK2S) ( }

1Ι(Μίβί + iKjS) - l/(M2j?2 + iK2S) (54)

Ι / ί Μ ^ ! - ί Χ ^ ) + 1Ι(Μ2β2 + iK2Sy

The limiting case of a homogeneous plasma is recovered by letting lx -► oo, whence

Δ = Μ1β1 4- iKtS

which is identical to Eq. (38) after dropping the subscript 1.

6. ARBITRARY INCLINATION OF MAGNETIC FIELD

The previous development of propagation in a magneto-plasma is fairly straightforward since the superimposed Magnetic field is parallel to the magnetic field of the wave.* Thus, within the broad confines of magneto-ionic theory there is no coupling between T.E. and T.M. modes. While this theory would appear to be rather restrictive it is just the transverse component of the terrestrial Magnetic field which is so important in propagation of v.l.f. radio waves to great distances. Nevertheless, it is desirable to consider the consequences of an arbitrarily dipping Magnetic field if a complete understanding of related phenomena is sought.

To simplify the discussion, the motion of heavy ions and the energy dependence of the collision frequency are neglected. The consequences of these assumptions are discussed in Appendixes A and B of this chapter. Unlike the analysis in the previous section, it is not desirable to choose a

* Note that the m in magnetic is capitalized when reference is made to the steady superimposed magnetic field to distinguish it from the magnetic field associated with the wave. This convention is adopted for the remainder of this chapter.

coordinate system aligned with the Magnetic field. Instead, an x, y , z coordinate system may be chosen quite arbitrarily. Then the dielectric tensor has the form

00 = | h* eyy h* I (55)

where, in general, the elements of the tensor are finite. Denoting the angle between the Magnetic field and the x, y , and z axes by ax, ay, and az, respectively, it follows, from Eq. (154) of Appendix A, that

! - , ! _ « ^ Γ < * - * > -rgC02

S g*1 (56a) ε0 ω(ω - ιν) |_ (ω - ivy - ω^ J

Sxy _ _ ωο Π(ω - iv)coH cos <xz - œjj cos ocx cos «y1 ε0 "" ω(ω - iv) |_ (ω — iv)2 - ω^ J

ßjcz __ ω ο Γ—ι'(ω — ί'ν)ωΗ cos ocy — ω^ cos αχ cos a J ε0 "" ω(ω - iv) [_ (ω - ÏV)2 - ω | J

and so on. If the Magnetic field were aligned with the z axis, cos ocx = cos OLy = 0 and cos az = 1, (ε) reduces to the skew-symmetric form given by Eq. (2).

Maxwell's equations can again be written

curl#=i(e)co£ (57) and

curl E = - ißmcoH. (58)

By eliminating H one can obtain a general equation for the vector electric field E in terms of tensor dielectric constant (ε) and the (scalar) magnetic permeability /im. To simplify the problem it is assumed that for a given region (e) and μΜ are constant. Also, without serious loss of generality, the Magnetic field is taken to be parallel to the y-z plane. Thus, cos ccx = 0 and cos ocz = sin a r We seek solutions which are proportional to exp(+//i/:z) where μ is a refractive index* and k = 2π/(wavelength). Consequently, the direction of propagation is the direction of the z axis. Then, from the second of Maxwell's equations

QiJe0)*Hx=TiiE, (ΐ^εο)*Η,= ±μΕχ

H2 = 0. (59)

* The symbol /χ should not be confused here with the magnetic permeability which, in this section, is designated by /im.


Introducing this into the first of Maxwell's equations, one readily finds that

(1 - μ2 - P)EX - iQ cos a(l - μ2)Εγ + iQ sin aEz = 0

- iQ cos a(l - μ2)Εχ - (1 - μ2 - P)Ey + 0 = 0

- iQ sin a(l - μ2)Εχ + 0 + (1 - P)EZ = 0 (60) where

P= ω \ , ρ= ω» ω(ω — iv)' ω — iv

and oc(= az) is the angle between the direction of propagation and the Magnetic field.

In order that there be a non-trivial solution for the fields, the determinant of the coefficients of Ex, Eyi and Ez, in the preceding set of equations, must vanish. This leads to a formula for the refractive index

2 = 1 2P(1 - P) μ 2(1 - P) - Q2 sin2 α ± {β4 sin4 a + 4Q2(l - P)2 cos2 a } * V j

This result, when written in more conventional notation, is the Appleton-Hartree formula, given by

M2 = 1 ^ —, ^τπ r (62) ^r /2 , f Fi/4

where

and

1 - ' z -r^Yr^±l( i - / - ,^ + r

ω ω ω

y r = sin a. ω

This equation for μ is the basis of the magneto-ionic theory which is the subject of an excellent monograph by Ratcliife [1959]. Its properties have also been discussed extensively by Budden [1961].

It is apparent that μ is double valued and thus two upgoing and two down-coming waves are possible. An important related quantity is the polarization ratio which is defined by

for the present choice of coordinate axes. From Maxwell's equations it may be shown that

p2 UlE + 1 = 0 (64)

and its solutions are

P - 2yL(i -x - iz)± {AYKX -x - izf + l] (65)

as indicated by Budden [1961]. Two important special cases are "longitudinal" and "transverse" propaga

tion. In the first of these YT = 0 and thus

whereas for the second case, since YL = 0, we find

/ * 2 - l - j ^ J 2 , P = ° (67) or

, Λ X(l-X-iZ) " '^- ( i -KXi- i r -^-y" p=0°· ( 6 8 )

It is seen that, for purely longitudinal propagation, the waves are circularly polarized and the refractive indices are different and both depend on the Magnetic field.

For transverse propagation the waves are linearly polarized. In the first case the electric vector of the wave is parallel to the Magnetic field whereas, in the second case, the magnetic vector of the wave is parallel to the Magnetic field. This purely transverse condition was adopted in the preceding section.

7. REFLECTION FROM A HOMOGENEOUS PLASMA WITH ARBITRARY MAGNETIC FIELD

We will now consider the reflection of a plane wave from a homogeneous electron plasma bounded by a plane interface. The superimposed Magnetic field has an arbitrary orientation. The solution outlined here follows the work of Bremmer. The method has also been used by Yabroif [1957], Johler and Walters [1960], and Crombie [1961] to obtain specific numerical results.

Choosing a Cartesian coordinate system (x, y, z), the ionized medium occupies z > 0 and free space z < 0 as indicated in Fig. 3. Without loss of generality, the electric vector of the incident field lies in the y-z plane. Thus, the incident field, for z < 0, is specified by the T.M. wave

Ey = C \ Ez= -S) x e - K ! e - " s ' (69)

riHx= - l j

Fio. 3. The incident, reflected and doubly refracted transmitted rays for a magneto-plasma occupying the space z > 0.

and Ex = Hy = H2 = 0. Here C = cos Θ and S = sin 0 where Θ is the conventional angle of incidence. The reflected wave, for z < 0, is now taken to be the super-position of a T.M. and a T.E. wave. These have the respective forms

Ey=-C

and

(70)

(71)

where ^R^ and ^RL are reflection coefficients. The first subscript of the reflection coefficients is chosen to indicate that the electric vector, of the incident wave, is parallel (i.e. ||) to the plane of incidence (i.e. y-z plane). The second subscript is chosen to indicate that the electric vector of the reflected wave is either parallel (i.e. ||) or perpendicular (i.e. l ) to the plane of incidence.

Within the plasma there are two components of the field, because of the double valued nature of the refractive index. For convenience, one of these is called the ordinary and the other the extraordinary. To describe these waves in the plasma, two auxiliary coordinate systems (x0, y0, z0) and (xei ye9 ze) are chosen such that z0 and ze are in the respective directions of propagation (of the wave normals) of the ordinary and extraordinary waves. Then the y0 and ye axes are chosen so that Magnetic field H lies in the (y09 z0) and (ye9 ze) planes. Now, since the phase variation of the incident wave is zero in the x direction, the variations of the transmitted waves in the plasma must also


be zero. Thus, the z0 and ze axes lie in the (y9 z) plane and they subtend angles θ0 and 0e9 respectively, with the z axis. For this development it is desirable to regard 0O and 9e as real angles.

Following the analysis of the preceding section, it is now a straightforward matter to write down the form of the solutions of the ordinary and extraordinary waves in the plasma. They are, for the ordinary wave,

- ik μοζ

EXo=-iA0R0

Ey0 = A0

Ezo = 6(1 - ^)_1(1 - /io)sin «o^o^o I x e

Hyo - - ΨοΑο^ο

where A0 is an undetermined coefficient and

- β sin2 a0 + [Q2 sin4 a0 + 4(1 - P)2 cos2 a0]1/a

(72)

* o = 2(1 - P)cos a0 (73)

For the extraordinary wave, the field components EXe, Eye etc., have the same form as the ordinary waves and are obtained by simply replacing 0 by e in Eq. (72). In this case, however,

R» = -Q2 sin2 ac - [Q2 sin4 ac + 4(1 - P)2 cos2 ag]1/2

2(1 - P)cos ae (74)

which employs the negative sign preceding the radical. Now, since the phase variations in the y directions must be the same, it

readily follows that sin θ = μ0 sin 0O = μβ sin 0e

which is nothing else than a statement of Snell's law. Thus, the directions of the z0 and ze axes are specified. The final solution is obtained by matching the y and x components of the total electric and magnetic fields at either side of the boundary z = 0. This leads to four linear algebraic equations to solve for the unknown coefficients A0, Ae, ^R^, and n^±. As shown by Bremmer [1949] the final results can be cast in the form

and

I I P I — — 2C (E0xHey — EexH0y)

C + A B

(75)

(76)

where

*-i A =

and

Εθ,:

n0,y ^β,γ

Hoty Hey

1

0

-CI

B = Εθ,χ Ee,x " 1

H0,x He,x 0

Ho,y He,y C

Here, E0x means the electric field in the x direction of the ordinary wave evaluated at the interface, etc. While the above development was carried out for real angles θ0 and 0e, the final results also hold for complex angles which is the case when μ0 and μβ are complex. An alternative method is outlined in Appendix C.

The specific forms of ^R^ and ^R± for this general case are not written out because of their complexity. The reader is referred to a number of recent papers [Yabroff, 1957; Johler and Walters, 1960; Johler, 1961], where the explicit formulae are given and numerical results are obtained for a number of interesting cases. The latter authors make use of the Booker quartic which eliminates the need of the auxiliary coordinate systems for the ordinary and extraordinary waves (e.g. see Appendix C).

A special case of some interest corresponds to vertical incidence (i.e. Θ = 0). Thus the (x0> o» z0)?iria{xe,ye, z j coordinate systems coincide because z0 = ze = z. Then, according to Bremmer [1949],

and

R = (μ<*μβ - lX*o - Re) - Q*o - μ,)(*ο + flg)cos(2jg)

IIÄJ. = -ifro - AÜ[2 + i(R0 + l*g)sin(2jg)]

(μ0 + 1)(μ. + l)(«o - Re) where ß is the angle between the x axis and Λ:0 (or xe) axis. Here

* o = - Q 2 sin2 a ± IQ2 sin4 a + 4(1 - P2)cos2 a]1/z

2(1 - P)cos a

(77)

(78)

(79)

where a(= a0 = ae) is the angle between the z axis and the Magnetic field. When α = π corresponding to a vertical Magnetic field, in addition to β = 0,

the preceding formula simplifies even further to

I lKl l — μ0μ£ - 1

0Ό + IX/*. + 1) (80)

and

A-±'(ATTO <"> where the + sign is to be used for P < 1 and the — sign for P > 1. This special case corresponds to normal incidence at the north magnetic pole when the Magnetic field is directed downwards. At the south magnetic pole, a = 0, and the sign convention for ^R± is reversed.

The extension of Bremmer's method to calculate the reflection coefficient from a stratified magneto-plasma made up of any number of homogeneous slabs is straightforward though extremely involved. However, by making use of a digital computer some important results have been obtained by Johler and Harper [1962]. Of course, when the Magnetic field is purely transverse, the impedance methods [Wait, 1961] are convenient. This special case was discussed in the preceding section.

8. DERIVATION OF APPROXIMATE REFLECTION COEFFICIENTS

Because of the greatly increased complexity of the solutions for reflection from a plasma with arbitrarily oriented Magnetic field, it is often desirable to invoke the quasi-longitudinal approximation [Booker, 1935] to the Appleton-Hartree formula for the refractive index μ. This is valid when

Y2r

2YL\ <\l-X-iZ\ (82)

where the symbols were defined following Eq. (62). It may also be written in the equivalent form

\Q sin2 al 2 cosa <\1-P\. (83)

From Eqs. (62) or (65), it is seen that μ2 can be expressed in the simplified form

»2 = 1 - ì I l7 (84) 1 — ιΖ ± YL and the polarization ratio by

ττ=±ί (85) hy

where the upper signs are to be taken together. It is evident that the approximation is particularly suitable when the field is steeply dipping and the direction of propagation within the ionosphere does not depart appreciably from the vertical.

Using the quasi-longitudinal approximation, Budden [1951] has shown how to obtain an approximate formula for reflection from a homogeneous ionosphere of the form illustrated in Fig. 3. The method is described briefly here.

The key step is that YL is assumed to be constant even though, strictly speaking, its value depends on the angles θ0 and 9e which are different in general. Furthermore, it is assumed that \iZ ± YL\ is large compared with unity. Both of these approximations become better as the frequency is lowered. Thus

rôsl-i-re±b (86) e ω

where « V - x - < ( m ω~(Ζ2+ YD* - ω(ν2 + ω2)* {* °

and YL <»L tan τ = — = —. Z v

Following Budden [1951], ωΓ and τ are assumed to be real constants independent of the angle of incidence 0. He further assumes that, in matching boundary conditions at the interface z = 0, the component of the electric fields, of the ordinary and extraordinary waves within the ionosphere, can be neglected. Again, if the direction of propagation within the ionosphere is nearly vertical, this would appear to be a satisfactory approximation.

After making the above described simplifications, Budden [1951] finds explicit expressions for yi^ and \\R± which characterize the reflected field for an incident T.M. wave. In addition, he gives the corresponding forms for ±RL and ±R^ which characterize the reflected field for an incident T.E. wave. In the latter case, the first subscript (i.e. ±) is to indicate that the electric vector of incident plane wave is perpendicular to the plane of incidence. The results, expressed in a form suitable for computation, are listed below.

A = [Ö*o + /0 (C 2 - C0Ce) + (μ0μ6 - 1)(C0 + Q C ] / D (88)

„Kj. = 2ÌC(AI0C0 - ^Ce)/D (89)

l J R„=2iC(^ 0 C e -^ e C 0 ) /D (90)

i * i = ί(μ0 + AO(C2 - C0Ce) - (μ0με - 1)(C0 + Ce)C^D (91)

D = (μ0 + /0 (C 2 + C0Ce) + (μ0μ6 + 1)(C0 + Ce)C where

and C = cos 0, C0 = cos 0O, Ce = cos 0e.

Some numerical values based on these formulas are available [Wait and Perry, 1957].

Now, for highly oblique incidence, the value of simplification can be made. For example, if \C2

C\ is small and some further

11Ä11 = -

\ _ (ihß, - l)(Cp + Ce) c (μ0 + ^e)C0Cg

ί [ (μ<*μβ + i)(Cp + ce) c

(jU0 + A*e)C0Ce

Furthermore, if |μομβ072| < 1,

Co + C.

c0ce

where

Al"

-exp(-2j8„C)

/Ό + μ* C0Ce

To the same approximation

R - fl 2 Co + C e c l 1 1 L μ0 + μ, c 0 c e J

S -exp(-2j8±C)

1 C0 + Ce

where

Also

and

ßi =

\\R1 =

1*11 =

μο + μβ c0ce

2iC c0ce

2iC C()Ce

Α^Ο^Ο ~~ μβ^β

μο + μβ

μο^β ■" / ^ e c 0

μ0 + μβ

(92a)

(92b)

(93)

(94)

(95)

(96)

(97)

It is immediately evident from the above that as Θ tends to π/2 (i.e. grazing incidence) the reflection coefficient nRn and ±RX are both approaching —1 whereas the conversion coefficients ||jRx and XR^ are both approaching zero. In this sense a sharply bounded ionosphere behaves as an equivalent isotropie medium for highly oblique incidence.

Some further simplifications are possible when the ionosphere is effectively a good conductor. For example, if

KM > 1 l4 = l - ι ·(ω»βίτ s - ΐ(ωΓΙω)&ίτ (98) μΐ^ 1 - ί(ωΓ/ω)β-'τ s - i(_œrlœ)e~h (99)

Consequently,

and

^(1 + 0(ω/ωΓ)* COS(T/2)(C2 - 1) ± 2*[1 - iœ/œ^C i J! ~ (1 + 0(ω/ωΓ)% COS(T/2)(C2 + 1) + 21/2[l + ico/co^C l '

Ä Ä - 2 ( œ K ) ^ l + QCrin(T/2) ί ìi "" (1 + <)(ω/ωΓ)% COS(T/2)(C2 + 1) + 21/2[1 + ίω/ωΓ]0 * ^ '

9. THE MODE SERIES FOR AN ANISOTROPIC IONOSPHERE

In this section the formalism for the mode theory is developed for a plane earth and an anisotropic sharply-bounded ionosphere. The geometry is thç same as in Section VI-4 where the ionosphere was assumed to be isotropie. In the present case the reflection coefficient [RtY is regarded as a matrix and written in the form

\\R\\ ±R\\ iRiY =

\\Ri JÄL. (102)

where the 2 primes are to indicate that it is a two-column matrix. The individual coefficients ^R^9 LR^9 ^Rx and ±R± discussed earlier, indicate the complex ratio of an electric field component in the wave after reflection to an electric field component in the wave before reflection. The first subscript denotes whether the electric field specified in the incident wave is parallel (||) or perpendicular (1) to the plane of incidence. The second subscript refers in the same way to the electric field in the reflected wave.

When the ionosphere becomes isotropie corresponding to a zero Magnetic field, the reflection coefficient in matrix notation becomes simply

IRiY = Ri

0 o Rì\

(103)

where Rt and Ä* are the complex scalar reflection coefficients for vertically polarized and horizontally polarized waves, respectively. The corresponding (matrix) reflection for the ground is

[KJ' = 0 (104)

The case of two successive reflections, the first from the ground and the second from the anisotropic ionosphere, is represented by the matrix

[ΛΑΓ = e*,·]- x [*,]" = ■Rullali R0l.R\\ Rg\\Rl Rg±R±

(105)


In the case of the isotropie ionosphere this reduces to

[*Λ·]'' = RgRi 0 (106)

0 R^R)] The arguments employed here are virtually identical to those of Budden who, however, assumes a perfectly conducting ground, such that

Rg=+l and Ä j = - 1 . In the previous formulation for a vertical electric dipole between the plane

ground surface and a sharply bounded isotropie ionosphere, the fields could be completely derived from an electric Hertz vector which has only a z component. Of course, if the source was not symmetrical it was necessary to introduce an additional component of the electric Hertz vector. When the upper boundary is anisotropie the single component Hertz vector is not adequate even for a vertical electric dipole source. This is not surprising since the T.M. type modes are coupled to the T.E. modes by the anisotropie boundary conditions.

Any electromagnetic field, in such a parallel plate region can be obtained from a superposition of T.M. and T.E. modes which are derived from electric and magnetic Hertz vectors. These are denoted individually by IT y and Πχ or, collectively by the matrix

[Π]'=ίΠ'Τ (107)

where the single prime is to indicate that it is a single column matrix. Π± which is a magnetic Hertz vector is often referred to as a Fitzgerald vector. Furthermore, the electric and magnetic fields can also be written as single column matrices in the manner

[£] ' = r*r k.

1

» LUT = "*« ßl

where £|l = (fc2 + grad div)nü

//|l = ίεω curl Π|, ηΗ± = (fc2 + grad div)n±

ηΕλ = — ίμω curl Π±.

(108)

(109) (110) (111) (112)

The intrinsic impedance η is introduced in the latter two equations to make Π n and Πχ of the same dimensions. The Hertz vector in matrix form corresponding to the primary excitation is then written

Γπ„ [Πρ]' =

0 (113)

R

where Πρ has only a z component Π<ρ). To match boundary conditions in the case of azimuthal symmetry it is only necessary that the vectors Ily and Π χ have a z component. The condition of azimuthal symmetry is achieved if the reflection coefficients themselves are independent of the azimuthal coordinate φ.

Formally the solution has the same form as the isotropie case if the appropriate reflection coefficients are now regarded as matrices. For example, for the space 0 < z < h the (matrix) Hertz vector

p i j ' = iÄl£[F(C)]"//<2>(fcSp) AC (114)

where

and where

[Ai] -[ti -with M = Ids

4πίεω

IF(Ç)Y = (e<fcCx + IRg]" e-<fcCz)(eifcC(h"zo) + [K,]" e"<fcC(A"Zo))

e ^ l - C ^ R J ' e - 2 ' * ' * ) * (115)

It should be noted that the denominator in the above expression is also a two-column matrix. Inverting this, following the usual rules for such operations, leads to

[F(C)]" = (eikCz + lRgY e-ikCz)(ll£)lNY{é2kCh + [R,]"}

for z0 = 0, where A2iCkh D P Ό Dh

e ~" llKll Ke J.KII K9 H R± Rg *2lCkh — ±R± Ri

and

[JV]" = r-UCkh e - λΚχ Kg ±K|, Kg

II R i Rg -UCkh

II 11 Rn

The corresponding residue series are thus given by

ΓΠ

n±

çUCpkh

p LdAldCjc=Cp #ip.

with

and G||p=/P(z)(e2iC'*',-AÎiAx)

GXp=ifP(z)RenR1.

(116)

(117)

(118)

(119)

(120)

The summation is over the poles of the integrand [F(C)]". Clearly this corresponds to the roots of the equation, Δ = 0, which are designated C = Cp.


It is understood that all quantities in the summand of Eq. (119) are to be evaluated at C = Cp. The "height-factors" fp(z) and fh

p{z) have the usual form, that is

2/p(z) = (A,)"* eiAC"2 + {Rg)v> e-'*c'*. (121) and

2fkp(z) = (Rh

gy v> eikC>z + (Rhg)i/2 e-ikC>z (122)

The above results reduce to those of Budden [1952] when the ground is perfectly conducting.

The modes excited in the waveguide can be logically grouped into two sets. The first has a T.M. (transverse magnetic) character and the second has a T.E. (transverse electric) character. To obtain the attenuation and the phase constants of these individual modes, it is adequate to consider the anisotropy as sort of a perturbation to the corresponding T.E. and T.M. modes for the isotropie case.

To simplify the discussion the ground is considered to be perfectly conducting. That is, Rg = 1 and Rg = — 1. The modal equation now becomes

(ei2'c* - ,,Α,,Χβ"«* + ,R,) + ,,ΚχχΚ,, = 0 . (123)

When mode coupling is disregarded this breaks into two equations

iRiC-l2kCh=1=e-i2*n ( 1 2 4 )

ΙΙΛΙΙ ,,R e~i2kCh = - 1 = _e-i2*m (125)

where n and m take integral values. As mentioned in the previous section the reflection coefficient for highly oblique incidence may be approximated by

„R„ £ -exp(-2/?„C) (126) and

1R1^-exp(-2/?1C) (127)

where to a first order, β{1 and ßL are independent of C. It thus follows that the first approximation (indicated by a superscript (1)) for the solutions of the modal equation are

n(n - i) kh - iß

for the T.M. modes, and

^ « ^ » - ^ ^ ( » - l ^ S . . . . ) (129)

for the T.E. modes. These have exactly the same form as when the ionosphere is assumed to be isotropie. The difference lies in the value of the coefficients /?H and ß± which are functions of the earth's Magnetic field.

C , sCi" = S h r ( » = l , 2 , . . ) (128)

A second approximation to the mode equations is obtained in the following way. The modal equation is rewritten in the two equivalent forms

l-„Ä,|e-|2fcC* = 25(C) (130) l + 1R1e-i2*c, , = 2y(C) (131)

where l l K i i K l l e

WQ=-\»ten. » (132) and

R r> ~-i2kCh i, n K l l e

JlkCh n e — iiiVii

2ΚΟ=-".«ί>" o · (133)

cp*cr=,iKn-v-::^> (m)

CjgCg»- " Γ , . · (135)

It is to be expected that <5(C) is a small quantity for the T.M. type modes and y(C) is a small quantity for the T.E. type modes. The second approximations then are obtained by replacing <5(C) by δ ( ^ υ ) for the T.M. set, and replacing y(C) by yiC^) for the T.E. set. Solving Eqs. (131) and (132) with these substitutions leads readily to

<n -j)-M(Cff>) kh - ij?„

for the T.M. mode types, and nm - iy(Cg>)

kh - ißL

These should be adequate solutions since |5 (^ υ ) | and | ν (^ υ ) | are small compared to unity for the important modes. In fact for most cases of practical interest, the first-order approximations should suffice.

To provide some idea of the character of the T.M. and the T.E. mode types excited by a vertical dipole source, the ratio of the tangential magnetic field in the two principal planes is considered. For the /7th mode, this ratio is given by

S - v , t ! ' for kp > 1 l f c n i | P

Le2iCkh_RH Κ Δ kCpdzJpy'\ (137)

Λ(ζ) > where it is understood that the reflection coefficients are to be evaluated at C = Cp. The preceding expression reduces to


which was given originally by Budden.* In general this ratio is small except near the T.E. resonance wherein the denominator becomes very small. For the important T.M. modes which are of low order, both Cp and ^R± are small compared to unity and thus the magnetic field Hp in the direction of propagation has a relatively small magnitude.

10. EFFECT OF EARTH CURVATURE

The formulation of the mode theory for a curved earth and an anisotropie sharply bounded ionosphere may be treated as an extension of the corresponding isotropie case. Again it is necessary to regard the reflection coefficients as matrices. As in Chapters VI and VII the earth is regarded as a homogeneous sphere of radius a and the ionospheric reflecting layer is of radius a + h. The ground is again characterized by a surface impedance Zg.

The local reflecting properties of the ionosphere are described by the reflection coefficient

\\R\\ 1ÄII [*,·]" =

L||K± 1^1 (139)

which is again a 2 x 2 matrix. Now, however, it is a function of the angle 0' which is the local angle of incidence. As in Chapter VII the parameter / is related to 0' by

(yo -O 1 / 2 = ( | ) , / 3 co s0 ' (140)

where y0 = (2/ka)i/3kh. The elements ^R^9 etc. of the matrix have the same form as in the corresponding planar problem discussed above. This is justified provided \(ka/2)l/3 cos 0'| is somewhat greater than unity. Thus the factor A(t), which occurs consistently in the curved earth theory with an isotropie ionosphere, is now replaced by the matrix

lA{t)Y s [RJ* exp[-i(4/3)(y0 - 0% " **/2] (141)

This is completely analogous to Eq. (20) of Chapter VII. The general mode equation for a curved earth with an anisotropie ionosphere

is written 1 - [Λ(ί)]"[Β(0]" = 0 (142)

where Vw'2(t) - qw2(t)l 0 [w\(t) - qWl(t)\

[B(i)]" = 0

Yw'2(t) - qw2(ty\ |wi(f) - qw^t)]

(143)

* [1952].

252

where

and


q=-Kkal2)'/'Ae,

4=-i(kal2)'/'Ahe,

\<Tg + isgœ/ \ Og + lEgCO/

withAj=(^±^-lf. \ ιε0ω /

Here σ0 and eg are the electrical constants of the ground. If ( — t) > 1, the ground reflection matrix can be approximated by

[ß(or = Rg 0

0 R* βχρ[ΐ(4/3)(-0% + ΐπ/2]

where "Z __, „* {-t)*-iq (-0*

^β = 7—Tv i 2 a n < l ^β = , ^1/ —.

(144)

(145)

On identifying (—t)'A with (ka/2)'/:> cos 0, Äff and /Jj can be identified as Fresnel reflection coefficients for vertical and horizontal polarization, respectively. For example,

* . -■ C-A„ C — Λ*

and Ιζ = £ - 4 In this case the modal equation may be written

\Rg(Q 0 cwxr 0 *!(C)

e - i / _ e-i2«n (146)

where

J = (4/3XJO - 0* - (4/3X-0* = ( ^ ) C ( ^ ) 3 - C3].

IfO>o — 0 > 1, or if | C21 P 2h/a, this reduces to the flat earth modal equation since C approaches C while Ϊ approaches 2kh.

As mentioned in Chapter VII, it is not usually possible to assume (—/) > 1 for v.l.f. waves and thus the more general form of the ground reflection matrix [B(t)Y must be employed. However, it is usually permissible to assume (y0 — t) ρ 1. Thus, the asymptotic form for the ionospheric reflection matrix [A(t)]" may be employed. Then on making use of Eqs. (141), (142), and (143) it readily follows that

[1 - „*„JLF. e-'/Kl - χΛχ**** e"'/] - l lÄ± 1Äl lÄfFfJÖF;e"i a / = 0 (147)


where w'2(t) - qw2(t)

'-*«-««[-ÌFÌFÌT1 and F g has the same form with q replaced by #. Only if (—t) > 1 can both Fe and Fg be replaced by unity.

The application of these results to obtain specific results is discussed in the following chapter where reference is also made to experimental data.

Appendix A

In this appendix a rather approximate theory of electric currents in a partially ionized gas is given. The gas is supposed to consist of electrons of mass me and of charge —e; and positive ions of mass mi and charge e. Since the gas must be, to a close approximation, electrically neutral, the number of electrons and positive ions per unit volume is the same; each is taken equal to N0. There are also Na neutral atoms or molecules per unit volume with mass nil indistinguishable from the positive ions. The electrons, positive ions and neutral atoms are regarded as three gases moving independently. The interaction between the electron and the ion gases is supposed to be smoothed into a continuous force. In fact only those forces are considered which result from macroscopic electromagnetic effects on charge and from friction-like action and reaction between the free charges and the uncharged background. Only mean velocities and forces are employed, and non-linearities are avoided by a perturbation treatment.

Further assumptions, adopted here, are that the velocity of the gas as a whole is zero, and that the gradients of the electron and ion pressures are zero. The removal of these latter restrictions would require that the problem be treated on the basis of magneto-hydrodynamics [Hines, 1953; Alfven, 1950]. Such an approach has been given by Spitzer [1956] for a wholly ionized gas and Cowling for a partially ionized gas [1957],

The mean velocities of the electron gas, positive ion gas, and neutral gas are denoted by ve, vf, and va, respectively. Since me <ζ mh the electrons lose, on the average, a quantity of momentum equal to their mean momentum me(\e — vu at each collision with the charged ions. If the mean time between successive collisions is vö1, then the momentum lost by electrons in collision with positive ions per unit volume is N0me(\e — vf)v0. Similarly, the momentum lost by electrons in collision with neutral atoms per unit volume is N0me(ye — ya)ve where vj"1 is the mean time between successive collisions of electrons with neutral atoms. Remembering that the momentum of the mass as a whole is zero, and since the mass of the electrons is negligible, it follows that

Νοτη^ + Ναπι·να = 0 (149) and thus ve = — avf where a = N0/Na.

Now the electromagnetic forces acting on an electron are —Q(E + μ0νβ x H0) where E is the electric field of the wave and H0 is the constant magnetic field superimposed on the system. (It is assumed that H0 is much greater than the magnetic field H associated with the wave.) The equation of motion for the electron gas may then be written

me dvjdt = meico\e

= - e [ £ ' + μ0\β x H0] - mev0(\e - vf) - meve(ve + αν,). (150)


The equation of motion for the positive ions can be obtained in an equally simple fashion. In this case, the momentum lost by the positive ions in collision with the electrons is —N0me(ye — Vi)v0 being equal and opposite to the quantity appearing above. The positive ions, in collision with the neutral atoms of the same mass, lose half their momentum relative to the neutral gas. Specifically, the momentum lost is ^Νοτηρ^Ι + α)νέ where vj~1

is the mean time between successive collisions of positive ions with neutral atoms. Therefore, the desired equation is

mf dv,/di = m 'caVf

= e[£ + μ0Ύί x H0] + mev0(v, - vf) - m^v^l + a)/2. (151)

Equations (91) and (92) may be solved for ve and V; in terms of E and other known quantities. The current density resulting from motion of the electrons and the ions is then equal to N0 e(vf — ve). Noting that the displacement current is ie0coE and the total current is ί(ε)ωΕ9 it is seen that the dielectric tensor (ε) can be obtained from the relation

i(e)coE= ιε0ωΕ + N0 e(v, - \e). (152)

Choosing H0 to be directed along the z axis, the dielectric tensor is found to have the form

(e) =liq ε' 0 ) . (153)

\0 0 e"/

The quantities ε', ε" and q are functions of me, mh ve9 vh v, N0, Na and H0. Explicit results for certain special cases are given in the body of the chapter. To simplify the notation there, the results are expressed in terms of the positive real quantities ω0 and coT which are defined by

œ^N0e2le0mi (154) and

ω Γ = -μ0Η0βΙηίί. (155)

Also, in discussing certain special cases in the text, the subscript on v is dropped.


Appendix B

A NOTE ON THE ENERGY DEPENDENCE OF THE COLLISION FREQUENCY

In the theory of ionospheric propagation of radio waves, it is nearly always assumed that the collision frequency of electrons with neutral particles is independent of electron energy. In fact, the Appleton-Hartree equations were developed on this basis. It has been suggested recently [Alpert et al 1953 ; Sen and Wyller, 1960; Phelps, 1960] that the calculation of the propagation constant for a weakly ionized medium should take into account this energy dependence. While the theory has been so generalized by Sen and Wyller [1960] and Johler and Harper [1962], it seems worthwhile to present a somewhat simplified account of the consequences of a linear dependence of the collision frequency on electron energy. Furthermore, this sheds some light on the validity of describing the electrical characteristics of the lower ionosphere in terms of a conductivity.

Allis [1956] and others have related the components of the dielectric tensor to integrals involving the angular frequency ω, the gyro frequency of electrons ωΓ, the electron plasma frequency, the electron density N9 the normalized electron energy distribution function/0, the frequency of mment-um transfer collisions of electrons with gas molecules v(w). When a constant and uniform magnetic field is applied in the z direction the dielectric constant of the ionized medium is a tensor of the form

/ε' -iq 0\

(ε) = ( iq e' 0 ) (156)

\ 0 0 ε"/ where

«' = ÌOL + βκ) q = ÜBL - eR)

ε" = εΡ. (157) In this, the dielectric constants eL9 eR and εΡ may be written in the form

sL = ε(ω + (oT) sR = ε(ω — ωτ) (158) εΡ = ε(ω)

where, according to Molmud [1959], 4π C°° u3/l of

/ ε ( Ω ) ω=σ ( Ω )=-τε^]0^κΐΩέα Μ (159)

In the above, σ(Ω) is a generalized conductivity which is a function of a generalized frequency Ω.

In the case where the electrons are in thermal equilibrium with the gas, the energy distribution of the electrons is Maxwellian and given by

f0 = (elnkT)*t-°»lkT. (153)

In the case of weakly ionized dry air, the collision frequency v(w) is now believed to be approximately proportional to the electron energy u. In fact, Phelps [1960] has shown that, for low energy electrons in nitrogen,

v(w) s 1.2 x 1 0 " 7 N ( N 2 ) M sec- 1 (161)

where u is in electron-volts/cubic centimetre. For present purposes, we will just set v(u) = au where a is a constant. Furthermore, a normalized collision frequency vx is chosen such that

Vi = akT/e (162)

where k is the usual Boltzmann constant, T is the absolute temperature in degrees Kelvin and e is the electronic charge.

Using the simplifications described in the preceding paragraph, it is seen that

This is essentially the formula given by Phelps [1960], Sen and Wyller [1960] and others. As they have indicated, the integral can be expressed in terms of the "semi-conductor integral" Ep(x) defined by

Ep(x) = - ± - f °V(a2 + X2)"1 e"a da. (164) KP-) Jo

These have been tabulated by Dingle, Arndt and Roy [1956] for integral and half-integral values ofp in the range —\ to +5. It easily follows that

σ ( Ω ) , !°?ϊ ίίΕ,/2(χ) - ixEy2(x)l (165) V l

where x = Ω/ν^ If, on the other hand, v(w) had been replaced by a constant v0, as is con

ventionally done, we would have arrived at the standard result

This can be written in the form

<τ(Ω) = ε 0 ω Γ — — (167) 1 + \p

where

ß = - and ωΓ = ^ . (168)

The parameter ωΓ occurs often in the theory of v.l.f. propagation which is usually formulated on the assumption of an energy independent collision frequency. Furthermore, at v.l.f. ω <ζ ωτ so that

β*± — . (169)

The parameter ωΓ/ν0 also occurs consistently in the presentation of theoretical results. Thus, it appears that a convenient way to illustrate the implications of the energy dependent collision frequency is to define effective values (cor)e and ße as follows

σ(Ω) = e0(œre) — ì — . (170) 1 + lße

Thus (œr)e = (Dricce(x) (171)

where

» , . - ^ (172) Vi

and

« (χ) = * £ * ( χ ) Π 73Ì

Also, it is seen that

where, as above, x = Ω./νί. Using the numerical values of E3/2(x) and Eypc) given by Dingle, Arndt and

Roy [1956], the values of <xe(x) and ße{x) are listed as a function of x and given in Table 1. It is seen that for small values of xf <x(x) asymptotically approaches the constant value of £, whereas ße(x) is asymptotically approaching the value 2x. On the other hand, for large values of x9 the respective asymptotes are 0.4 and x/2.5.

If the collision frequency were chosen to be independent of energy, the corresponding values oce(x) and ße(x) would be simply 1.0 and x, respectively. It is thus concluded that a linear energy dependence for the collision frequency is not going to lead to any essential modifications to the theory of v.l.f. propagation. In fact, most of the numerical results on the characteristics of the v.l.f. modes can be adapted directly to the energy dependent case if œr and the ratio ωτ/ν (or œjv) are given their more general meaning.

n < H

3 H

w^r^vo^vo<s*-H

r*-ooooooorn»-<f^Ti-fÔN

r--oovoTj-^HO

oQO

oor«i«n«OTi·

^^fOT

tr^rn<stô

^^'Ôo

o«o

inrtT

tcnT

-HO

Nr^-«o

cocN

OO

NO

ot ,^vo

«OT

j-(N

OO

OO

^r^

cn

»o

vo

r^<

»O

N^

cn

«o

r-ON

rnv

oO

Tfo

ofS

vo

oN

cn

r-^H

«o

ON

^

VO

\0^

mT

tNO

\Mn

fON

OO

Oh

\ûiO

'tf<1

NM

-H^

^^

HO

OO

OO

O

öö

öö

öö

öö

öö

dö

öö

öö

öö

öö

öö

öö

oö

öd

öd

ö

oo

qq

^N

Tf

ô

oo

ri^

q«

no

>o

oo

oq

qq

oo

oo

oo

qq

q O

OÒ

Ò6

dd

ÒÒ

«'H

FH

(SN

fOfn

Tt^

\o^

oo

o\O

^N

fnT

j:io

vo

t^Ò

Appendix C

APPLICATION OF THE BOOKER QUARTIC TO CALCULATION OF REFLECTION COEFFICIENTS

Although there is no reason to doubt the validity of the reflection coefficients given by Eqs. (75) and (76), their derivation based on Bremmer's method does require the use of auxiliary coordinate systems. The geometrical meaning of two right-handed systems is not clear when the angles Qe and θ0 are complex which is certainly the case when the collision frequency v is finite. A more attractive method is based on the quartic equation introduced into magneto-ionic theory by Booker [1939].

To illustrate the method, the sharply bounded homogeneous model is used. The relevant coordinate system is shown in Fig. 4. The homogeneous

FIG. 4. Coordinate system for the sharply bounded homogeneous ionosphere.

plasma occupies the space z > 0. A plane wave is incident from the free-space region below and a typical field component is described by

F =/exp[— ik(x sin ßt sin af + y sin ß{ cos a,)] exp[— ikz cos /?,] (175)

where fis independent of JC, y\ and z, and where <χ( and /}, are angles. Thus, the direction cosines of the wave normal are sin β{ sin α,, sin ßt cos a, and cos ßt. A field component of the reflected wave has the same form except that the latter exponential factor is exp[+ ikz cos /?,·]. The steady Magnetic field is again taken to be parallel to the (yz) plane and thus the direction cosines of the H0 vector are 0, sin j3, cos ß.

Within the plasma, the field components must have the form

G = g exp[— ik(x sin ßt sin o + y sin ßt cos oc<)]exp[— iDfcz] (176)


where D is a complex constant which is chosen so that Im D > 0 to assure attenuated waves as z -► oo. It is clear that if the medium is bounded, terms containing exp(+iDkz) must also be included.

Now, Maxwell's equations in the electron plasma may be written ΟΙΓ1£ ·= -ίμηωΗ (177) curl H = - Ney + ΐε0ωΕ (178)

where μη is the magnetic permeability, e is the electronic charge, v is the drift velocity of the electrons, N is the electron density, and ε0 is the dielectric constant of free space. Also, the equation of motion for the electrons is

icomy = — ε£— mvv — μ„(βγ χ H0) (179) where H0 is the steady Magnetic field. On making use of the form of Eq. (176), these latter three equations can be used to obtain three linear equations involving Ex9 Ey9 and Ez within the plasma. In matrix form this set reads

1 - a 2 -

aTD +

s -h* s(s2 - ft2)

ihT

aTD — ihT

52"^P aLD +

hThj

aLD +

s2-h2

s(s2 - ft2)

1-al-D2-

araL +

s - f t 2

- f t 3

aLaT - -j

\-D2-a\-

s(.s2 - ft2)

- f t 2

hj

s(s2 - ft2)J E,

=0

(180)

where s = (1 - iZ)IXt ft = YIX, hL = h cos ß,hT = h sin $

X = Ne2lme0co2 = ωΐ/ω2 = (normalized plasma frequency)2

y = /t0ei/0/oom = normalized gyro frequency Z = ν/ω = normalized collision frequency

aL = sin ßi cos oc,, aT = sin ß, sin a,·, a = sin /?,. In order that E be finite the determinant of Eq. (180) must be zero. Thus

b4D* + b3D3 + b2D2 + b^D + b0 = 0 (181) where

b4 = s(s2 - ft2) -s2 + ht b3 — 2aLhLhT

b2 = 2s[(l - a2)A2 - (s - l)(s - a2s - 1)] - Aftl - a2) - (2 - a2)h2L

i>i = -2(1 - a2)aLhLhT

b0 = (s- 1){[(1 - a2)s - l ] 2 - (1 - a2)2h% cos2 a,} - ( 1 - a2)[(l - a2)s - lftft2- sin2 a, + ft£) (182)


The four complex values of D obtained from this quartic represent the two upgoing and two downgoing characteristic waves which can propagate independently in the medium. Values of D with positive real and negative imaginary parts represent upgoing waves.

The fourth-order equation in D given above has the same form as the Booker [1939] quartic. As soon as the complex values of this quantity are known, the determination of the reflection coefficients is quite straightforward since explicit forms for the field components tangential to the boundary are easily obtained. For details the reader should consult papers by Johler and Walters [1960], Crombie [1961], Field and Tamarkin [1961], and Johler and Harper [1962].

REFERENCES

ALFVEN, H. (1950) Cosmica! Electrodynamics, Oxford University Press. ALLIS, W. P. (1956) Handbuch der Physik, 21, 413, Springer-Verlag, Berlin. AL'PERT, IA. L., GINZBURO, V. L. and FEINBERG, E. F. (1953) The Propagation of Radio

Waves, Moscow. BARBER, N. F., and CROMBIE, D. D. (1959) v.l.f. reflections from the ionosphere in the

presence of a transverse magnetic field, / . Atmos. Terr. Phys., 16, 37. BOOKER, H. G. (1935) The application of the magneto-ionic theory to the ionosphere,

Proc. Roy. Soc, Sen A, 150, 267. BOOKER, H. G. (1939) The propagation of wave packets incident obliquely on a stratified

doubly refracting ionosphere, Phil. Trans., Α-237, 411. BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier, Amsterdam, Netherlands. BUDDEN, K. G. (1951) The reflection of v.l.f. radio waves at the surface of a sharply bounded

ionosphere with superimposed magnetic field, Phil. Mag., 42, 833. BUDDEN, K. G. (1952) The propagation of a radio atmospheric II, Cavendish Lab., Cam

bridge, 43, 1179. BUDDEN, K. G. (1961) Radio Waves in The Ionosphere, Cambridge University Press. CHAPMAN, S., and COWLING, T. G. (1939) The Mathematical Theory of Non-Uniform

Gases, Cambridge University Press. COMPTON, K. T., and LANGMUIR, I. (1930) Electrical discharge in gases, Rev. Mod. Phys.,

2, 124. COWLING, T. G. (1945) Electrical conductivity of an ionized gas in a magnetic field, with

applications to the solar atmosphere and the ionosphere, Proc. Roy. Soc, A-183, 453. COWLING, T. G. (1957) Magnetohydrodynamics, Interscience, New York and London. CRAVATH, A. M. (1930) The rate at which ions lose energy in elastic collisions, Phys. Rev.,

36, 248. CROMBIE, D. D. (1961) Reflection from a sharply bounded ionosphere for v.l.f. propagation

perpendicular to the magnetic meridian, / . Res. Nat. Bur. Stand., 65D (Radio Prop.) 455-464.

DINGLE, R. B., ARNDT, D. and ROY, S. K. (1956) Semiconductor integrals, Appli. Sci. Res., B6, 155-164.

FAINBERG, Y A . B., and GORBATENKO, M. F. (1959) Electromagnetic waves in plasma situated in a magnetic field, / . Tech. Phys., U.S.S.R., 29, No. 5, 549.

FIELD, E. C , and TAMARKIN, P. (1961) v.l.f. ionospheric reflection coefficients, / . Geophys. Res. 66, 2737-2750.

GINZBURG, V. L. (1948) The Theory of Propagation of Radio Waves in The Ionosphere [in Russian], Gostekhizdat.

HINES, C. O. (1953) Generalized magneto-hydrodynamic formulae, Proc. Camb. Phil. Soc, 49, Pt. 2, 299-307.

JOHLER, J. R. (1961) On the analysis of l.f. ionospheric radio propagation phenomena, / . Res. Nat. Bur. Stand., 65D (Radio Prop.) 507-530.


JOHLER, J. R., and HARPER, J. D. (1962) Reflection and transmission of radio waves at a continuously stratified plasma with arbitrary magnetic induction,/. Res. Nat. Bur. Stand., 66D (Radio Prop.) No. 1, 81-99.

JOHLER, J. R., and WALTERS, L. (1960) On the theory of reflection of l.f. and v.l.f. radio waves from the ionosphere, / . Res. Nat. Bur. Stand., 64D (Radio Prop.) 269-285.

MOLMUD, P. (1959) Langevin equation and the a.c. conductivity of non-Maxwellian plasmas, Phys. Rev., 114, 29-32.

PHELPS, A. V. (1960) Propagation constants for electromagnetic waves in weakly ionized dry air, / . Appi. Phys., 31, 1723-1729.

RATCLIFFE, J. A. (1959) Magneto-Ionic Theory and Its Applications, Cambridge University Press.

SEN, H. K., and WYLLER, A. A. (1960) On the generalization of the Appleton-Hartree magneto-ionic formulas, / . Geophys. Res., 65, 3931-3950.

SCHELKUNOFF, S. A. (1943) Electromagnetic Waves, Van Nostrand, New York. SPITZER, L. Jr. (1956) Physics of Fully Ionized Gases, Interscience, New York and London. WAIT, J. R. (1961) Some boundary value problems involving plasma media, / . Res. Nat.

Bur. Stand., 65B (Math, and Math. Phys.) No. 2, 137-150. WAIT, J. R., and CARTER, N. F. (1960) Field strength calculations for e.l.f. radio waves,

Nat. Bur. Stand., Tech. Note No. 52 (PB-161553). WAIT, J. R., and PERRY, L. B., (1957) Calculations of ionospheric reflection coefficients at

v.l.f., / . Geophys. Res., 62, 43. WHITMER, R. F. (1959) Principles of microwave interactions with ionized media, Microwave

J., 17 and 47. YABROFF, I. (1957) Reflection at a sharply bounded ionosphere, Proc. I.R.E., 45, 750-754.

Additional References GUREVICH, A. V. (1957) On the effect of radio waves on the properties of a plasma, Soviet

Physics, / . Exp. Theo. Phys., 3, 895. SUHL, H., and WALKER, L. R. (1954) Topics in guided wave propagation through gyro-

magnetic media, Bell Syst. Tech. J., 33, 579. VAN TRIER, A. A. (1953) Guided electromagnetic waves in anisotropie media, Appi. Sci.

Res. B3, 305. WAIT, J. R. (1960) Propagation of electromagnetic waves along a thin plasma sheet,

Canad. J. Phys., 38, 1586-1594. WAIT, J. R. (1961) The electromagnetic fields of a dipole in the presence of a thin plasma

sheet, Appi. Sci. Res., Sec. B, 8, 397-417. WAIT, J. R„ and SPIES, K. (1960) Influence of earth curvature and the terrestrial magnetic

field on v.l.f. propagation, / . Geophys. Res. 65, No. 8, 2325-2331. WIEDER, B. (1962) Microwave propagation in an overdense bounded magneto-plasma,

Ph. D. Thesis, Faculty of the Graduate School, University of Colorado.

s

Chapter IX

V.L.F. PROPAGATION—THEORY AND EXPERIMENT

Abstract—Approximate techniques for solving the v.l.f. modal equation are described. Essentially, the idea is to expand the logarithm of the reflection coefficient in a power series of C, the cosine of the angle of incidence. Using this approach, curves of attenuation rates for a wide variety of conditions are obtained. Both the terrestrial Magnetic field and the earth's curvature are included in the analysis. The results are then compared with experimental data obtained from many sources beginning with the early field strength data of Round et al. [1925] to the most recent "sferics" data of Taylor [1961].

1. INTRODUCTION

In the preceding chapter the basic theory of propagation in a waveguide bounded by anisotropie media was outlined. The form of the modal or resonance equation was seen to be quite analogous to the waveguide with isotropie walls. It is the purpose of this chapter to describe simplified methods for solving the modal equation when the Magnetic field is included. Applications of the results to v.l.f. radio propagation are also included and comparisons are made with experimental data. The nonreciprocity which exists between east-to-west and west-to-east propagation is also discussed.

Much of the numerical work described in this chapter is due to the author's colleague Kenneth P. Spies.

2. APPROXIMATE SOLUTIONS OF THE MODE EQUATION

2.1 Alternate Expansion for the Reflection Coefficient To facilitate solving the mode equation for v.l.f. propagation in the earth-

ionosphere waveguide, it is convenient to express the ionospheric reflection coefficients \\R\\ and LRL by representations having the form

R = - e x p f ^ C + <x2C2 + a3C3 + ...] (1)

where the (complex) constants alf α2, α3, ... depend on the ionospheric properties but not on C. At highly grazing angles C is small and the first term suffices. This limiting form was used in the previous chapter to obtain

264

V.L.F. Propagation—Theory and Experiment 265

a simple solution for the modal equation. More generally higher terms can be found in the following way. Since

R = elog* (2)

the exponent log R is expanded in a Taylor series about C = 0 to obtain

K = e x p [ ? oa ^ ] (3)

where

a *=è^ ( l o g A ) (4a)

2.2 Application of the Q.L. Approximation Employing the Q.L. (quasi-longitudinal) approximation of ^R^ given by

Eq. (88), for a homogeneous and sharply bounded ionosphere, of the previous chapter, one finds

cc0 = in (4b)

_ 2μ0μβ Γ μβ μ0 ] '" o + ^U^-l)+V(^o2-l)J (4C)

J*î 2μ0μβ

2μ0μβ VOiS - lXyOiJ - 1) «2 = Ì T T - "7772 Tw772 Γ (4d>

24L /lofi. j £ - l j

2μ0 + μβ^(μΙ-ί)[μΙ-ί μ\ - lj l J

where μ0 and μβ are the refractive indices of the two magneto-ionic components. Within the same Q.L. approximation,

±Α± = β χ ρ ^ Σ ο Α ^ ] (5) where

and

ßo = ι'π, jffjL = ——, β2 = α2 /*oMe

It is interesting to observe that the coefficients a2 (and also a4, a6, ...) and ß2 (and also ß4,ß6, ...) are finite. Ιΐμ0 = μ6 = /ι, corresponding to the absence of a terrestrial magnetic field, these even-ordered coefficients would be zero.

To illustrate the nature of the modes in the earth-ionosphere waveguide, it is assumed that only the first two terms in the expression for (| R y are required. Furthermore, coupling between the T.M. and T.E. modes is neglected, the ground is taken to be perfectly conducting and the earth is assumed to be flat. Thus, using Eq. (128) of the previous chapter and noting that ocl = — 2/?|(, one has for the T.M. modes,

c « = ο ? ϋ Ι · ■ s - = e 1 - c 2 » ) * = ί - c«i2 (6> 2kh + IOLI

for n = 1, 2, 3, ... . Using the simple formula given above, curves of the attenuation constant

— Him Si for the dominant mode are shown in Figs. 1 and 2 for H( = h/X = kh/2n) ranging from 2 to 6. The parameters B and </>L are defined by

ω(ν2 + ω£)%

tan 0L = — :

1 Ιω \ _ lue ω H\cuJ corh9 œr

longitudinal gyro frequency collisional frequency

FREQUENCY, kc/S

ωΐ

FIG. 1. Attenuation as a function of frequency for the dominant mode when the magnetic field is zero. These curves are for a FLAT earth.

FREQUENCY, k c / S

10 12 14 16 18 20 22 24 0.5

0.4

x

x 0.2. I

0.1

0 2 3 4 5 6

H

FIG. 2. Attenuation as a function of frequency for the dominant mode when the magnetic field is included and the Q.L. approximation is valid. These

curves are for a FLAT earth.

The attenuation constant expressed in decibels per 1000 km of path length is also shown on the curves where h is chosen to be 70 km. In Fig. 1, where </>L = 0, the magnetic field is neglected whereas, in Fig. 2, 0L = 60 degrees corresponds to cojv = ^ 3 . The similarity of the curves in Figs. 1 and 2 indicates that the attenuation of the dominant mode is not appreciably affected by the presence of a magnetic field. Of course, if 0L were somewhat greater than 60 degrees, the influence of the magnetic field becomes appreciable ; however, for v.l.f. propagation this condition is not applicable since v is of the order 107 in the D region where v.l.f. waves are reflected and coL would always be less than 107.

It is observed in Figs. 1 and 2 that the dependence of the attenuation on the value of B is quite noticeable. Generally, the attenuation becomes greater as B is decreased—at least for the range of B shown. At first glance this behavior may appear to be somewhat surprising as smaller values of B are associated with higher conductivities. However, it should be remembered that for highly oblique incidence, the reflection coefficient \\R\\ is closer to — 1

J I I L

when the refractive indices μ0 and μβ are reduced. For example, in the isotropie approximation

ι*ι » -exp("7(^T)C) <7> which clearly has the required behavior over the range of variables shown in Fig. 1. Thus, contrary to widely held opinions, increased electron densities in the D region may be associated often with higher attenuation. However, a more important factor is the influence of the reflecting height A. Generally, for constant values of cor and angular frequency ω the attenuation is increased with decreasing height A.

2.3 Application of the Transverse Condition The same approach may be used to obtain results for the attenuation in

the flat earth-ionosphere waveguide with a purely transverse magnetic field. In this case

11 11= exp[fQafcCfcJ (8a)

where, on making use of Eq. (39) of the previous chapter,

a0 = in (8b)

2[(l + 3 ) 2 - y 2 ] [(l + <5)05 + < 5 2 - y 2 ) ] * - i y a i = ~η , ^,s , *2 .,2v,*—77. ( 8 c )

a2 = 0 (8d)

\(δ + δ2- γψ 1 Λ 3 r(5 + ,32-y2)* . ] \ a3 = naH(i + 3)2-y4 (1 + *)» + Ί + α ι )

(8e)

α4 = 0. (8f )

Here δ = i — where ωΓ = ω^/ν

/

and γ =

ω H— |tan φτ\ for east-to-west propagation ω,

ω Itan φτ\ for west-to-east propagation. Mr

In the above it has also been assumed that ω <ξ ν which is nearly always valid for v.l.f. radio waves in the lower ionosphere.


It is noted that the even ordered coefficients are zero. In general this follows from the condition that \\R\\(C) = [^R^(-C)]"1 for a purely transverse magnetic field. Thus, to a fairly high order of accuracy

( 2 η - 1 ) π C""2feÄ + ia1· W

Using this formula with the value of ax given by Eq. (8c), attenuation curves are shown in Fig. 3 for B a value of 0.1. Here

1 ω 2nc 2ncv H cor hcor ha>l

and i i i \ωτ\ gyro frequency tan ώτ\ = -—- = .

1 ' v collisional frequency Again, a frequency scale in kilocycles per second and an attenuation scale in decibels per 1000 km is appended for a height h of 70 km. It is evident that a purely transverse magnetic field has a pronounced effect on the attenuation of the dominant mode. Consistently, the attenuation for propagation from east-to-west is greater than for propagation from west-to-east.

2.4 Extension to Arbitrarily Dipping Magnetic Field The results indicated graphically in Figs. 1-3 are based on a number of

simplifications. Among these is the neglect of coupling between T.M. and T.E. modes via the conversion coefficients ^R± and ±R^. Using the perturbation method discussed in the previous chapter, this effect can be accounted for. To within the accuracy of the graphical plotting of Figs. 1-3, the results would not be influenced. Furthermore, the retention of terms proportional to a2, a3, ... in the expansion of log ^R^ only lead to minor change in the form of the results in Figs. 1-3. More important are the restrictions on the orientation of the terrestrial magnetic field. For an arbitrarily dipping magnetic field, and when the Q.L. approximation is not valid, it is necessary to employ the full wave solutions discussed in the previous chapter.

For expediency we shall assume that in the general case

Π*Β S - e * c (10) for the important modes. To estimate a, the numerical results of ^R^ evaluated by Johler [1961] are used. His calculations were carried out for a (real) angle of incidence of 82 degrees, for various magnetic dip angles / and at various directions of propagation φα (measured clockwise from north). Since |jjR|| ei2* is equal to Johler's Tee9 one finds that

R e a = l ^ l a n d l m a = ^ % ^ . (11) cos 82° cos 82° v

FREQUENCY, kc/S

10 12 14 16 18 20 22 24

o 0.3

o o o

o

2

<

2 3 4 5 6 H

FIG. 3. Attenuation as a function of frequency for the dominant mode when the Magnetic field is purely transverse.

Then, if coupling between T.E. and T.M. modes is again neglected, the modal equation reduces to

JlkhC - A = 0 (12) or

* - & ^ - . « · . *

One would expect this to be a satisfactory approximation when Re Cn is near cos 82 degrees and Im Cn is small. For the important modes in v.l.f. propagation, this is quite appropriate.

The results are now normalized by introducing the factor

P = Attenuation with magnetic field Attenuation without magnetic field [Im SJ with magnetic field (13) [Im S J without magnetic field

The results are shown in Figs. 4, 5, and 6 where P is plotted as a function of the azimuth angle φα for dip angles of 0 degrees, 45 degrees and 84.3 degrees,


P I.Ob

respectively, and/ = 10 and 22 kc/s. In all these curves the electron density N is 103 electrons/cm3, the collision frequency v is 2 x 107 sec"1, and the strength of the terrestrial magnetic field is 0.5 Gauss. The case for N = 3 x 103 electrons/cm3 when / = 0° is shown in Fig. 7.

When the azimuth angle φα is 0 degrees, propagation is towards the north whereas 180 degrees corresponds to propagation towards the south. It is evident that in these two cases, the magnetic field hardly influences the


1.2

1.1

P 1.0

0.9

0.8 120 180

FIG. 6

l·

L·

1

N = (IO)3 1 = 84.3° n = l

1 1 1

[f = 22kc|

^ W~-'\n i r\

1 1 1 1 I 1

- = = s

1 J

240 300 3Θ0

FIG. 7 FIGS. 4, 5, 6, 7. Ratios, of attenuation with the Magnetic field to the attenuation without the Magnetic field, as a function of direction of propagation

(e.g., φα = 90 degrees corresponds to propagation from west-to-east).

attenuation at all. On the other hand, at φα = 90 degrees and 270 degrees, corresponding to propagation towards the east and west, respectively, P departs significantly from unity. The effect becomes less noticeable as the dip angle / approaches the vertical.

A good check on the method at / = 0 degrees is to* employ Eq. (8c) which is valid for φα = 90 degrees and 270 degrees (i.e. purely transverse magnetic field). The values of P so obtained are in good agreement with the curves in Figs. 4 and 7.


02

01 h

ro O x 0.0 2

x 2

-0.2

0.1

0.0

-01

[-

H

1

|f = lQkcl

|f=i2kc|

|f = 22kc[

1 1

^|f = i4kd ^ ^ 3

1 1 1 1 1

N= (IO)3 1 = 0° n= |

1 1 1 ! 60 120 180 240 300 360

J-

1 ι

|f = IOkc'i y\i = \2kc\

!f=22kc[

I 1 _ 1 J J I I

N =(IO)3 1=84.3° n=l

1 1 1 1 Ü CO l?0 180 ?40 300 360

A, FIGS. 8, 9, 10. Dififerences between phase velocity with magnetic field and

phase velocity without magnetic field.

The influence of the magnetic field on phase velocity has not been considered explicitly. For the v.l.f. range its effect appears to be very small. A convenient parameter is the quantity M which is the difference between the relative phase velocity with the magnetic field H0 included and the relative phase velocity without the magnetic field. Thus

M-ès] -es] <1 4 ) with Ho without fio

This quantity is shown in Figs. 8,9, and 10 for the same conditions as Figs. 4, 5, and 6. It is quite clear that the magnetic field has an exceedingly small influence on the phase velocity in the v.l.f. portion of the radio spectrum.

2.5 Inclusion of Earth Curvature in the Analysis The influences of the terrestrial magnetic field and the sphericity of the

earth are now simultaneously considered. A complete rigorous treatment of the problem would require the intractable solution of a very complicated boundary value problem. However, since the terrestrial magnetic field is a slowly varying function of position over the earth's surface it may be considered locally as a constant. Furthermore, it can be expected that coupling between T.M. and T.E. modes is very small since the product ||jRx LR^ has a magnitude of the order of 10"3 or 10"4 for angles of incidence around 80 degrees. Thus, for long distance propagation the effective value of the ionospheric reflection coefficient can be expressed as

H*l, = - e " C ' (15) where C = (C2 + 2h/a)i/2 is the cosine of the (complex) angle of incidence at the ionosphere and at is the coefficient discussed above. The relevant mode equation for a perfectly conducting spherical earth is thus given by

, , / - 2 2/i\3/i . / 2 2ft\% f w'2(t) ,A ^ π Λ /Λ^

ika(c+T) + ι«Λ° +τ) + n o g ^ - ( 4 ^ 1 ) 2 = 0 (16)

where / = —(ka/2)2/*C2 and n is an integer. This equation is simply a special case of Eq. (36) of Chapter VII when q = 0. Here, the influence of the magnetic field is contained implicitly in (xi.

While Eq. (16) may be solved in the same manner as discussed in Chapter VII, it is desirable to make a further simplification in order to facilitate computation. The first approximation consists of replacing the slowly varying term i log [H^CO/WKOI by nß which is its value at t = 0 (or C = 0). Secondly, it is assumed that \C2\ <ζ 2h/a for the other factors in the modal equation. Thus

(~2 2hYA (lhV2\< 3 c2 1


and (^Î^YHU The resulting mode equation can thus be solved for C2 to give

^ (12n - 5)π/6 - (2kal3)(2hla)K - ia^h/a)* ka(2hla)* + ( i a ^ ^ f t / a ) - * * l '

As a check on the errors introduced by the approximations the relative velocity 1/Re Sn for n = 0 and a! = 0 is calculated from the "exact" Eq. (16) and the approximate Eq. (19). The results are shown in Fig. 11 for h = 60,

0.010

L

\ 1 |\ [ \

\

COMPARISON OF EXACT ( ) AND APPROXIMATE ( ) METHODS OF SOLUTION

u^n \\ 1 \ \ \ \ \ \ ^ \ \ \ \ \

^ >\ \ ^\. - \ \

- \ \ , > χ

^ Λ. ^4j | r \ ^ ^ ν "»^^"^»^^ 1

Γ V νχ ^^r:^r-H L vv χ ^ ν , """""" l

\ \ v C ^ ; | Γ ^^^ ^*^*^^^ \ >v ***· ■^***·***«^^

N s * S s \ Λ- * 1 \- ^ % |— "" -»^s^^

^^^q 1 . . „J 1 " J L - - J ! J L — X i

0.008

0.006

0.004

0.002

~ 0.000 CE

-0.002 h

-0.004 h

-0,006 h

-0.008

-0.010 8 l? 16 20 24 28

FREQUENCY, kc /S

FIG. 11. Comparison between accurate or "exact" modal equation and the approximate form used for most of the calculations in this chapter. These

curves are for a CURVED earth.


80, and 100 km. The corresponding curves are sufficiently close to justify the use of the approximate solution of the mode equation for rough estimates.

Calculations of the attenuation coefficient -Him Sn for n = 1 are carried out from Eq. (19) using the Q.L. approximation for o as specified by Eq. (4c). As before, H = A/A, B = 2nc/(corh) and tan φι = cojv. The results are shown in Fig. 12 where the attenuation scale in decibels per 1000 km and the

FREQUENCY, kc /S

10 12 14 16 18 20 22 24

FIG. 12. Attenuation for the dominant mode as a function of frequency when earth curvature is accounted for.

frequency scale in kilocycles per second correspond to A = 70 km. To permit this convenient type of parametric plot, it is necessary to take A/a to be a constant (here chosen as 10~2).

It is apparent from the curves in Fig. 12 that the magnetic field has a relatively small effect. However, the differences between the sets of curves for 0L = 0 and φι = 60 degrees is more marked than in the corresponding results for the flat earth (e.g. compare with Figs. 1 and 2). The reason for the difference is connected with the steeper angles of incidence at the ionosphere when the curvature is considered.

3. MEASURED FIELD STRENGTH VS. DISTANCE DATA AT V.L.F.

We are now in a position to compare some of the calculated characteristics of v.Lf. propagation with measured data. No attempt will be made to discuss the experimental techniques and the interpretation of the data as the subject has been covered adequately in a number of papers.*

I M I 1 1 1 1 [ M l 16.6 kc

Howoii—Woke — Tokio: Sept. 1954 Son Diego — Howoii: Dec 1954 Tokio-Woke: Aug. 1955

(Daytime)

1 1 1 5 6 7 8 1.000

Distance (km) 4 5 6 7 8 MOO

5

<£ a>

σ or J J5 jy "ai E > a> o

ou

70

60

50

40

50

?0

10

i l l !

^ w ^ \ S ^ M ^ / " N

\n

-

1 ! ! . 1 !

i i i | : : ; , ι ; i 16.6 kc

Woke-Howoii . 6 Aug. 1955

Howoii —Son Diego: II Aug. 1955

(Daytime)

l ^ ^ ^

! : , . i , : : ! i l i

I

A

- i

-

-

1 3 5 6 S 6 7 8 KUXX) 7 8 IJOOO 2 3

Dis'cjrice ( k m )

FIGS. 13a and b. A sketch of the field strength vs. distance data obtained by Heritage, Bickel, and Weisbrod in an aircraft.

* See for example the June 1957 (v.l.f.) issue of the Proceedings of the I.R.E. and the Record of the Symposium on v.l.f. Radio Waves, Boulder, Colorado, Jan. 1957.


Quite recently some reliable experimental data for v.l.f. propagation to great distances was obtained by Bickel, Heritgage and Weisbrod [1957]. They recorded v.l.f. field intensities in an aircraft at distances up to 7600 km from the transmitters. Some of their measurements are shown in Figs. 13a and 13b which were all taken in the daytime over the Pacific Ocean. The transmitter was the v.l.f. station NPM in Hawaii operating at 16.6 kc/s. The field is expressed in terms of decibels above 1 μν/m for 1 kw radiated. The corresponding curve from the mode theory (taking h = 70 km and B = 0.1) along with a quasi-empirical formula proposed by Pierce [1952] are shown in Fig. 14. The inverse distance field is shown as a solid straight line.

80 | , , , j | | | | | 1 1 1 1 — | | | | I

16 .6 kc

11 I I 1 I 1 I 1 i 1 l l l L _ i I 100 2 3 4 5 6 7 8 1.000 2 3 4 5 6 7 8 10.000

Distance (km)

FIG. 14. A sketch of a calculated field strength based on mode theory and an empirical formula of J. A. Pierce. The straight line is the unattenuated or

"inverse distance" field strength. (Compare with Figs. 13a and 13b.)

The general agreement with the experimental data and the experimental formula of Pierce is quite good. It is interesting to note that even for distances as great as 3000 km the second and higher order modes may contribute to the field.

Some very early v.l.f. measurements were made by Round, Eckersley, Tremellen and Lunnon [1925] during a round-the-world cruise in 1922 and 1923. The field strength measurements were combined with directional observations on signals from v.l.f. transmitters then situated in various parts of the globe. For convenience in comparing their measured field strength E it is desirable to replot the quantity 20 log E 4- 10 log (d/a) as a function of distance in kilometers. The results of Round et al. have been presented in this manner in a recent review paper [Wait, 1958]. The slope of the curves is


a direct and simple measure of the attenuation rate. The outward voyage from Liverpool to Australia, via the Panama Canal, was on the "S.S. Dorset," and the return voyage, via the Suez, was on the "S.S. Boonah." Some examples from these data are shown in Figs. 15a, 15b, and 15c. The abscissae are the great circle distance, between transmitter and receiver, expressed in thousands of kilometers (denoted k, km or K). The ordinates are the field amplitudes normalized as mentioned above. An all daylight condition for the path is indicated by a dot whereas a mixed day-night path or an all-night path is indicated by a x. On each curve a straight line is drawn through the points which appear to be most compatible with the daytime points. The attenuation rates, obtained from the slope of the straight lines, are indicated in units of db/K (decibels per thousand kilometers).

Certain features of Round's et al data are very significant. The attenuation rates for the European stations on the outward voyage across the Atlantic are consistently of the order of 3 or 4 db/K. In these cases, the direction of transmission is essentially northeast to southwest. For the same portion of the voyage, the attenuation rates from the American stations were less than 2 db/K. On the basis of this set of data, it is possible to say with some certainly that transmission from east-to-west is poorer than from west-to-east, at least in the North Atlantic. This apparent non-reciprocity is in accord with mode theory as discussed in Section 2.

For the measurements of the American stations, in the South Atlantic and through the Panama Canal to the Pacific, the attenuation rates are about 3 db/K. Here the propagation is sometimes overmixed land-sea paths. In a few cases elimination of long stretches of land decreased the attenuation by as much as 2 db/K. This is also in accord with theory. In general, however, the phenomena are much more complicated since land/sea boundaries and other geographical discontinuities may produce higher modes.

The field strength values (of the European stations) on the return voyage from Australia aboard the "S.S. Boonah" indicated that the attenuation rates were of the order of 2 db/K for the portion of the paths over the Arabian Sea and the Indian Ocean. At the shorter distances the transmission paths were mixed land and sea and these were accompanied by higher attenuation rates.

For both the experimental data of Heritage et al. and Round et al there is no pronounced dependence of attenuation with frequency for the range 16-25 kc/s. However, Round's limited data at 12.8 kc/s for the station Bordeaux (LY), the attenuation was about 4.0 db/K. Such a behavior is quite consistent with mode theory as indicated by Fig. 4a of Chapter VII.

As can be seen in Figs. 15a and 15c, the night-time data points are somewhat scattered. However, in Fig. 15a for 22.7 kc/s there is a definite tendency for field strength at night to be lower than in day-time, although the attenuation rate was about the same or slightly lower. This effect was repeated in other

T


db

50

4 0

3 0

20

10

0

1 | 1 |

X _ X

•

- · · • · X

X

~" X

• Doy Values * Night Values

l ! ι l _

Γ - " ! i i I " - ! J 1 1 1 J 1 Γ

BOLINAS, SAN FRANCISCO (KET) SS " Dorset"

22.7 kc

· · · — * · * *

~~— ^ · · • ^ Γ— • x

X X X X X X

X X

J I 1 I I 1 1

• X

*

t

./2.9db/K

X X X

J 1 I 1 L

1 1

-J

J 4

• 1 X J

XX

—j

1 1 7 8

d in kkm

(a)

10

db

50

40

30

20

10 h

i ! 1 r

• Day Values x Night Values

LYONS (YN) SS "Boonah"

19.3 kc

• ·— I.edb/K-

-i 1 r

J I I i I L 10

Ί r

12

d in kkm

(b) FIGS. 15a, b, and c. Normalized field strength of early data of Round, et al,

as a function of distance in thousands of kilometers.

data of Round et al in the frequency range between 20 and 30 kc/s. The particular phenomena are entirely consistent with the behavior of the excitation factor Ai shown plotted in Fig. 7 of Chapter VII. For a change of height from 70 to 90 km, At is decreased by about 12 dB. Apparently, at

V.L.F. Propagation—Theory and Experiment 281 60

50

40 h

30

20

10

1 1 1 1 1 1 -

r— —^_ · l * ^ " ? r - - J ^

• · * ^ — -Γ~ ' ^ ^ X

L 1 X

Γ

L

i Γ · Day Values

* Night Values

I 1 ! I i L

1 1 1 1 1 1 1

^l.8db/K

~^-~^Z^~ "~^"———£___·

'—- · · •X · ^ ^ " - » ^ - ^ f

X £ • X

X

BORDEAUX (LY) S.S."Boonah"

16.0 kc

i I i ! I 1 '

π 1 1 1 —

• ^2.9 db/K -~z * ^ - ^ « ^ · ·

X · - ^ ^ ^ · X """"

X

J 1 ! I

Ί i

Ί

H

w· J ^^x/x 1

H

J - j

1 1 2 4 6 8 10

d in kkm Fig. 15(c)

night, the dominant mode (n = 1) is not being effectively excited. In fact, a typical difference between the night-time and day-time excitation factors appears to be of the order of 15 db for the 22.7 kc/s transmission shown in Fig. 15a. The effect is also present at 16.0 kc/s as seen in Fig. 15b, but here the difference between day and night levels is only about 5 db. Again, this is typical of other data of Round et al. for frequencies of this order.

4. MEASURED PHASE CHARACTERISTICS OF V.L.F. CARRIERS

Until recently, experimental data on phase velocity of v.l.f. radio waves has been lacking. This is a pity since this is a crucial characteristic of the mode theory. Fortunately, the U.S. Navy Electronics Laboratory has now obtained data which permit the absolute determination of the phase velocity [Pierce, et al, 1962; Casselman, et al, 1959]. The observations were made by transmitting one-second bursts of an unmodulated carrier from a master station. When the master station was silent, similar transmissions were made at the same frequency at two slave stations. The relative phases of the master and the two slave transmissions were observed at two receiving sites. The results of these experiments have been published by Pierce, Casselman, Tibbals, and Heritage [1962]. Since the subject is of great importance a comparison of their data with mode theory is given here. For sake of completeness a concise description of their technique is also given.

In the first set of experiments, in the spring of 1959, the master station was at Haiku, Hawaii, and a slave station was at San Diego, California. The


receiving site was located at Wahiawa, also in Hawaii. The frequencies employed were 10.2, 11.2, 13.2, 14.2, and 15.2 kc/s. In the second set of experiments, in early 1960, a second slave station was located at Forestport in New York state and an additional receiving site was located near San Diego. The frequencies were now 9.2, 10.2, and 12.2 kc/s.

The data obtained in such experiments define only the phase, φ, of one signal with respect to another. Now, the corresponding phase velocity v is related to φ by the equation

2nf— = 2mπ + φ v

where d is the total distance of travel and m is some unknown integer. An initial estimate of m can be made by taking the phase velocity v to be the same as the velocity of light. The remaining ambiguity can be resolved by plotting derived values of v as a function of frequency for various values of m. When this is repeated for several values of d it becomes quite apparent which sets of points should be chosen. Essentially, this is the technique used by Pierce, Casselman, Tibbals, and Heritage [1962].

The situation is illustrated in Figs. 16a and 16b where possible values of the phase velocity deviation (v/c — 1) are plotted vs. frequency. The small circles correspond to the path from Hawaii to San Diego to Hawaii. The dots correspond to the path from Hawaii for Forestport, New York, to Hawaii. The crosses correspond to the path Hawaii to Forestport, to San Diego. Also shown on Figs. 16a and 16b are some points which represent data taken on a ship moving between California and Hawaii in August 1959. The distance to San Diego in hundreds of kilometers is indicated beside each point. Because the ship's position was not known accurately, this set of data is not as precise.

Curves of (v/c — 1) derived from mode theory are now also drawn on Figs. 16a and 16b. The full effect of earth curvature is taken into account by using Airy integral approximations for the relevant spherical wave functions as discussed in Chapter VII. The ionosphere is assumed to be sharply bounded and its lower edge is located at heights of 70 and 90 km as indicated in Figs. 16a and 16b. In both cases ωΓ is assigned the value 2 x 105 and the σβ may be replaced by oo since the lower boundary is sea water. The effects of the earth's magnetic field are neglected in the calculation since its influence on phase velocity is expected to be very small.

It is quite apparent from studying Figs. 16a and 16b that the curve for h = 70 km is quite a good fit for the data in the day-time, whereas the curve for h = 90 km is more appropriate for the night. In fact, it would be very difficult indeed to find any other smooth curves which could be drawn in a satisfactory manner through the data points. Ideally, one should have coincidences between the dots, circles, and crosses for the correct value of


o o

0.5 0.4 0.3 0.2 0.1

X

— 0

>io -0.1 -0.2

-0.3

-0.4

-0.5

_ _ -

-

_ -

V 1

•

%

•

o

' . "

•

%

•

o 1*

l u

\ o

o

o I

1

8 •

#I9

•

9

. · 2 5

1

0

o

lo

o

o

, 1

1

o

■""- -S—.

o

.. 1

~~

_. -

-—

-

9 10 II 12 13 14 15 16 FREQUENCY (kc/s)

FIG. 16a. Phase velocity as a function of frequency for day. The curve is based on mode theory (for h = 70 km, ωΓ = 2 x 105, og = oo, and n = 1) and the points are obtained from the experimental data from the U.S. Navy

Electronics Laboratory.

0.5 0.4 0.3 0.2 0.1

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

-

Γ

- \

—

-

p γ — 1-

— i

• o

•

X

( 3 s" *

•

X

• o

•

i 1

~ i

* 8

•

•

8 O

X

1

~Ί o

o

ο ^ -

O

O _J

~Ί ¥17" ê o

• • 35

è o

m

- * & •Il

i · 3 5

•

i · 3 5

I

o

o

o

o

o

J

■ ^

o

o

0

o

o

1

^

o

o

o

o

o

o 1

A

| -j

-J

*~!

-]

η -]

-î 8 9 10 II 12 13 14

FREQUENCY (kc/s) 15 16

FIG. 16b. Phase velocity as a function of frequency for night. The curve is based on mode theory (for h = 90 km, ωΓ = 2 x 105, ag = oo, and n = 1) and the points are obtained from the experimental data from the U.S. Navy

Electronics Laboratory.

v/c — 1. Unfortunately, the variability of the relative phase is large enough to destroy any close coincidences. Furthermore, the theoretical curves refer only to the first order or dominant mode. In actuality, there are many modes present and, only at very great distances, the higher modes may be neglected. The presence of such modes is to render the phase velocity a function of distance. Thus, perfect coincidences would not be expected between the three kinds of points in Figs. 16a and 16b even if there were not any time variability.


Another effect of higher modes is to introduce an undulation in the dispersion curve. There is some suggestion in Fig. 16b that the circles would fall on such a curve. However, the smallness of this undulation is actually strong evidence that only one mode is really significant at these ranges.

It can be concluded that there is fairly good agreement between measured phase velocity and the mode theory as developed for a curved earth-ionosphere waveguide. Furthermore, the heights of the equivalent reflecting layer are fairly consistent with other results.

Recently, Pierce and Nath [1961] have studied the dependence of the phase velocity at 10.2 kc/s on the relative amount of land over the path. They find for the day-time data that v/c changes from 1.0030 for an all sea path to about 1.0025 for an all land path. This is seen to be in excellent agreement with Fig. 5a of Chapter VII if the land conductivity is about 10 mmho/m. At night they find that v/c is about 0.9993 and no significant dependence on the relative amount of land is evident. If anything, there is a slight reverse trend which indicates that the phase velocity is very slightly increased for an all-land path. Mode theory would not predict such a behavior, although the difference between phase velocities over land and sea should be much less at night because the mode is beginning to have the "earth detached" character which was discussed in Chapter VII.

5. MEASUREMENTS OF DIURNAL PHASE SHIFTS AT V.L.F.

Another important quantity to measure at long distance from a v.l.f. transmitter, is the phase difference of the signal between day and night. With the advent of highly stabilized carrier frequencies, considerable attention has been paid recently to the measurements of diurnal phase shifts. It is of interest to interpret these data in terms of mode theory.

To predict the diurnal phase shift At in microseconds from the phase velocity r, one may simply use the formula

cAt c c v(h2) v(hi) d vQii) v(h2) ~~ c c

for a height change from ht to h2. Here c = 300 m//isec is the velocity of light. On this basis the height increment hl — h2 = Ah can be deduced. The results are shown in Table 1 where the source of the experimental data for very long ranges is acknowledged. With the exception of the new results from A. G. Jean, the data are the same as those listed by Westfall [1961]. For the data used here the range always exceeds 3500 km, so the higher order modes do not play a role. It is seen that the deduced height changes are in the range from 14 to 21 km. For the values shown, the daytime height has been assumed to be 70 km since this represents the most reasonable value in the light of previous


Table 1. Diurnal phase shifts for 10-20 kc/s radio waves over path distances of 3851-11,000 km

Freq. (kc/s)

16

16

10.2

11.2

19.8

18

10.2

10.2

18

18

18

16

Path and distance-*/ in km

Rugby-New Zealand* 11,000

Rugby-Cambridgeb

5200 Haiku-San Diegoc

4180 Haiku-San Diegoc

4180 Haiku-San Diegod

4180 Panama-San Diegod

4670 Haiku-Boulderc

5360 Haiku-Forestport, N.Y.C

7833 Panama-Haikud

8417 Panama-Forestport, N.Y.d

3851 Panama-Boulder

4300 Rugby-Bouldere

7500

Path composition

percent land

—

0

0

0

0

90

25

50

0

0

30

40

Diurnal phase shift (/asec)

70

34

53

46

26

29

61

87

50

28.5

27.8

42

Δ/ι

16.5

16.9

21.0

20.5

16.0

17.0

18.0

17.5

15.5

18.5

16

14

cat d

1.91

1.96

3.8

3.3

1.87

1.93

3.4

3.33

1.79

2.22

1.94

1.66

a Crombie et ah, [1958] d US NEL phase shift measurement. b Pierce [1957] e A. G. Jean (private communication) c Extracted from U.S. NEL Radux-Omega

data

work. In any case, the deduced values of Ah are not very sensitive to the assumed daytime height. In fact, for the same reason it appears to be almost impossible to make a reliable estimate of absolute height from the diurnal shifts.

It is interesting to note that the results for 10.2 and 11.2 kc/s yield somewhat greater height increments than those deduced for 16, 18, and 19.8 kc/s. A more elaborate model is required to take account of this effect.

6. SFERICS AND MODE THEORY

The attenuation rates in the v.l.f. band can also be obtained from atmospherics waveforms. These signals, or sferics as they are called, emanate from lightning discharges. One method is to record and analyze a large number of

atmospherics at a single station and assume that, on the average, the radiation spectra of the source are the same. Using care in the selection of data, some very useful results have been obtained by this method [Morrison, 1953; Chapman and Macario, 1956; Chapman and Pierce, 1957; and Pierce, 1961]. Some of these are shown in Fig. 17.

A much better method of utilizing atmospherics for propagation studies is to record the electric field waveforms of the same lightning discharge at two or more stations [Bowe, 1951 ; Taylor and Lange, 1958; Taylor, 1960a and b]. After the Fourier spectra Ε{ω, d) are calculated, it follows that the attenuation in db/1000 km is given by

α ( ω ) = ^ ; [ : 20 log ^ - 1 0 log t ^ ' ' Ε(ω, d2) ' sin(dila)j

where the distance d2 is greater than άγ. In this case there is no need to assume a source spectrum provided the source is equivalent to a vertical electric dipole.

The resulting attenuation as a function of frequency is presented in Fig. 17. The data of Taylor [1960b] and Taylor and Lange [1958] were smoothed to minimize the effects of waveguide modes of higher order than one, the dominant mode. Thus, the attenuation rates determined by various workers utilizing atmospherics are representative of the dominant mode and are reasonably consistent. They are also in good accord with theory if earth curvature is accounted for as discussed in Chapter VII.

o o o

o l·-< z>

l·- \

Ί — I I I I I 1 I CHAPMAN AND MACARIO, DAY

CHAPMAN AND MACARIO, NIGHT

X - — x - ECKERSLEY, DAY

x x ECKERSLEY, NIGHT

O O TAYLOR, DAY, LAND, FOR EAST TO WEST

• · TAYLOR, DAY, S E A , FOR EAST TO WEST

x y TAYLOR , DAY, SEA, FOR WEST TO EAST

TAYLOR AND LANGE, NIGHT, SEA,

FOR EAST TO WEST

J I I 1 I I 10 20

FREQUENCY,kc/S 50

FIG. 17. Attenuation rates deduced from various experimental data.


In the band of frequencies from 10 kc/s to 30 kc/s, the attenuation rates vary from approximately 4.5 db/1000 km to 1.0 db/1000 km. This agrees with the attenuation rates computed from the C.W. transmissions of Round, et al. previously mentioned. Attenuation rates deduced by Eckersley [1932] from C.W. data are shown in Fig. 17. The various methods of determining attenuation rates give compatible results.

The determination of phase characteristics of v.l.f. waves from atmospheric waveforms, recorded simultaneously at spaced stations, has also been carried out [Jean, Taylor and Wait, I960]. Using currently available procedures, it is not possible to determine absolute phase velocities from the waveform data. Rather, the technique leads primarily to the dispersive properties of the propagation medium (i.e. how the phase velocity varies with frequency). C.W. (continuous wave) techniques are really more suited for studying phase velocities as described above.

REFERENCES

BICKEL, J. E., HERITAGE, J. L., and WEISBROD, S. (1957) An experimental measurement of v.l.f. field strength as a function of distance using an aircraft, Report 767, U.S. Navy Electronics Laboratory, San Diego, California.

BOWE, P. W. A. (1951) The waveforms of atmospherics and the propagation of v.l.f. radio waves, Phil. Mag., 42, 121.

CASSELMAN, C. J., and HERITAGE, D. P., and TIBBALS, M. L. (1959) v.l.f. propagation measurements for the Radux-Omega navigation system, Proc. I.R.E., 47, 829-839.

CHAPMAN, F. W., and MACARIO, R. C. V. (1956) Propagation of audio frequency radio waves to great distances, Nature, 177, 930.

CHAPMAN, J., and PIERCE, E. T. (1957) Relations between the character of atmospherics and their place of origin, Proc. I.R.E., 45, 804.

CROMBIE, D. D., ALLAN, A. H., and NEWMAN, M. (1958) Phase variations of 16 kc/s transmissions from Rugby as received in New Zealand, Proc. Instn. Elect. Engrs., B, 301-304.

ECKERSLEY, T. L. (1932) Studies in radio transmission, / . Instn. Elect. Engrs., 71, 405. JEAN, A. G., TAYLOR, W. L., and WAIT, J. R. (1960) v.l.f. phase characteristics deduced

from atmospheric waveforms, / . Geophys. Res., 65, No. 3, 907-912. JOHLER, J. R. (1961) Magneto-ionic propagation phenomena in low- and very-low-radio-

frequency waves reflected by the ionosphere, / . Res. Nat. Bur. Stand., 65D (Radio Prop.), No. 1, 53-66.

MORRISON, R. B. (1953) The variation with distance in the range 0-100 km of atmospheric waveforms, Phil. Mag., 44, 980.

PIERCE, E. T. (1961) Attenuation coefficients at very low frequencies (v.l.f.) during a sudden ionospheric disturbance (S.I.D.), / . Res. Nat. Bur. Stand., 65D (Radio Prop.), No. 6, 543-546.

PIERCE, J. A. (1952) Sky wave field intensity: Low- and very-low frequencies, Tech. Report No. 158, Cruft Laboratory, Harvard University. (For daytime propagation over sea water, Pierce's formula reads E = 231(P/sin 0)1/2 exp[-(O.l/)w0] when P is the radiated power in kilowatts, Θ = d/a,f is the frequency in kc/s, and u = 0.9).

PIERCE, J. A. (1957) Intercontinental frequency comparison by very low-frequency radio transmission, Proc. I.R.E., 45, 704-803.

PIERCE, J. A., CASSELMAN, C. J., TIBBALS, M. L., and HERITAGE, D. P. (1962) The velocity of propagation of v.l.f. radio waves, Proc. I.R.E., 50. (The information in this paper was communicated to the author by M. L. Tibbals in 1960, who also kindly granted permission to quote the experimental data.)


PIERCE, J. A., and NATH, S. C. (1961) v.l.f. propagation, Annual Progress Report No. 60, pp. 1-6, Cruft Laboratory, Harvard University.

ROUND, H. J. T., ECKERSLEY, Τ. L., TREMELLEN, K., and LUNNON, F. C. (1925) Report on measurements made on signal strength at great distances during 1922 and 1923 by an expedition sent to Australia, / . Instn. Elect. Engrs., 63, 933-1011.

TAYLOR, W. L. (1960a) Daytime attenuation rates in the v.l.f. band using atmospherics, / . Res. Nat. Bur. Stand., 64D (Radio Prop.), 349.

TAYLOR, W. L. (1960b) v.l.f. attenuation for east-west and west-east daytime propagation using atmospherics, / . Geophys. Res., 65, 1933.

TAYLOR, W. L. (1961) v.l.f. Propagation—I, Lecture No. 10 (Ionospheric Propagation) Nat. Bur. Stand. Course in Radio Propagation.

TAYLOR, W. L., and LANGE, L. J. (1958) Some characteristics of v.l.f. propagation using atmospheric waveforms, Proc. Second Conference on Atmos. Electricity, p. 609, Pergamon Press, London.

WAIT, J. R. (1958/III) A study of v.l.f. field strength data: Both old and new, Geofis. Pur. Appi. {Milano), 41, 73-85.

WESTFALL, W. D. (1961) Prediction of v.l.f. diurnal phase changes and solar flare effect, J. Geophys. Res., 66, 2733-2736.

Additional References AL'PERT, YA. L. (1956) Lightning and the propagation of audio-frequency electromagnetic

waves, Uspekhi Fiz. Nauk., 60, 369. CROMBIE, D. D. (1960) On the mode theory of v.l.f. propagation in the presence of a trans

verse magnetic field, / . Res. Nat. Bur. Stand., 64D (Radio Prop.), No. 3, 265-268. JEAN, A. G., LANGE, L. J., and WAIT, J. R. (1957) Ionospheric reflection coefficients at v.l.f.

from sferics measurements, Geofis., Pur. Appi., 38, 147-153. POEVERLEIN, H. (1958) Low-frequency reflection in the ionosphere, I. / . Atmos. Terr. Phys.,

12, 126. TAYLOR, W. L., and JEAN, A. G. (1959) Very low frequency radiation spectra of lightning

discharges,/. Res. Nat. Bur. Stand., 63D (Radio Prop.), 199. WAIT, J. R. (1958) Propagation of very-low-frequency pulses to great distances, / . Res.

Nat. Bur. Stand., 61, 187. WAIT, J. R. (1961) A diffraction theory for l.f. sky-wave propagation, / . Geophys. Res., 66,

No. 6, 1713-1724. WAIT, J. R., and MURPHY, A., The geometrical optics of v.l.f. sky-wave propagation,

Proc. I.R.E., 45, 754. WAIT, J. R., and PERRY, L. B. (1957) Calculations of ionospheric reflection coefficients at

very low radio frequencies, / . Geophys. Res., 62, 43. WAIT, J. R., and SPIES, K. (1960) Influence of earth curvature and the terrestrial magnetic

field on v.l.f. propagation, / . Geophys. Res., 65, 2325-2332. WHITSON, A. L. (1961) Research on v.l.f. propagation in arctic regions—geophysical effects,

Final Report, Stanford Research Institute, Menlo Park, California, Air Force Cambridge Research Laboratory 124.

Added in Proof BAIN, W. C. (1952) The propagation of very-low-frequency radio-waves, Ph.D. Thesis,

University of Aberdeen, Scotland. BARRON, D. W., (1961) Numerical investigation of the theoretical problems arising in the

study of the ionosphere, Ph.D. Thesis, Cambridge University. CROOM, D. L. (1961) Atmospherics and the propagation of long waves, Ph.D. Thesis,

Cambridge University. SAKURAZAWA, A., ASAI, J., and ISHII, T. (1961) Results of measurements of field intensity

of VLF radio waves over great distances, J. Radio Res. Lab. (Japan) 8, 425-539. [An attempt is made to explain diurnal changes of attenuation rates in terms of waveguide mode theory. Unfortunately, equation (2) in the paper is not valid at v.l.f. since L is not small compared with unity.]

WESTCOTT, B. S. (1962) Ionospheric reflection processes for long rad io- waves,/. Atmos. Terr. Phys. 24, 385-399 and 619-631.

Chapter X

E.L.F. (EXTREMELY LOW FREQUENCY) PROPAGATION—THEORY AND EXPERIMENT

Abstract—The mode theory of propagation of electromagnetic waves at extremely low frequencies (e.l.f.) (1.0-3,000 c/s) is treated in this chapter. Starting with the representation of the field as a sum of modes, approximate formulas are presented for the attenuation and phase constants. Certain alternate representations of the individual modes are mentioned. These are used as a basis for describing the physical behavior of the field at large distances from the source, particularly near the antipode of the source. At the shorter distances, where the range is comparable to the wavelength, the spherical-earth mode series is best transformed to a series involving cylindrical wave functions. This latter form is used to evaluate the near field behavior of the various field components.

The eifect of the earth's magnetic field is also evaluated using both quasi-longitudinal and transverse approximations. In general it is indicated that if the gyrofrequency is less than the effective value of the collision frequency, the presence of the earth's magnetic field may be neglected for e.l.f. When this condition is not met the attenuation may be increased somewhat. The influence of an inhomogeneous ionosphere is also briefly considered and, finally, the propagation of e.l.f. pulses is treated.

1. INTRODUCTION

The propagation of electromagnetic waves at extremely low frequencies (e.l.f.) has received considerable attention recently [Schumann, 1952; 1954a and 1954b; Wait, 1957; and Liebermann, 1957]. Unfortunately, the experimental data in this frequency range (1.0-3,000 c/s) is still rather limited [Chapman and Matthews, 1953; Hepburn and Pierce, 1953; Willis, 1948; Aarons and Henissart, 1953; Benoit, 1956; Goldberg, 1956; Aarons, 1956; and Watts, 1957]. There is little doubt, however, that the bulk of the energy is transferred via a waveguide mode. The bounding surfaces of this waveguide are the ground and the lower edge of the ionospheric E region.

An important feature of e.l.f. propagation is that the distance between source and observer may be comparable to the wavelength. For example, at 300 c/s the wavelength is 1000 km. In fact, many of the experiments are carried out at distances of this order and account must be taken of the near field effects.

2. BASIC THEORETICAL MODEL

The assumed theoretical model is taken to be a homogeneous conducting spherical earth of radius a surrounded by a concentric conducting ionospheric

289

shell of inner radius a + h. It is convenient to introduce the usual spherical coordinate system (r, θ, φ). Using this model and assuming a vertical electric dipole source located at Θ = 0, r = a, the expression for the vertical electric field, En was derived in Chapter VI. For convenience it is rewritten below.

£ ^ Σ , / - ^ ί 1 ) ρ ϊ ( _ ς 0 8 θ ) (1) .2kha2

n=o sin νπ where / is the average current in the source dipole; as is the length of source dipole ; η is the intrinsic impedance of air space £ 120π Ω ; Pv ( — cos Θ) is the Legendre function of argument — cos Θ and (complex) order v; v + \ = kaSn where Sn9 for n = 0, 1, 2, ..., is determined from the boundary condition and described below; and

i + khcf ^ 1 for "-1·2·3··· If the observer is assumed to be on the surface of the earth, the individual terms in the series correspond to the waveguide modes. In the general case, the factor Sn is obtained from the solution of a complicated transcendental equation which involves spherical Bessel functions of large argument and complex order.

Certain aspects of this problem have been discussed by Schumann [1952] and also by the author [1957]. For a homogeneous earth and a homogeneous ionosphere, both assumed isotropie, it was shown in Chapter VI that Sn may be approximated by

where -(^ntW(1+4w). 2 \'A

(2)

Δ Nt + N.

in terms of the refractive indices, Ng and Ni9 of the earth and the ionosphere, respectively. This equation is valid subject to the condition|A|/:A < 1. The sign of the radical in the above equation is chosen so that the real part is positive. The values of Sn are then located in the fourth quadrant of the complex plane. When n = 0, the above simplifies to

V Γι · Δ Γ (3)

Now, since \A\kh <ξ 1, the radical in Eq. (2) can be expanded for n > 0 to yield

V4

for « = 1,2,3,... (4) „ l\ (nn\2 .2Δ s"=[l-[kh) ~lTh

E.L.F. Propagation—Theory and Experiment 291

3. ANTIPODAL EFFECTS

For purposes of computation of the mode series, several simplifications can be made. The asymptotic expansion for the Legendre function, given by

Pv(-cos 0) s ( - 4 - ^ ) 1 / 2 c o s f(v + *Χπ - 0) - £1 (5) \πν sin 0/ L 4J

is valid if | v | > 1 and 0 not near 0 or π [Magnus and Oberhettinger, 1949]. Thus, the modes are simply proportional to

(^cos[fc«Sn(*-ö)-g which apart from a constant factor can be identified as the linear combination of two peripheral waves. These have the form

1 c-ikaSn9

(sin 0)1/2

and 1 -ikaSn(2n-e) in 12

(sin0)% where 0 < π.

These waves are travelling in opposing directions along the two respective great circle paths αθ and α(2π — 0), from the source to the observer. It is noticed that there is a π/2 phase advance which the wave travelling on the long great circle path picks up as it goes through the pole 0 = π. The linear combination of these two travelling waves forms a standing wave pattern whose distance ôm between minimums is approximately given by

kömRtSn = n or <5m = Λ/(2Re S„) (6) subject to

Im Sn <ReSn.

The preceding asymptotic expansion for Pv(—cos 0) is not usable at and in the vicinity of the pole 0 = π. A suitable representation for these was given in Section 7 of Chapter VI. It was shown that the field in the neighborhood of the pole is proportional to the Bessel function

J0lkaSn(n - 0)] which is finite at 0 = π.

4. EARTH-FLATTENING APPROXIMATION

The possibility that the curvature of the earth may be neglected at e.l.f. is now investigated. To simplify field calculations at relatively short distances,

it would seem desirable to transform the mode series to a form where the first term corresponds to the model of a flat earth and the succeeding terms are corrections for curvature. This approach has been used by Pekeris [1946] and more recently by Koo and Katzin [1960] in their investigations of microwave duct propagation. From their work, it may be shown that

P v ( -cos0)

sin νπ S H<0

2>(fcS„p) ±@'mu. ,Ρ) (7)

+ terms in (ρ/α)4, (pià)6, etc., where v + \ = kSnp and p = αθ. Η^2) and 7/£2) are Hankel functions of the second kind of order zero and two, respectively. When the great circle distance p is reasonably small compared to the earth's radius a, only the first term in the expansion need be retained.

For convenience in what follows, it is desirable to express the field components as a ratio to the quantity

E0 = i(nlk)Iàs ( e - ^ / p , in s 120π); E0 is the radiation field of the source at a distance p on a perfectly conducting ground. Thus, for both the source and the observer near the ground, it is not difficult to show that

E2 = WE0

where

where

and

where

l y s - ™ e ^ ^ W ^ ) n n = o

EP=-SE0

S£^e*'£<5AM2)(fcS„p) iV a ft n=0

(8a)

(8b)

Ηφ=-ΤΕ0/η

Γ ϊ - > e" ' Σ ônSnH[2\kSnP) = -NgS n n=o (9)

where terms containing (p/a)29 (p/a)4 have been neglected.

When kp > 1, corresponding to the "far-zone," the above expressions may be simplified since the Hankel functions may be replaced by the first term of their asymptotic expansion. This leads to the compact result

W

S

T

^ί[2πρ/λ-π/4] V Â (ΛΜ) „to " ■sw, i-ilnSnp/k (10)

which is valid for p > λ. As expected, the ratio of W to T for a given mode is Sn which for low order or grazing modes is of the order of unity. The ratio of S to Γ, quite generally, is — l/Ng which is very small compared to unity; in fact, it vanishes for a perfectly conducting ground as it must.

The results for the horizontal electric dipole may be stated in a similar manner. A complete discussion of the theory was given in Chapter VI. The expression for the vertical field Ez of the horizontal dipole / as which is located at the origin and oriented in direction φ = 0 may be written in the form

E2 = SE0 cos φ (11) where S and E0 are defined above. By using the reciprocity theorem, this result can also be deduced directly from Eq. (9). Comparing the above formula for Ez (for a horizontal dipole) with the corresponding formula for Ez9 for a vertical dipole, it may be easily shown that

Ez (for horizontal dipole) ^ cos φ / 1 \Vî ^ cos φ Ez (for vertical dipole) S 1Ç \ N*J S Ng

( }

being valid in the asymptotic or far zone. In terms of the ground conductivity ag and dielectric constant eg9

1 / ιε0ω Y„(tW>\*m Ng \σβ + iegco) \ ag )

This is a small quantity. The above formulas for W, S, and T are valid only if p <4 a. If the first

curvature correction term is included, it is a simple matter to show, in the "far zone", that this amounts to multiplying the right-hand side by the factor

1 ! (P\2 i 0 2

12 \a) 12 Starting from the mode series for a spherical earth [Wait, 1960a] and using the far field approximation for the Legendre function, it turns out that W, S, T have precisely the same form as in equation (10) except that the factor (0/sin ΘΥ2 occurs in the right-hand side. Noting that

/ Θ \1/a Θ2

- r - r S 1 + — + terms in 04, 06, etc. \sin 0/ 12

it is apparent, to a second order, that the derived curvature correction is the same as from the Hankel function series.

Another question concerning earth curvature is the approximation of the radial function in the solution. In particular, it is implied in the present analysis that the radial wave impedances are the same as if the lower edge of the ionosphere is a plane boundary. On physical grounds, one would not


Ί — i — i — i — i — i — i i i i r

ωΓ=οο h = 90km

DISTANCE, />(km)

(a)

0.03

0.02 [

0.011

1 I I I I I I I I Γ

I I 1 I I 200 400 600 800 1000 1200 1400

DISTANCE, />(km)

1 1 1 I I 1600 1800 2000

(b)

FIGS, la, b, c. The normalized magnitude of the magnetic field as a function of distance from the source (p is expressed in kilometers in both ordinate and

abscissa).


LLT Q Z>

to

3.0

2.0

1.0

0.7

0.5

0.3

0.2

0.1

0.07

0.05

0.03

0.02

0.01 200 400 600 800 1000 1200 1400 1600 1800 2000

DISTANCE, p (km)

(c) FIG. lc.

expect this to be an important factor since the depth of penetration, into the ionosphere, of most of the e.l.f. energy is small compared with the radius of the earth. An analytical argument is given in the Appendix of this chapter which supports this contention.

l ' i l i

h \ \^^ F\ \ ^ -

H V ^ — -

—-1 / \ ~ 1 / ^ "*" -_ \J —

L__J__L_I ] _

1 1 1 1 I 1 i

" l^_50ç/s_

■ loo

200

" ___400_____

~ 800___^

~~^——^J6oo_^

J _ J ! ! ' 1 1 1

I 1 i 1 Ì

CUr = 5 x IO5

h = 90 km

1 1 1 1 I

1 1

T~

_.-. -_.

' H — _J

-. A

1 |

5. DISTANCE AND FREQUENCY DEPENDENCE

Certain features of the mode series are best illustrated by a graphical plot of field strength vs. distance under various values of the parameters. Since Ng > Ni it may be safely assumed for e.l.f. that Δ = l/JVf. Then on the assumption that the ionosphere is behaving like a conductor, it is possible to write

where

N,

ωΐ ω=— =

-('-5Γ (13)

(plasma frequency)2

collision frequency

The above relation for Nt is an approximation which is valid when the collision frequency v between electrons and neutral ions is much greater than the angular frequency ω. Since v is of the order of 107 in the lower E region, this

u


is certainly valid at e.l.f. The (angular) plasma frequency co0 is given by

2 e2N e0m

where N is the electron density, e and m are charge and mass of the electron, and ε0 is the dielectric constant of free space. ωΓ, which is proportional to the ratio N/v, is believed to be of the order of 105 in the D layer or the lower region of the E layer. At least this appears to be the effective value deduced from v.l.f. observations for highly oblique incidence as indicated in the previous chapter. At e.l.f. it appears that the waves are reflected at higher levels in the ionosphere and the effective value of ωΓ is somewhat larger, being possibly of the order of 106.

To indicate the variation of the field as a function of distance, the magnitude of T and W are shown in Figs, la to lc for a range of values of œr and h = 90 km. The curves denoted ωΓ = oo correspond to a perfectly conducting ionosphere. These are presented for purposes of comparison with the set for the more reasonable values of œr which are 2 x 106 and 5 x 105. Actually, as plotted, \τ\ is divided by the square root of the distance p (in kilometers) since the curves then become linear at larger distances. The slope of these linear parts of the curves is proportional to the attenuation rate (in db per 1000 km) as usually defined.* It is seen immediately that at the shorter distances, the curves are no longer straight. This certainly indicates that considerable caution should be exercised in computing attenuation rates from spectral analyses of "sferics". For example, it is only when the distance p exceeds about one-sixth of a wavelength, is it permissible to assume that log (\T\/yfp) or log(|#^| x yjp) vary in a linear manner with distance p. Similarly, the phase of Τ(οτ Ηφ) varies in a linear manner only when the distance exceeds about one-half wavelength. Similar remarks apply to the electric field variations with distance.

To shed further light on the nature of the e.l.f. fields in the waveguide, the radial impedance of the wave is now considered. It is noted that

which is by definition the (normalized) impedance of the wave looking in the radial or p direction. First it should be noted that for kp > 1,

W 1

T

* Here the antipodal or "round-the-world*' signal is neglected. Except for the idealized case ωΓ = oo, this would not modify these curves for the distance range indicated. (see Section 3.)

and for kp 4 1 and p < h, w ~ l

Y = Jkp'

For the intermediate and interesting range, kp is comparable to unity and p is somewhat greater than h. Using the numerical values of T and W mentioned above, the ratio | W/T\ and the phase (lag) defined by arg T — arg W are plotted in Figs. 2a and 2b as a function of distance for various frequencies.

50 70 100 POO 300 500 700 1000 2000

D I S T A N C E , / > ( k m )

FIG. 2a. The magnitude of the radial wave impedance.

As before, h = 90 km, but only one value of ωΓ is chosen (i.e. 5 x 105). More extensive calculations [Wait and Carter, 1960] show that the ratio W/T is very insensitive to the value of ωΓ. The impedance ratio W/T is independent of the frequency spectrum of the source provided, of course, it may be represented by an equivalent vertical electric dipole.

<

a:

< oc < Lü co < X DL

50 70 100 200 300 500 700 1000

DISTANCE, yo(km)

FIG. 2b. The phase (lag) of the radial wave impedance.

2000

6. NEAR FIELD BEHAVIOR

At the very short distances where p is of the order of the ionospheric reflecting height (i.e. 90 km), the mode series representation for the fields is very poorly convergent. However, alternate representations are available which converge very rapidly in this case [Wait, 1960a]. For ωΓ = oo, these are

2, ik(rm-p)

and

m = 0 rm

7 ki\)

(15)

(16)

*-ifc(rm-p)

where

rm = [P2 + (2mh)2y\ ε0 = 1, sm = 2 (m # 0).

Individual terms in these series correspond to the contribution from images of the source dipole. These particular expressions are exact on the assumption of perfectly conducting planes: (i.e. ωΓ = oo and ag = oo). It is known, however, from the mode calculations that at short distances (i.e. for p < 100 km), the numerical values of the functions T and W are only very slightly dependent on the finite value of œr and ag. Thus (15) and (16) are generally applicable.

To illustrate the nature of the fields at short ranges, the magnitude of T and W are shown plotted in Figs. 3a and 3b. The solid curves correspond to Eqs. (15) and (16) for h = 90 km. The dashed curves correspond to Eqs. (15) and (16) when h = oo; this is the same as taking only the first term in the series. At the very short ranges, it is seen that the solid and dashed curves merge together. Apparently in this region, the reflecting layer at 90 km can be ignored. At the greater ranges, the Tand Wfunctions both approach unity for the case of no reflecting layer (i.e. h = oo). Corrections for earth curvature resulting from diffraction would, of course, modify these dashed curves at

DISTANCE, p (km)

FIG. 3a. The magnitude of the normalized magnetic field at short distances. The broken curves correspond to no reflecting layer.


5 ? 10 20 30 50 70 100 200 300- 500 700 1000

DISTANCE, p ( km)

Fig. 3b. The magnitude of the normalized electric field at short distances.

ranges exceeding about 500 km. It may be concluded from the information plotted in Figs. 3a and 3b that only for extremely short ranges (i.e. less than about 30 km) is it permissible to neglect the presence of the ionospheric reflecting layer in field strength calculations at e.l.f.

7. EFFECT OF THE EARTH'S MAGNETIC FIELD

In the preceding it has been tacitly assumed that the ionosphere is behaving as an isotropie homogeneous conductor. The presence of the magnetic field renders the ionosphere anisotropie and some modification of the analysis is

required. When the earth's magnetic field is steeply dipping, it is appropriate to invoke the Q.L. (quasi-longitudinal) approximation of Booker [Ratcliffe, 1959]. In this case the refractive index is double valued and thus,

where

Γ Ω Ί1/a

Ν , £ 1 - i — exp(±ir) (17)

œL longitudinal component of gyro-frequency tan τ = — =

v collision frequency and

Ω = < . ' (ν2 + ωί)*

Employing the Q.L. approximation, the reflection coefficient for a sharply bounded homogeneous ionosphere was derived in Chapter VIII, following the work of Budden [1951]. Therefore, for present purposes,

Vi

cos τ/2. (18)

Since ΩΓ = œr cos τ it is possible to write the results in the same form as the isotropie ionosphere if we set

Δ s 1 Ä (19) Ng W) e f f (N^rr * '

where i % «"*-ΡΕ·Γ

and COS T

(w^ff=^(^W2 (20)

are the effective values of Nt and ωΓ, respectively. In other words, the effective conductivity of the ionosphere is modified by the factor cos T/(COS τ/2)2

which varies from unity to zero as τ varies from 0 to π/2. Thus, the attenuation is increased as a result of a steeply dipping (or vertical) magnetic field.

It is now possible to say that

Attenuation with magnetic field cos (τ/2) , x Ξ ~ i-L : (21) Attenuation without magnetic field (cos τ)Υ2

If τ < 1 (i.e. coL <ζ ν), this ratio becomes unity and the influence of the earth's magnetic field vanishes. At the level in the ionosphere where e.l.f. waves are reflected, it is expected that a>L and v are comparable, both being of the order of 106> Assuming that they were actually equal, τ becomes 45

degrees and the ratio of the attenuation rates is 1.05. Thus, it is only when œL is greater than v does the earth's magnetic field appreciably influence the attenuation. The ratio is plotted in Fig. 4 as a function of tan τ or cojv. This illustrates the situation clearly.

ΤΤΤΊ

(Q.L. APPROX.)

FIG. 4. The earth's magnetic field correction to the attenuation factor for the quasi-longitudinal approximation.

The preceding results are subject to the validity of Booker's quasi-longitudinal approximation of the Appleton-Hartree equation [Ratcliife, 1959]. The validity of this approximation requires that

4

4œWL < \ co2 ω]

(22)

where œL and ωτ are the longitudinal and transverse components of the (angular) gyrofrequency. Clearly, this condition is violated when the transverse component of the earth's magnetic field is large such as for propagation along the magnetic equator. This case, however, was discussed in detail in Chapter VII where explicit results for the reflection coefficient at a sharply bounded ionosphere with a purely transverse magnetic field were

given. Adapting the results to e.l.f. it is not difficult to show from Eq. (37) of Chapter VIII that Δ is now given by

where

Z = 1 + — l + i —-M-ii— --L i _

V <orJ \ ωΓ ν* ω,) ν \ ωΓ/

(23)

(-3'- ω | ω 2

2 2 ν ω:

The real and imaginary parts of Δ are plotted in Figs. 5a and 5b as a function of the frequency parameter 5 x IO5 x f/cor where fis expressed in kc/s and œr is in (sec)"1. The various values of coT/v are shown in the curves; positive values of the ratio correspond to propagation from east-to-west while negative values correspond to propagation from west-to-east.

< O

< Lü OC

0.01 1 1 I 100 200

J I I 1 1 I 500 1000

f x 5x ic r IN c/s

FIG. 5a. The real and imaginary parts of the factor Δ as a function of normalized frequency (for a purely transverse magnetic field).

en

s >-en < o <

f X 5 x l 0 5 IN C/S

FIG. 5b. The real and imaginary parts of the factor Δ as a function of normalized frequency (for a purely transverse magnetic field).

The attenuation of the dominant mode in nepers per unit distance is given by -kS0 where S0 is given by Eq. (3). Then, using the above results for Δ the attenuation coefficient a, expressed in db per 1000 km of path length, is plotted in Fig 6a for h = 90 km. It is immediately evident that the attenuation is only slightly modified by the presence of a transverse magnetic field unless, of course, ωτ is considerably greater than v. The phase velocity of the dominant zero-order mode, relative to free space, is simply 1/Re S0. Denoting

this dimensionless quantity by ß it is plotted in Fig. 6b for h = 90 km. It is apparent that the presence of a transverse magnetic field does have some influence on the phase velocity.

e _*: O O o

00.

>-~ o 3 LU > LU < X Q_

LU

>

2 h

ö H

< Z>

0.5

~T

—

u

Γ 1

1 1 1 | 1

cur = 5 x IO5

h=90km

i i i l i

1 1 1 1 1

0 > ^ ^ ^ ^

1 I I I !

1 1 1

i^

-H

1 1 1 50 100 200 500 1000

FREQUENCY, c /S

50 70 200 300

FREQUENCY, c/S

500 700 1000

FIG. 6a, b. Attenuation and relative phase velocity for a purely transverse field.

A reasonable conclusion from the above is that only the vertical component of the earth's magnetic field is really significant in e.l.f. propagation. Furthermore, the ionosphere is effectively an isotropie conductor even in the presence of the earth's magnetic field.

8. EFFECT OF AN INHOMOGENEOUS ATMOSPHERE

The electron density in the actual ionosphere usually increases with height in the E region. Furthermore, the collision frequency decreases with height. Thus, the effective value of the refractive index cannot be assumed constant. An approach is to let the refractive index increase or decrease from some initial value N at height A in a monotonie fashion. In this section the influence of the earth's magnetic field is neglected. Choosing an exponential variation, the refractive index as a function of height is explicitly

iV(z) = 1.0 for 0<z<h

= N exp[(z - ft)//] for z > h (24)

where / is a scale factor. If / > 0, the refractive index is increasing with height and, if / < 0, the refractive index is decreasing with height. It was shown in Chapter VI, Section 11, that

s°si-MI+-fr} <25)

where

and

where I0, Ilf K0, and Ki are modified Bessel functions with their conventional meaning.

The preceding results are valid for \N\ > 1 which is well satisfied at e.l.f. To the same approximation

Ν4-'!'ΓΒ(!')ν"" <28) where œr = ωΐ/ν in terms of the plasma frequency ω0 and collision frequency vatz = h. It then follows that the imaginary and real parts of the propagation constant kS0 are given by

imks°—à{£yp(x) (29)

and Reks^k+M^Tnx) m


where

P(x) = | Q | s i n ^ - a r g ß ) v / 2 (31)

P'(x) = | ß | c o s g - a r g ß ) V 2 (32) and where

χ = \ΜΝ\ = \Μ\(ω,Ιω)ν>. The functions P(x) and P'(x) approach unity if x is sufficiently large. This limiting case corresponds to the homogeneous ionosphere. Thus, P(x) is the modification of the attenuation and P\x) is the modification of the phase resulting from a nonhomogeneous ionosphere. It may be observed that the attenuation is generally lower if the refractive index is an increasing function with height. On the other hand, the attenuation is increased where the refractive index is decreasing with height.

To illustrate the behavior of the attenuation rates as a function of frequency, they are plotted in Figs. 7a and 7b in terms of db/1000 km of path length for a frequency scale from 50 c/s to 1.5 kc. Values of the scale distance / are indicated in kilometers. The height h of the lower edge of the ionosphere is taken to be 90 km. The values chosen for c5r are 105 and 106 which are not inconsistent with previous work.

3

2

E o i 8 0.8

^ 0.6 3 0.5

0.2 0.05 0.08 0.1 0.2 0.3 0.4 06 08 I 15

FREQUENCY, kc

FIG. 7a. Attenuation at e.l.f. for an exponential profile, h = 90 km and ώΓ = 106.

It is seen that the curve for / = oo corresponding to a homogeneous ionosphere has a slope of \ as it should. Depending on the sign of / and its magnitude, the slope may be modified considerably for the inhomogeneous ionosphere. The dotted curve in Fig. 7b corresponds to night-time experimental data from Chapman and Macario [1956] who obtained it from the spectral analyses of large numbers of atmospherics recorded in London.

1 I I I I I Γ

^ = -20

i=IO

Clearly this experimental curve appears to fit fairly well on the / = 30 km curve, at least in the range 100 c/s to 1.5 kc. There is no reason to expect any better fit than this because of the idealized profile assumed. Furthermore, the atmospherics analyzed by Chapman and Macario were quite often near the receiving location in terms of a wavelength.

E OC O O O

JO •o Ö

0.050.06 0.08 0.1 0.2 0.3 0.4 0.6 0.8 I 1.5

FREQUENCY, kc CALCULATED MEASURED (CHAPMAN AND MACARIO)

FIG. 7b. Attenuation at e.l.f. for an exponential profile, h = 90 km and ώτ = IO5.

A somewhat more refined analysis of the exponential profile has been given recently by Galejs [1961]. In his theory the lower edge of the ionosphere is not abrupt.

From the work of Galejs it is indicated that the sharply bounded model is not a completely adequate representation for an actual ionospheric profile. This aspect of the problem was discussed recently by the author [Wait, 1962] where a perturbation method is used to account for the presence of a transition region at the lower edge of the ionosphere. In most cases the sharply bounded model (with a continuous variation above) can be located so the effect of the transition region is minor.

Very recently, Harris and Tanner [1962] have considered the problem of the continuous conductivity profile. They make a number of approximations in the formulation which are stated to be applicable in the e.l.f. range. Particular attention is paid to the regions where the refractive index is not large compared with unity. In common with Galejs [1961] they contend that this region must be accounted for in the interpretation of experimental data at frequencies below 100 c/s.

P Γ

— —

-

Ji - -20

-30

-50

-100

00

100

50 '

30 '

^ = 20

_i

i—: i i i | r~

y/ s y? y y y Ss y

y y s y y y / y

y y * y y y y

y 'y y ' *y y

y y V y ' <9/ y

l/l ,1 1 1 I_

i 1 1 1 ! Ì 1 1 1

__S^ ^-^■^--^$3

^ - ^ ^ ^ x ^ ^ —-—-^^-$^^^v^J

^ j S y\

^^^^£\s /'\ ^ y ^ y ^ X ^ /

/ j ' y ' /

/ / h=90km

/■ r = ' ° 5

— ■A

: >j

_ 1 I I ! I t i l i

One possibility for e.l.f. field-strength calculations is to return to the harmonic series representation of the field. A series of this kind involves a sum of the form Y,f(n)Pn(cos Θ) where n takes all integral values. This was discussed briefly in Section 10 of Chapter VI. Recently Johler and Berry [1962] have undertaken to program series of this kind which are applicable to a spherical earth surrounded by a concentrically stratified plasma. Despite the poor convergence of the series they obtained numerical results by using a large scale computer. The curious and unexplained feature of some of their results is that the computed field strength vs. frequency curves exhibit sharp minima which are separated by small intervals. This and other anomalies can probably be attributed to the great difficulties inherent in such calculations.

It appears that further experimental and theoretical work on the attenuation of e.l.f. radio waves is necessary to resolve some of the apparent discrepancies in recent work.

9. PROPAGATION OF E.L.F. PULSES

In the foregoing discussion, it has been assumed that the source is time harmonic. In most cases of practical interest, the current in the source dipole is of a transient nature such as a surge. The radiated fields are also transient in nature. While the resulting waveforms may be transformed to the frequency plane via spectral analyses, it is often more convenient to study the waveforms themselves [Schumann, 1956; Liebermann, 1956; Wait, 1958; and Tepley, 1959].

The source is assumed again to be equivalent to a vertical electric dipole on the earth's surface, but now its current-moment is a function of time and is denoted p(t) which is zero for t < 0. At a distance p along the earth's surface, the resulting vertical electric field is ez(t) and the horizontal magnetic field is Ηφ(ί). The Laplace transform of the current moment is denoted P0(s) or more explicitly »<#

P0(s) = Lp(t)= le-*/**) dt (33)

where s is the transform variable and may be formally identified with ι'ω. A1SO E2(s) = Lez(t) (34) a n d ΗΨ00 = ΙΛΨ(0 (35) where L is the Laplace transform operator defined above. Now it is assumed that the important frequencies in the spectra are sufficiently low that only the zero-order mode need be retained. Thus

310

where


*-H(5)T (37)

and where K0 and Kt are modified Bessel functions of argument sS0p/c. These may be asymptotically represented by

and ™«(έ)"·"[ι+ΜΜ (38)

(39)

for large x, where A0 = -1/8, B0 = 9/128, Ax = 3/8, Bl = -15/128.

Thus

- é ©¥«*'·«· — sSop/c

1 +

1 +

s ρ s2 V7

s p s2 \p / (40)

where x has been replaced by sS0p/c in the exponent, but is replaced by sp/c elsewhere. This is justified since S0 is near unity. In fact, the binomial expansion

S0 = l + 2h (5T-im.+k®'£T+~ <4i> may be truncated beyond the second term in most cases of interest, although the third term is retained since it does not complicate matters.

The current moment is now taken to be an impulse, that is

KO = PoS(t)

where δ(ί) is the unit impulse function. Thus

Po(s) = Po-

The transforms may now be readily inverted

e°(0 ~JL/±y/2-^exDr^f£oil x -»ffcj(0 " 4πΛ UP/ ß% * W W J

where

and


ρ ^ = 2 ( ϊ )"«ρ(-4ί) <«>

P,m - 4 ( 1 ) * e x p ( - i ) - 2 i , , « rfc^) (45)

''-'-*^",Α=έ(ίΓ· It is immediately seen that if (ct'/p) <ζ 1, the field responses are both

proportional to P(T). To illustrate the nature of this function, it is plotted in Fig. 8 as a function of Γ. To facilitate the application a multiple scale is included which relates actual time t' in micro-seconds with the parameter T for distances p of 1000-4000 km. The height of the ionospheric reflecting layer is taken as 70 and 90 km. The quantity B indicated on the curves is related to the ionospheric conductivity by

βθσ^ corh

The values of B shown, namely 0.05 and 0.1, are typical. For h = 90 km, these correspond to ωΓ values of 4 x 105 and 2 x 105, respectively.

The responses ez{t) and Ηφ(ΐ) to a general source p(t) can be expressed in terms of the impulse responses e°(t) and Aj(/), respectively, by using the convolution theorem. Thus

ez(t) = - f' p(t - φ°2(τ) άτ · u(t - pic) (46) Ρθ J p/c

and similarly for /ιφ(ΐ). For example, if a special analytical form for p(t) is chosen such as

the convolution integral may be evaluated to give

where

y* = β* + (fitj* = ^ (ff + (6îm)*.

Wh) (48)

x


The function P(T) was defined above and is the same as the one plotted in Fig. 8. Now, however, T is to be identified with t'/y rather than t'/ß. The special form for p(t) given above is a unidirectional pulse which reaches its

FIG. 8. Transient response of the zero-order mode for an impulsive vertical dipole source. Often this is called the "slow tail" as it is preceded by an

oscillatory portion containing energy in the v.l.f. spectrum.

maximum value at / = tm and decays to 7 per cent of its maximum value at / = 3tm. If the duration, 3tm, of the source pulse is much less than ß the response ez{t) approaches e°{t). This is not surprising since p(t) approaches p0ô(t) if tm approaches zero.

10. INTERPRETATION OF HEPBURN'S EXPERIMENTAL DATA

We are now in a position to employ these results to interpret experimental data on the "slow tails" of atmospheric waveforms [Hepburn, 1957]. A convenient parameter is the separation in time between the start of the high-frequency oscillatory part of the wave-form and the maximum of the first half-cycle of the slow tail. Actually, in Hepburn's work this "slow-tail separation" is taken as the interval from the triggering of the oscilloscope sweep to the maximum point of a sine wave which is fitted to the first part of the slow tail. It is now noted that the argument T, in the response functions PJT), is given by

_t' _ t' τ = Γ ϊ^+ ^ί 2 (49)

where and δ = 6f„

and δ is a measure of the width of the source pulse. This immediately suggests that the square root of the slow-tail separation should be approximately a linear function of distance p. To test this thesis, Hepburn's data are replotted

0.05

< CE

S.

g

~ 1

__

-

-

-

— r i — τ " ~Τ 1 Τ~~Ί

° } F R 0 M HEPBURN'S DATA

x *

y/*.

>r ^-—"—ο

L_J I L_

χ y^C

X >r

ο J^&^^

_3 l i t .

'

' χ _

-

-

J 0 1 2 3 4 5

ρ Ι Ν IO3 km

FIG. 9. The square root of the separation between the oscillatory head and the maximum of the slow tail as a function of range. (ts is in seconds.)

as shown in Fig. 9. As is indicated, straight lines may be drawn through the experimental points for both day and night. (Each of these data points represents the mean of a large number of individual observations.) That these

experimental points lie on a straight line is a fairly good confirmation of the validity of the waveguide model for e.l.f. propagation.

From the theory presented here the maximum value of P(t'/y) occurs when t'jy is 0.09. Denoting this time by ts, it follows that

liti ωγ-or

= 0 . 0 9 ^ - 1 · (50)

(i5y/2 = 0 . 3 [ £ - V + <5,/2 v s I2h ω^

On identifying ts with the slow-tail separations plotted in Fig. 9 it is a simple matter to deduce effective values of ωΓ by measuring the slopes of these straight lines. Assuming that h = 70 km for daytime and h = 90 km for night-time, it readily follows that

ω/day) ^ 1.2 x 105

and cor(night) £ 3.5 x 105.

These are the same order of values deduced from v.l.f. field-strength data [Wait, 1957]. In terms of conductivity these become

<j(day) s 1.0 x 10"6 mho/m

tf(night) s 3.0 x IO"6 mho/m.

It is interesting to note that Hepburn [1957] deduces the values 2 x 10" 7 and 4 x 10"7 mho/m for day and night, respectively.* His analysis, however, was based on a theory of Hales [1948], which is not really applicable to broadband waveforms. On the other hand, Liebermann [1956] deduces the values 1.1 x 10" 6 and6 x 10"6 mho/m for day and night, respectively. These are in fairly good agreement with the present analysis, although the experimental conditions are quite different.

On the assumption that the source pulse is equivalent to the function p(t) it is possible to deduce the effective value of δ or tm from the intercept on the ordinate in Fig. 9. This yields a value of <5 of 4.9 x 10"3 sec. The particular conclusion to be drawn from this deduction is that certain lightning strokes have a slowly varying current component whose time constant is of the order of several milliseconds. This fact is consistent with the observations of Pierce [1955a and b, 1960] that there may be pronounced slow changes of the electrostatic field after the main stroke in the discharge. It is probable that most of Hepburn*s data correspond to discharges of this type. The slow tail radiated from a discharge, which does not have the lingering or slowly varying component, would be of much smaller amplitude.

* The values of conductivity deduced by Hepburn [1957] are about a factor of 2 less than those deduced by Hepburn and Pierce [1953].

In the present theoretical study it has been assumed that the e.l.f. signal propagates entirely in the space between the earth and a concentric ionosphere. It is quite possible that whistler type propagation may also be important. Furthermore, sources of e.l.f. energy may be present in the exosphere due to the possible existence of ionized hydrogen whose gyrofrequency is of the order of 900 c/s [Dungey, 1955]. Other types of ions such as sodium may also be significant as pointed out by Aarons [1956]. Despite these extraneous factors, the evidence that the "slow tails" are propagated in the earth-ionosphere waveguide is overwhelming [Wait, 1960b].

It is the opinion of the author that the only really questionable assumption is the neglect of heavy ions on the constitutive properties of the lower edge of the E region or the top of the D region.

11. INFLUENCE OF HORIZONTAL CURRENTS

Although most cloud-to-ground and intracloud lightning discharges may be represented by a vertical dipole, it is believed that certain cloud-to-cloud discharges are better represented by a horizontal electric dipole [Norinder and Knudsen, 1959]. The excitation of the earth-ionosphere waveguide by a horizontal dipole has been treated in considerable detail elsewhere [Wait, 1960a]. In the e.l.f. portion of the spectrum the height of the discharge is very small compared with the wavelength, and thus the transform of the field is given adequately by

EM s ^ ( g ) Λ sP(s) e - 8 « " ^ / * , ) * cos φ (51)

where ag is the conductivity of the ground, ε0 is the dielectric constant of free space, and φ is the angle subtended by the horizontal dipole and the direction to the observer. For a source pulse, p(t\ defined by Eq. (47), it is known that

P(s) = p0exp(-<5V/2).

The transform for the vertical electric field may now be readily inverted to give

where, as before,

t' = t-plc.

Since lightning discharges, in the general case, would seldom be purely vertical or horizontal, it is of interest to consider the response to an inclined electric dipole. Furthermore, in view of the small dimensions of the discharge


paths in terms of a wavelength, it is permissible to represent the inclined channel by superimposed vertical and horizontal dipoles. For example, if the current moment of the vertical electric dipole is p(t\ the current moment of the horizontal electric dipole is gp(t\ where g is assumed to be a positive or negative number. (In the most general case, g could also be a function of time.) For lightning strokes with long horizontal sections, g could be a large number.

Invoking the above principle of superposition, it readily follows that the far-zone expression for the inclined dipole is given by

where

z(0 = ^ (£) iP(T) + GQ(T)-]y-MO (53)

T=t'/y. The direction φ = 0 corresponds to the direction of the horizontal dipole component.

To indicate the behavior of the waveform the quantity

S(T) = P(T) + GQ(T)

1 + lGl

is plotted in Figs. 10a and 10b for both positive and negative values of G. In both parts of the figure the vertical scale is normalized so the maximum value is unity. The special case G = 0 corresponds to a vertical dipole source,

1 1 ! I I I ! I I J I 1 M I] II

cu

0

-20

-40

-60

-80

-loo;

1 I (b)

-

-

J 1

M M i l t -·" ' 1

^Ύη \ Ό/ 1 \ °/ / \ '/ / \ \ /V \ V J Ί

.J. J.I 1 1 II I

■1 I I 1 I 1 11

: ! [ M i l l

-

-

A

0.01 0.02 0.05 0.1 0.5 1.0 2.0 0.01 0.02 0.05 0.1 0.5 1.0 2.0

FIGS. 10a and 10b. Transient response of the zero-order mode for an inclined dipole source. When G = 0 the dipole is vertical and when G = ± co the

dipole is horizontal.

E,L.F. Propagation—Theory and Experiment 317

whereas G = ± oo corresponds to a purely horizontal dipole. Such a range of values can be expected since lightning discharges may be vertical, horizontal, or inclined, and may have any polarity [Pierce, 1955b; Norinder and Knudsen, 1959].

The results shown in Figs. 10a and 10b are specifically for the source function p(t). If δ = 0 the source is the impulse ρ0δ(ί) and the corresponding response is still given by Eq. (53), but now

.y = ft IW/ft and

G = g(e0corlagY/2(hlp) cos φ.

This special case was given previously [Wait, 1960c]. The modification of the pulse shape of e.l.f. waveforms as a result of the

inclination of the current channel would appear to be an important factor in the interpretation of experimental data. For example, the observed reception of a pulse which has a second half-cycle of the same magnitude as the first half-cycle could be reconciled with an inclined discharge channel with an effective value of G of about ±0.2. As is seen in Fig. 10b, a third half-cycle of relatively small amplitude may also be produced if G is in the range from about —0.1 to —0.2. Slow-tail waveforms consisting of more than two half-cycles have been observed on certain occasions [Tepley, 1959], and it is possible that some of them may be due to the inclination of the discharge channel. It is more probable, however, that multiple or pulsating strokes in the discharge process are more effective in producing waveforms that have additional half-cycles. This would be particularly true when the sources are lightning strokes over the sea, since then the coupling from the horizontal currents to the e.l.f. waveguide mode is very weak.

It can be seen that a value of G of ±0.1 requires that the ratio, g, of the horizontal to the vertical currents is of the order of a 100 for land and as much as 104 for sea. Such nearly horizontal discharges in nature are probably very rare. However, they could be produced by an artificial generator or by feeding a long grounded cable.

There is one important distinction between the effects produced by an inclined channel with a unidirectional current pulse and a vertical channel with an oscillatory or pulsating current source. In the former case, the slow-tail waveform has a dependence on direction (i.e. azimuth), whereas, in the latter case, there should be no appreciable dependence. No doubt, this important distinction could be resolved if simultaneous observations of slow-tail waveforms were made at widely separated stations.


Appendix

SURFACE IMPEDANCE OF A SPHERICALLY STRATIFIED CONDUCTOR

In this chapter the concept of surface impedance at some height h in the upper atmosphere is utilized. It is implicitly assumed that this value does not depend on earth curvature. Physically such an assumption is expected to be valid when the penetration of the waves is small compared with the radius of the earth. However, at e.l.f. one may question the validity of this condition since the depth of penetration can be very large. Consequently, it is worthwhile to obtain a rigorous formula for the surface impedance at the lower boundary of a concentrically stratified conductor.

With respect to the usual spherical coordinate system (r, 0, φ\ the lower boundary of the reflecting layer is at r = b. The outward looking impedance, for T.M. waves, is defined by

zin )=+f; l (54)

for spherical harmonic waves of order n. (These waves behave with Θ according to the factor Pn(cos 0)). Beyond r = b, the medium is considered to be homogeneous out to r = b + h^ with electrical properties σί9 ε1 and μ0. For r > b + hu the medium is taken to be homogeneous with electrical properties σ2, ε2 and μ0.

Although we can solve this problem in a straightforward manner using boundary matching techniques, it is desirable to apply Schelkunoff 's non-uniform transmission line theory as in Chapter VIII. Thus the medium between r = b andr = b + hY can be regarded as a non-uniform transmission line. The characteristic impedance of the line looking outward is

where Kn is a modified spherical Bessel function of order n defined by

^„(z)=(|y/2iC„+,/2(Z) (56)

in terms of the modified (cylindrical) Bessel function Kn+Vz of order n -f £. In the above

ηι = ( ^ ° ω ) and γχ = O o ^ + ι^ω)]^

are the intrinsic properties of the medium.

In a similar fashion the characteristic impedance looking inward (i.e. towards the center of the earth) is

Wir) where

1π(ζ) = (τΓ/ η + ι / 2 ( ζ ) · The line is now considered to be terminated by an impedance

H a F ^ l = ** + **>· (57) Then, from the theory of non-uniform transmission lines, it follows that

ZfO = M(b) \ + qAf° (58) 1 + qhAhBh

where the q's are reflection coefficients at r — b + ht and the A's and B's are transmission factors. The subscripts e and h are to indicate that the quantities pertain to the electric or magnetic fields, respectively. The specific forms of these quantities are

= ijPjb + ht) - 1/Af(6 4- fei) q° ~ \jP{b + ht) + i/M(b + hi)

P(b + Λ,) - M(b + hi) <ih = P(b + hi) + M(b + hi)

_ bfrMb + hj)-] =(b + hjl'JjyM e (b + hi)R'„lyiby e bì'Mb + hj}

= bfLn[yj(b + hj)] ^ (b + hJljjyM " (b + hJ&JyW " bÎMb + hiï'

This is the rigorous solution of the problem. When Z\n) replaces Z\q) of Chapter VI the formal solution given there is fully applicable to a spherical earth surrounded by a concentrically stratified ionosphere. The extension to more than two layers is simple. For example, in the case of three layers we need only replace the function P(b + hA) by the appropriate two-layer impedance function. Thus, the solution for m + 1 layers is readily written down in terms of the solution for m layers.


To obtain the required simplifications it is necessary to replace the spherical Bessel functions by their Debye or saddle approximations. To obtain these directly it is noted that since R(z) satisfies

d2£„(z) dz2

the function M(r) satisfies

[.♦*£>>

M2(r) + ni àM{r) Vi àr

1 + n(n + 1)

7Ì + 1)1 2 H"1

(59)

(60)

The Debye approximation corresponds to the neglect of the term involving the derivative of M(r). This is valid provided R e ^ r ) > 1 and |yj.r|2 is somewhat greater than n(n + 1 ) . Thus

Similarly, "^hWi (61)

These approximate forms are equivalent to taking the WKB solution of equation (59) for Kn or in. Thus, it easily follows that the transmission factors have the forms

A„e"

b

6 + /l!

ΛΓ1 e"

6 A r ' e " · "

Α β * + * ΐ Λ β - *

(62)

(63)

(64)

(65)

where

and

(•yi(*+*i)r „(„ + 1)1%

Λ . =

n(n + 1)

inni« + 1)

2».2 rît

Therefore, within this approximation, AeBe^e-2*»^AhBh.

The planar approximations are obtained under the assumption that h1 < b. Then, writing n(n + 1) = —S2ylb2 one easily finds that

Z(») ~ „ [i _ ΊΪ sA 72 bi " y°S] (66) Z ' = f l \ l

V 2 Ò 2 r 2 _ 2s2-,./2 W >

72 L7i ~ 7(Λ J

This is identical to assuming that the medium is behaving locally as a planar stratified medium. In this case S can be identified as the sine of the (complex) angle of incidence in the medium below the reflecting layer at r = ft. It is evident that for any applications in this chapter the planar assumption is well justified.

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WAIT, J. R. (1957) On the mode theory of v.l.f. ionospheric propagation, Rev. Geofis. Pura. Appi., 36, 103-115. (Paper presented at International URSI Conference of Radio Wave Propagation, Paris, Sept. 1956.)

WAIT, J. R. (1958) Propagation of very-low-frequency pulses to great distances, / . Res. Nat. Bur. Stand., 61, 187-203.

WAIT, J. R. (1960a) Terrestrial propagation of v.l.f. radio waves, / . Res. Nat. Bur. Stand., 64D, 153-204.

WAIT, J. R. (1960b) On the theory of the slow tail portion of atmospheric wave-forms, / . Geophys. Res., 65, 1939-1946.

WAIT, J. R. (1960c) Mode theory and the propagation of e.l.f. radio waves, J. Res. Nat. Bur. Stand., 64D, No. 4, 387-404.

WAIT, J. R. (1962) On the propagation of v.l.f. and e.l.f. radio waves when the ionosphere is not sharply bounded, / . Res. Nat. Bur. Stand., 66D (Radio Prop.) No. 1, 53-61.

WAIT, J. R., and CARTER, N. F. (1960) Field strength calculations for e.l.f. radio waves, Nat. Bur. Stand. Technical Note No. 52 (PB-161553).

WATTS, J. M. (1957) An observation of audio-frequency electromagnetic noise during a period of solar disturbance, / . Geophys. Res., 62, 199.

WILLIS, H. F. (1948) Audio frequency magnetic fluctuations, Nature, 161, 887.

Additional References CROMBIE, D. D. (1958) Differences between east-west and west-east propagation of v.l.f.

signals over long distances, / . Atmos. Terr. Phys., 12, 110. DEAL, O. E. (1956) The observation of very low frequency electromagnetic signals of

natural origin. Unpublished Ph.D. Thesis, University of California, Los Angeles, Calif.

DUFFUS, H. J., NASMUTH, P. W., SHAND, J. A., and WRIGHT, C. S. (1958) Subaudible geomagnetic fluctuations, Nature, 181, 1258.

GALLET, R. M. (1959) A very low frequency emission generated in the earth's exosphere, Proc. I.R.E., 47, 211.


HINES, C. O. (1957) Heavy-ion effects in audio-frequency radio propagation, / . Atmos. Terr. Phys., 11, 36.

HÖLZER, R. E., and DEAL, O. E. (1956) Low audio-frequency electromagnetic signals of natural origin, Nature, 177, 536.

JEAN, A. G., MURPHY, A. C , WAIT, J. R., and WASMUNDT, D. F. (1961) Observed attenuation rates of e.l.f. radio waves, / . Res. Nat. Bur. Stand., 65D (Radio Prop.) 475-479.

SCHUMANN, W. O. (1957) Über elektrische Eigenschwingungen des Hohlraumes Erde-Luft-Ionosphare angeregt durch Blitzentladungen, Z. Angew. Phys., 9, 373.

SMITH, E. J. (1960) The propagation of low-audio frequency electromagnetic waves, Ph. D., dissertation, University of California at Los Angeles.

STOREY, L. R. O. (1956) A method to detect the presence of ionized hydrogen in the outer atmosphere, Canad. j . Phys., 34,1153. (Discusses the effect of protons on the propagation of whistlers in the exosphere, collisions are neglected.)

WAIT, J. R. (1958) An extension to the mode theory of v.l.f. ionospheric propagation, / . Geophys. Res., 63, 125.

WAIT, J. R. (1962) On the propagation of e.l.f. pulses in the earth-ionosphere waveguide, Canad. J. Phys., 40 (in press, Oct.). (This paper contains additional comments on the enhancement of the negative tail.)

WATSON-WATT, R. A., HERD, J. F., and LUTKIN, F. E. (1937) On the nature of atmospherics, Proc. Roy. Soc, London, A, 162, 267.

WATT, A. D. (1960) e.l.f. electric fields from thunderstorms, / . Res. Nat. Bur. Stand., 64D (Radio Prop.), No. 5, 425-434.

WATT, A. D., and MAXWELL, E. L. (1957) Characteristics of atmospheric noise from 1 to 100 kc, Proc. I.R.E., 45, 787.

Chapter XI

ASYMPTOTIC DEVELOPMENT FOR GUIDED WAVE PROPAGATION

Abstract—The relationship between mode theory and ray theory is discussed in a general manner for stratified media.

1. INTRODUCTION

The physical connections between mode theory and ray theory are often regarded in a mysterious fashion. Probably, the difficulties arise from over-extending physical concepts to explain rather complicated phenomena. Thus, it would appear that an analytical discussion of the problem is worthwhile.

Attention is confined to horizontally polarized waves in this chapter, although the extension to vertical polarization is briefly mentioned. The starting point is the exact representation of the field in terms of a complex integral. The integrand is then expanded as a summation which leads, in an asymptotic sense, to a ray theory. The development leads readily to a concise treatment of caustics in ray systems. The application to tropospheric radio propagation is mentioned briefly.

2. FORMULATION OF PROBLEM

The model adopted for the present discussion is a horizontally stratified medium consisting of two inhomogeneous half-spaces separated by a homogeneous space. The situation is illustrated in Fig. 1 with respect to a cylindrical coordinate system (ρ, φ9 ζ).

m//k//m//m/////////////////////////////////////w

^mmmtmimm?mmmmmmm^. FIG. 1. Two inhomogeneous half-spaces separated by a homogeneous slab.

324

Asymptotic Development for Guided Wave Propagation 325

The relative refractive index N(z) is arbitrary for z < 0 and z > h, but is normalized to unity for 0 < z < h. The magnetic permeability of the whole space is taken as μ and is constant everywhere. The source, for convenience, is taken to be a magnetic dipole located at z = z0 and coaxial with the z axis.

In view of the obvious azimuthal symmetry, the fields can be derived from a magnetic Hertz vector with only a z component φ. Thus, the electric field (which has only a φ component) is given by

Εφ = ιμω(ΟφΙδρ) (1)

and the magnetic field components are found from

δ2ψ dpdz

HP = T7T- (2)

and H>={k2 + Q* (3)

where k is the constant wave number for the homogeneous space. The quantity, ψ, which can be described as a potential for the space 0 < z < h can be made up of solutions of the equation

It is evident that these have the form exp(±ikz)H(

02)(kSp)

where C = (1 — S2)Vl may take any value provided the imaginary part of S < 0. The latter condition, coupled with the use of the Hankel function of the second kind Htf\kSp) assures that the fields are not infinite as p -► oo.

The fields in the inhomogeneous half-spaces, for z < 0 and z > h, must also vary as HtfHkSp) and thus they satisfy the equation

[^ + *2(Φ = ° (5) where q2(z) = k2(N2(z) - S2).

3. THE COMPLEX INTEGRAL REPRESENTATION

It is now possible to construct the required form of the solution in the homogeneous region in terms of the normal wave impedances at the interfaces z = 0 and z = h. First, we note that the field in the space 0 < z < h can generally be written

φ=\ lA(C)e-ikCz + B(C)t+ikc<-]H{2\kSp) dS (6) J - o o

Now, near the source the potential φ must behave in the manner ç-ikR

^ ^ = M _ _ ( 7 )

where M is the strength of the source and R = [p2 + (z — z0)2]i/2. As is well known, the primary potential ψρ can be written in terms of the integral

ikM C + 0° r = _ l K M \ c-ikC\z-2o\H(2)(kSp){SlC) d S

2 J -oo (8)

Now, the normal wave admittances at the interfaces are defined by

+Η„/£Ψ1= 0 = Yi(Q (9) and

-ΗΡΙΕφ-ί2=„ = y2(o (10)

where Yt and Y2 are functions of C which, itself, may be regarded as a cosine of the (complex) angle of incidence of the waves in the homogeneous region. The forms of Y1 and Y2 may be obtained from the theory for plane waves incident on a single half-space as treated by the methods of Chapters III and IV.

By making use of Eqs. (1), (2), (9), and (10), the boundary conditions may now be concisely stated in the following manner

and \^=^ikYi{C) at Z = ° (U)

i ψ = - ikY2(C) at z = h. (12)

Then, without much difficulty, the solution for φ is found to be given by

t.-a»iycm.*,)s-£ a« where

[eifcCz + Ri Q-ikcz^eikC(h-2o) + fl2 e-ifcC(*-zo)-j e ^ l - K A e - 2 ^

and

«■-Ιτ^ (15a> - R-OT (15b)

This result is valid in the region 0 < z < h. An alternate form for F(C) may be obtained by replacing z by z0.

^(W "" JkCh/A r> n ~-2ikhC\ ( ^ )

It is convenient to introduce two new functions, Fi and F2, as follows

F^C) = F(C)]I=0 = ettC _ ^iJ?^ ç-2mC) (16a)

and p i n . w n l [ e ^ ' + ^ e - ^ X l + J t J F 2 ( C ) - F<Ql ·-* - ^ ( i - R A e - ^ * (16b)

Since ^ is continuous, the potential in the lower and the upper half spaces must have the respective forms

ikM C + °° S dS Φ Y J *Ί(9Λ(Ζ. Q«i)2)(fcSp) — (17)

for z < 0, while

* = - — I F2(Qf2(z, QHtfXkSp) — (18)

for z > h, where Λ(0,Ο = 1 and /2(fc,C) = l.

In view of Eq. (5), it is apparent that/ t and/2 must satisfy [d2/dz2 + g2(z)]/lf2(C) = 0 (19)

in the respective regions, and furthermore, they must give rise to vanishing fields as z -» |oo|.

Actually, this is the complete formal solution of the problem. The results for two homogeneous half-spaces, of refractive indices Ni and N2, bounding central homogeneous region can be retrieved as a special case by simply the using the following forms:

Υχη0 = (Ni - Ξψ

Υ2η0 = (Ni - S2f2

f^C) = exp[ifc(Nj - Ξψζ] and

hip) = exp[-ik(iV2 - S2)Hz - h)l

It should be noted that in the general case the functions F(C), Ft(C), and F2(C), have poles where

l-JR1R2e-2,'fc/,c = 0. (20) Solutions of this equation yield the characteristic values of C which correspond to the waveguide modes. In fact, as discussed in Chapter VI, the integral for ψ may be evaluated by deforming the contour to enclose these poles. In adopting this procedure one must also take account of branch line integrations as mentioned before.

Y


Before passing, it should be mentioned that the particular model shown in Fig. 1 can be used to describe a region which is inhomogeneous for all z. Thus, the width of the homogeneous region is simply allowed to vanish so that the two half-spaces now have a common interface which also contains the source. For a magnetic dipole this limiting process causes no difficulty, however, it is convenient to consider the limits in the following order: z0 -> A/2 and A -► 0. Thus, the form of the potential for a generally inhomogeneous medium is given by

ikM f + °° S dS *=-— J Ji.2(Qfi.i(*,QHV\kSp)— (21)

where

In Eq. (21),/ l f2 is the appropriate solution of the equation

[^2 + β2(ζ)]/ι.2 = 0 (23)

which has the initial condition that

/1>2(0,C) = 1.

4 THE MODE REPRESENTATION

As discussed in considerable detail in Chapter X, the complex integral, of the form given by Eq. (21), may be expressed as a sum of residues of the poles with due regard to the contributions from the branch line integrals. On the assumption that the latter are negligible, the field may be expressed as a sum of modes in the form

ψ S nkM Σ Π -. =f H(o\kSnP) (24) n= - o o \ O 1 I

LâCF1>2(C)/1(2(z,C)Jc=cM

where the Crt's are a solution of the equation Ä1(C)K2(C) = l = e-2nin (25)

where n may be any integer.* The latter root-determining equation may be

♦ In writing Eq. (24), use is made of the fact that

d _ S d

described as a "transverse resonance condition". To demonstrate this, it is possible to rewrite it, by making use of Eqs. (15a) and (15b), in the form

Y^C) + Y2(C) = 0 (26) This states that the sum of the (outward) normal admittances at z = 0 must vanish for the existence of a mode. The concept of a "transverse resonance condition" has been utilized very effectively by Zucker [1960] and his colleagues in a study of surface wave problems. At a later stage we shall return to a study of the transverse resonance condition and its application to radio propagation in natural ducts such as in the troposphere and the exosphere.

5. RAY THEORY AND SADDLE POINT APPROXIMATIONS

In the remainder of this chapter the complex integral representation developed above is used as a basis to obtain wave corrections to ray theory. For this purpose some simplifications are made to facilitate the discussion. Specifically, it is assumed that the source is located at the center of a symmetrical layer. Thus in Eqs. (22) and (23), Rt and R2 are identical and may be replaced by R, and the subscripts 1 and 2 may also be dropped elsewhere. Thus, the potential φ can be written

The factor (1 — R)'1 can be written provisionally as an infinite geometric series* and, thus

**kM £ φ„ (28) m = 0,l ,2, . . .

where /(z, C)[l + R(C)W(QHtf\kSp) ^β. (29) 2 J _ ,

The refractive index N(z) is now taken to be a smooth monotonically decreasing function as |z| increases from zero. Without loss of generality, N(z) is taken as unity for z = 0, as stated previously. Furthermore, it is assumed that the variation is sufficiently slow that the phase integral approximations for the wave functions are valid. Thus, for 0 < z < zt it is possible to use the approximate form

/^H-' - H"dz)+'· M-2i S? d ' M + f /,'« d2)] /(z. c ) = | —I 7 γτ, c

w / 1 + i e x p l - 2 i qdzj (30)

* This approach has also been used in ionospheric radio propagation (Wait, 1961).


where q = k[N\z) - S2^

q0 = k[N2(0) - 52]1/2 = fcC and zY is defined by

ΛΓίζ,) = S.

For the region 0 > z > —zu the same expression holds if z is replaced by —z. In both cases z must not be near zx as discussed in Chapter XI. The range of validity can easily be extended by using a more elaborate WKB form. For the present discussion, this would be an unnecessary encumbrance.

To within the same phase integral approximation, the reflection coefficient is given by

R(C) s i exp(-2i Ziq dz). (31)

A further simplifying approximation is to replace the Hankel function H{2)

by the first term of its asymptotic expansion. This is valid provided \kSp\ > 1 over the important range of the integration. Thus,

HVXkSp)2(J^\-us'. (32)

On utilizing the preceding approximations, it readily follows that

ΦΜ = ΦΙ + Φί, (33) where

qdz + kSp\

qdz + kSp)

- m \ + 2m I q άζ + kSp ) I dS (34a)

and

(m 4- 1) j + 2(w + 1) | q dz + kSp \ dS. (34b)

As pointed out by Doherty [1952] and Brekhovskikh [1960], these integrals are in a form which can be evaluated by the method of stationary phase, provided certain restrictions are met. The phase in the first integral is given by the function

?, z, S) = + q jr

w(p, z, S) = +\ q dz - m - + 2m qdz-\- kSp. (35a) o

The point of stationary phase occurs where

dw(p, z, S) dS = 0. (35b)

For a given value of p and z, this defines the value S0. Since the major contribution to the integral occurs near S = S0 it is convenient to express w as a Taylor expansion about S0. Thus

w(p,z,S)£w0 + ^ ( S - S 0 ) 2 (36)

where w0 = w(p, z, S0)

and 32 1

wS = ^ w ( p , z , S ) J . S = 5o

Furthermore, near the point S0, the factor (Sq0/C2q)i/2 in the integral of Eq. (34a) is slowly varying and may be taken outside the integral and replaced by

MM" where C. = [N2(z) - 5j],/2N_1(z).

Thus

*s iGsAw)v*D'~,,""~"β- <37) The integral is now deformed to the steepest descent path by introducing the new variable x via the substitution

WoXS-So)2= - 2 * * 2 (38)

and requiring that x range from — oo to + oo through real values. In the present problem it is assumed that no singularities of the integrand are crossed in this deformation. Then, on noting that

i: t-**dx = Jn (39) J — oo

it is readily found that

[ c η Vi e~iwo. (40)

A very similar expression is obtained for φ^.

(41)

6. RELATION TO GEOMETRICAL OPTICS

The expressions developed by this stationary phase approach turn out to be very closely related to the fields computed on the basis of pure geometrical optics. This aspect of the problem is now discussed.

First, it is noted that the stationary phase condition, given by Eq. (35b), can be written

f " Spdz Γ« S0dz P Jo LN\z) - S?]* + Jo LN\z) - Sg]* *

Similarly, for the integral ψ*, the condition becomes

fri S0dz fz S0dz

'-«m+1)l [ ^ ) - sa*-J„ [*»(*)-sa*· (42)

These are nothing but the ray tracing equations which for a given value of S0 trace out curves (on a surface) in the p, z coordinate system. S0 can be interpreted as the sine of the angle 0o which the ray subtends with the vertical. On the other hand, m is the number of times the ray crosses the surface z = 0. The situation is illustrated in Figs. 2a and 2b for m =2.

} /

0

~0o V

i^T^ / *<

/ 1 **. t s f Ì ■*■ ■* - o\b o i

(a)

\

\

| P < / > . J )

1/ X

/I

(b)

— / > -

~ A / ) ( ^ , S 0 )

Ρ ( Λ ? )

■δ ( s 0 ) - \-~ùpi}tS0)

FIG. 2a. Ray for φ2α. FIG. 2b. Ray for ^ .

In the two cases, which correspond to the ray for φ\ and φ02, respectively, the

ray starts off at 0 and reaches P via two swings across the zero axis. The two cases are distinguished as to whether P is before or beyond the maximum of the last half-cycle. In terms of S(S0), the distance between cross-over points on the axis, and Δρ, the horizontal projected distance from the last crossover point to P, the ray equations become

and

where

and

(a)

(b)

p = m<5(S0) + Ap(z, S0)

p = (ro + 1)<5(S0) - Δρ(ζ, S0)

<5(S0) = 2 tan θρ dz

Δρ(ζ, S0) = tan θρ dz.

(43a)

(43b)

(44)

(45)

The emergence of the tan θρ follows from the definition of Cp given before which results in the identity

tan θρ = (1 - Cl)* So

iN\z)-SW (46)

Fio. 3. The elemental portion of the ray.

The simple sketch in Fig. 3 indicates the relation between dp and dz, for an arbitrary point (p, z) on the ray. Noting that dp = tan 0P dz, the geometrical interpretation of equations (44) and (45) is obvious.

The phase (lag) of the potential function φαΜ is the

quantity w0. Explicitly, this is written

Wo = fckp + 2m Γ lN\z) - S3]* dz - m Ï + f * [ΛΓ2(ζ) - S*]* dzj. (47)

Geometrically it is seen that S0p is the horizontal component; the integral from 0 to ζγ is the effective vertical excursion for the complete half-cycles, while the integral from 0 to z is the vertical distance to the point P from the horizontal axis z = 0. Thus, the quantity, w0, with the exception of the muß, is the phase as predicted by pure geometrical optics. The factor muß is a phase advance which can be associated with the reflection at the m "critical reflection levels" at z = ±zx.

A similar interpretation can be attached to the phase (lag) for the potential function φ0

η. It is equal to

fc[s0p + 2(m + 1) ί V 2 ( z ) - SU* dz - (m + 1) ^ - [ W w ~ so]*d*]

(48)


which again is predictable on the basis of geometrical optics with the exception of the phase advance (m + 1)π/2.

The amplitude factor multiplying the term exp(—iw0) in Eq. (40) also has an interpretation in terms of geometrical optics. To demonstrate this, one notes that

+ fcp

and

(49)

(50)

The latter equation is simply the equation of the ray path. Thus, in general, we can write

w' - k[p - p(z, S)] (51) where

P(z, S) = | j [ | ^ q àz + 2m j ' q dzj (52)

is to be distinguished from p which is the horizontal coordinate of the observer. Then

d2w d w" = l^=-kës^S) (53)

and

oS iso co ™ ]θ=θο

remembering that S = sin Θ and S0 = sin 0O. Thus, on inserting this form for WQ into Eq. (40), one finds that

W = p So (55) \PNCpidp(z, S)idor}9m9o\

An identical expression holds for φ„. It is possible to define a focusing factor/by dividing Ν\φ^\2 by (kR)~2

which is the value corresponding to \φ\2 if the medium were homogeneous throughout (with N = 1). Thus

R2S0 f-Pcpidp(z, θ)ΐδθΐθ=θ0

(56)

is the ratio of the two power densities. It is instructive to derive the latter result directly from geometrical arguments such as those described by Bremmer [1958].

The situation is illustrated in Fig. 4. A number of the rays emanate from 0 and one of these is shown by the curve OPQ and is specified by the initial


FIG. 4. The ray construction which illustrates the focusing phenomenon.

elevation angle ψ0. The coordinates of the observer are at P. It is now imagined that the initial ray is changed by an infinitesimal amount άψ0. This is accompanied by a change of P to P'. Now the horizontal displacement between P and P' is

(IL* and therefore, the distance between the rays is

βΡ' = dp sin ψρ

where φρ is the elevation angle at P. The cross-sectional area, dör, of the tube is clearly given by

da = 2πρ x QP'

(57)

(58)

= 2 H ( s i n ^ ) ( | ) ^ d ^ 0 .

Now, if the total flux emanating from 0 is / then the flux supplied to this particular tube is

di = ^(cos ψο) ά\Ι/0ρ(ψ0).

where ρ(φ0) is a factor to account for the prescribed directionality of the source (for an omnidirectional source, ρ(φ0) = 1 ·) The flux density is then given by

di I cos i/ o 1 ST 4πρ

(59)

Now if the medium were homogeneous throughout, it is clear that d/ / da AnR 2Ρ(Ψο) where R =p +z . (60)


Then by definition, the focusing factor, / , is the ratio of the right-hand sides of the two preceding equations. Thus

/ = R2 cos φ0

p s i nM!L (61)

Noting that θ + φ = π/2, where Θ is measured from the vertical and φ from the horizontal, it is seen that cos φ0 = sin 0O = S0 and sin φρ = Cp.

It may be noted that the focusing factor becomes infinite if WQ = 0 or if (ορ/δφ)ψο = 0. In this case the geometrical optics are not valid. To study this case we must re-examine the complex integral representation.

7. TREATMENT AT THE CAUSTIC

The phase integral representation for the rays has the form

In the previous section it was indicated that the integral could be evaluated by the method of stationary phase. This method is valid provided WQ is small compared with the higher derivatives of w0. Near a caustic point or surface this condition is violated. The caustic of a ray stem occurs when both

w' = ^ w ( p , z , S ) = 0 (63)

and

w" = — w(p,z,S) = 0. (64)

An equivalent definition of the caustic is

P-HZ.S) and ^ = 0 . (65)

To obtain an asymptotic form which is valid at or near the caustic surface the phase function w is expanded about the point S = St where

w"(Si)=i/**,S)l =0. (66)

Thus w(p, z, S) s wiSi) + »'(SMS - Si) + W'XS^S - S,)3/6 (67)

where higher derivatives are neglected.* Then, again taking slowly varying quantities outside of the integral, it follows that

i / 2 \V21 S \^2 f + 0°

«^few (c s?) '""""].„«pi-*·™*-« + W'iS^S - StfW} dS (68)

c^iN^-siy^N-1^)5.

w'(St) = k\j> - p(z, St)2

P(z, S i ) = 7^ qdz + 2m\ qâz\ aò LJ o Jo Js=s,

It should be noted that for any value of p and z, S1 is defined such that W\SX) = 0. However, νν'Ο^) is only equal to zero when the point (p, z) is actually on the caustic.

A new parameter, t, is now introduced which is related to S1 by

where

Now, in general

where

Thus

(69)

(70)

(71)

i = ± 21/V(S1)| w'^S^-K

= ±21^[p~p(z,S1)]|ww(S1)|-1/3 (72)

The plus sign is to be employed when w'"^) > 0, and the negative sign when w'XSi) < 0. Thus, without difficulty one finds that

^ s 2 K ^ ^ ) " | w ' ' ' ( S l ) | " , / 3 e ' i w ( S ' M t ) where v(t) is the familiar Airy integral defined by

*)=7;J">s(if+s ,)ds· A focusing factor g can then be again expressed as the ratio of iV|</>J,|2 to the free space field (kR)~2. Thus

(73)

(74)

~25^

where R p2 + z\

\kR2St

\pCxCp V-v\i) I

ss2 J s = S l

(75)

* In some cases it may be necessary to consider the term proportional to νν4(5Ί). This situation is discussed by Pearcey [1946].

It is clear that at the caustic, t = 0. Thus, the potential at a distance Δρ from the caustic may be written

^ ( Δ ρ ) = Κ ( 0 ) ^

where

t= ±2'/>k2Ap\ d2p(z, S)

dS2

(76)

(77) S=Si

where the + sign is to be used when d2pjdS2 < 0, and the — sign when d2p/3S2 > 0.

To illustrate the behavior of the field near the caustic, the function v(t)/v(0) is plotted in Fig. 5.

FIG. 5. Behavior of the field near the caustic.

8. APPLICATIONS TO TROPOSPHERIC PROPAGATION

The methods developed above find extensive application in radio propagation, both in the ionosphere and the troposphere. In the latter case, the analysis becomes somewhat simplified in view of the near grazing character of the rays and the nearness of the refractive index to unity. As an interesting illustration of principles, a horizontally stratified troposphere is considered whose modified or effective refractive index N(z) is near unity. Thus, the formulas for ray tracing given by

can be approximated to

p{z ,<Αο)=Γ

2-1 Vi sS]

[2(N(z) - 1) + iAS] 2-|1/2

(78)

(79)


where φ0 is the initial grazing angle of the ray. Actually, this is a special case of Eq. (41) with m = 0. The extension to the multiple-hop case is obvious from the forms of Eqs. (41) and (42).

The focusing factor/, under the same conditions, is then given by

j„_ Ρ(ζ>Ψο)

Ψ&) F**'*)] (80)

which is an adequate approximation to Eq. (61). Near the caustic the focusing factor/, as defined above, becomes very large and should be replaced by the Airy integral representation. For the present case, it may be written

_ 5 Λ fcp(z, » M Q ^ 9 = 2 / 3 F^27

where φρ(ζ) is the grazing angle of the ray at the observer, Ι/Ί is defined by

ρ(ζ,ψ)] = 0

t = +2V> and

k2Ap I &p(z, ψ)

ΨΙ | # 2

where Δρ is the horizontal distance measured from the caustic. (Note that 4- or — sign is to be used when the second derivative of p is negative or positive, respectively).

Formulas such as these mentioned above have been used by Doherty [1952] and Wong [1958] and others to discuss anomalous conditions in airborne propagation in the troposphere. They appear to offer a satisfactory explanation for prolonged decreases in the amplitudes of microwave fields which have been observed by Bean [1954].


The preceding development provides a satisfactory demonstration of the connections between mode theory and ray theory for a horizontally stratified medium.

Although the results were developed for horizontal polarization, the formal extension to vertical polarization is obtained simply by replacing q by Q where Q is given by Eq. (76) of Chapter IV.

For highly grazing rays and sufficiently high frequencies, the difference between q and Q may be ignored and thus the focusing effects discussed in this chapter are essentially independent of wave polarization.


REFERENCES

BEAN, B. R. (1954) Prolonged space wave fadeouts at 1046 mc, Proc. I.R.E., 42, 848-853. BREKHOVSKIKH, L. (1960) Waves in Layered Media, Academic Press, New York. BREMMER, H. (1958) Propagation of waves, Encylopedia of Physics, Springer-Verlag,

Berlin. DOHERTY, L. H. (1952) Geometrical optics and the field at a caustic, Ph.D. Dissertation,

Elee. Eng. Dept. Cornell University, Ithaca, N.Y. PEARCEY, T. (1946) The structure of the electromagnetic field in the neighborhood of a

cusp of a caustic, Phil. Mag., 37, 311. WAIT, J. R. (1961) A diffraction theory for LF sky wave propagation, /. Geophys. Res., 66,

1713-1724. WONG, M. S. (1958) Refraction anomalies in airborne propagation, Proc. I.R.E., 46,

1628-1638. ZUCKER, F. J. (1960) Chapter in Antenna Engineering Handbook (edited by H. Jasik),

McGraw-Hill.

Chapter XII

SUPERREFRACTION AND THE THEORY OF TROPOSPHERIC DUCTING

Abstract—The theory of propagation in a spherically stratified medium is considered. The profile of modified refractive index M(h) is allowed to have a minimum with height. Particular attention is paid to the case when M(h) may be approximated by a parabolic form. Here the analysis closely follows the recent work of Fock, Weinstein, and Belkina in the U.S.S.R. Other approaches such as the Eckersley phase integral method and Furry's mode theory for the bilinear profile are also considered.

1. INTRODUCTION

In Chapter V the theory of diffraction by a sphere was considered. The refractive index outside the sphere was allowed to be a monotonically decreasing function of the radius. Under such an assumption the diffraction phenomenon bore a close similarity to the case where the sphere was surrounded by a homogeneous medium. Another assumption of the previous formulation was that the transmitter height was taken to be zero. In the present chapter, the refractive index is taken to be some smoothly varying function of radius and both source and observer are allowed to be arbitrary. The permitted forms of the refractive index variation are specified below in detail. Particular attention is paid to the case when the profile of refractive index may be approximated by a parabola. Here the analysis follows the work of Fock [1950] and colleagues [Fock, Weinstein, and Belkina, 1956-58]. Other approaches such as the phase integral method of Eckersley and the mode theory of Furry are also considered.

The results given in the present Chapter are of special interest at u.h.f. (ultra-high frequencies) where the troposphere is of prime importance. However, it is interesting to see that the general theory and the behavior of the fields are quite similar at v.l.f. where the ionosphere controls the propagation.

2. FORMULATION

Without much loss of generality the source can be considered to be a radially oriented electric dipole located at r = a + h0> Θ = 0 with respect to a spherical coordinate system (r, 0, φ). As before, the surface of the spherical

341


earth is r = a and it is assumed that its interior electric properties can be characterized by a constant surface impedance Z. Thus, the fields can again be derived from a single scalar function C/as indicated by Eqs. (1), (2), and (3) of Chapter V.

It is convenient to replace the function U by an attenuation factor V by the substitution

U = a(fl sin 0)* V ( 1 )

where αθ is the great circle distance, measured along the surface of the sphere, connecting the source and the observer. To further simplify the discussion, the following dimensionless quantities are introduced

ka\v> v 0 ~®

and

(2)

,o = (j|)V (3)

ΗέΓ -ΜέΓ** (4)

( ka\i/3 Z V v *>-120*· (5)

Thus it immediately follows from the results of Chapter V that

V = 2(πχ)1/> β"ίΛ/4Ψ (6) if Ψ satisfies

or

—Y — + k2[— + K(z) - 1 Ψ = 0 dh2 a 3Θ \a w /

(D' (K(h) - 1)

(7)

02Ψ 0Ψ ì?-Ìte+iy + r(yW = 0- (8)

In the preceding equation,

r(y) = (^) W(h) - 1) (9)

2\ ' / j

in terms of the refractive index N(h) or the relative dielectric constant K{h).

T-i — + p(yyv-o. (il)

Superrefraction and Theory of Tropospheric Ducting 343

It is now convenient to set

p(y) = y + KJO

Furthermore, since Λ{Λ) and N(h) are very near unity, it is desirable to write /ka\y*

Ky) = 2 ( -J M(/I)X10- 6

where Μ(Λ), the modified index of refraction, is given by

M(/i)^106iiV(ft)-l4--).

The equation for Ψ can now be written compactly

dy2 l dx Also, in view of the boundary condition

ΕΡ=-ΖΗΦ at r = a, (12) it follows that

^ + $Ψ = 0 for y = 0. (13) dy

It is also necessary to specify the behavior of Ψ for small values of x. However, in analogy to Eq. (11) in Chapter V,

V = [s2 + (h - ft0)2]* + R» [s2 + (h + ft0)2]* ( 1 4 )

for short distances where R0 is the appropriate Fresnel reflection coefficient. Thus

Ψ = 2(πχ)4 +y0 + y + 2iqxe J (15)

for x -» 0 while keeping h/s and h0/s < 1.

3. THE ASYMPTOTIC SOLUTION

The solution of the differential equation for Ψ is obtained by setting Y«y(i,jOe""fc* (16)

where 0 + lp(y)~QY = o. (17)

Independent solutions of the latter equation are denoted byf^y, i) and/2(y, i). These are chosen so that they reduce to the Airy functions w^t — y) and w2(t — y\ respectively, when p(y) = y.

The appropriate asymptotic (WKB) forms for/i and/2, valid for large y or large negative t, are

My' ° = E P ^ ) - V exp [" ' Γ[ρ(τ) " ']* H (18)

f*(y> o = Q ^ ) - V e x 4 + i Γ[ρ(τ) - °*dt] · (19)

Proceeding in a manner analogous to that used in Chapter V one finds that the integral form of the solution is given by

and

*-U' t-'*'F(t,y0,y,q)àt (20)

where

F(t, y0, y, 1) - - ^ [My, 0 - / ; ( 0 > 0 + , / l ( 0 ( 0/ l ( y ' °J (21)

for y0 > y, and the contour is to encircle the singularities of the integrand in a clockwise sense. In the above

/ i(0,0 = ^ / i ( y , 0 l (22)

and similarly for/2(0, t). Also

^i,2(0 =/i(0, 0/2(0, 0 - / 2 (0 , 0/i(0, 0. (23)

4. THE SPECIAL CASE OF A NORMAL ATMOSPHERE

Borrowing the terminology from that used in radio propagation, the case where p(y) increases monotonically with height corresponds to the "normal" atmosphere. This follows since the modified refractive index given by

Μ(Λ) = ±{κ(Λ) -1+^) ΐ0 6 (24)

is a monotonically increasing function of h under standard conditions. In this case,/! and/2 can be approximated by Airy functions w^y) and w2(y), respectively, where the argument y is given by

[ W)- ' ]*< i* -K-y) % (25a) to


or

where τ0 is defined by / ;

[ ί - ρ ( τ ) ] * α τ = Κ?)1Λ

ί-ρ(το) = 0.

(25b)

As discussed in Chapter IV, this extended WKB approximation is valid provided y is approximately a linear function of y near τ0. Therefore, since

and noting that ys[p'(t)]>i(T0->0

dy dy dy iyy n dy

it follows that suitable solutions are

Λ = (-|)\(ν), /2 = (-|)\(ν)

dA dy

(y)

(26)

(27)

(28)

(29)

The multiplicative constants (—d^/dy)'/2, etc. when chosen in this way lead to the simple Wronskian relation

Wl2(t) = 2x (30)

which, to within this approximation, is independent of /. Thus, in place of Eq. (21),

'■■1(TO* x vv2(y) - (%> wiOO + d—)w2(y)

wi(r)

y = 0

(31)

for y0 > y. Here y0 is defined in terms of y0 by

f 0>(τ)-ί]* dt = $(-*>)*·

The formal solution is thus given by

ψ = -Lb e-*'F di

(32)

(33)

or, in terms of the attenuation function, by

V = (-YL· e~ixtF dt. (34)

The integral can be evaluated by deforming the contour around the (complex) poles of the integrand. Thus

V^(2nx)^c'in^YJRm (35) m

when Rm is the residue at the pole tm. The poles tm are found from

/ί(θ,0 + «Λ(θ.Ο = ο (36)

or, in the case of the ' 'normal' ' atmosphere by

[wi(y) + i(^)wi(y)] o = o. (37)

5. REDUCTION TO RAY THEORY FOR "NORMAL" ATMOSPHERE

To provide some physical insight, a geometrical interpretation of the integral representation for the "normal" case is now outlined. As it turns out, the validity of such an interpretation is restricted to the line-of-sight region. In this case, the principal part of the integration corresponds to large negative values of t where also y and γ0 are both large and negative. Thus

Wi(y) S e-'*/4(-y),/4 e"'%<-rt3/2 (38) and

w2(r) s e+i^4(-y)1/4 e+i*<-?>3/2. (39)

The formula for F, given by Eq. (31) can thus be written in the asymptotic form

F agBK»-«iw-«g» hhfW t ) - 'i'4 H

(40) The integral representation can be decomposed into two terms as follows

V=Vd + V, (41)


where /ix\ K f+ 0°

e~ixtFddt (42)

«Fr at (43)

and where Fd and Fr are the two parts of F as given above in Eq. (40). Vd corresponds to the attenuation function for the direct wave and Vr corresponds to the reflected wave. The respective phase terms are denoted by

*W0 = xt + I ίρ(τ) - f]* άτ (44)

and i ΩΓ(ί) = xt + [ρ(τ) - ί]1/2 di + [ρ(τ) - ί ] * dt. (45)

Jo Jo The integrals Vd and Fr are in a form suitable for evaluation by a stationary phase method. The stationary points are

^"-'fii^-1 (46) and

fy άτ Çyo άτ w ) =HoE*FäHoiK^p- 0 · (47) These equations can be interpreted as ray tracing formulas in geometrical optics. The first one corresponds to the direct ray connecting source and observer, whereas the second one corresponds to the ray which is reflected from the surface of the sphere. In the geometrical sense, — t can be identified with the parameter (fca/2)% cos2 Θ where Ô is the initial angle which the ray makes with the vertical as it leaves the source point. [Strictly speaking, this is the angle in an equivalent horizontally stratified medium whose refractive index is M(A)]. The factor

q + *l>(0) - Φ q ~ i[p(0) - Φ

can thus be interpreted as the Fresnel reflection coefficient. Here [p(0) — t]Vz

can be identified with (ka/2)i/3 cos 0r where 0r is the angle of incidence of the ray reflected at the surface. It should be noted that the present asymptotic development requires that (—0 is reasonably large compared with unity. Thus, the geometrical arguments are only valid if (ka/2)i/3 cos 9r is somewhat greater than unity. Consequently, geometrical-optical behavior can be expected to break down when the reflected ray is near grazing incidence.


6. EXTENSION OF THEORY TO INCLUDE SUPERREFRACTION

In the case of superrefraction the modified index of refraction will not be a monotonically increasing function of height. In the simplest case, corresponding to a single duct, there will be a minimum of the curve of M(h) vs. h. The location of this minimum is designated Amin. Then p(y) will also have a single minimum and it is located at ymin where

2 \,/3

={L·)' The equationp{y) — t = 0 now has two roots which are located at y = yi and y = y2. When / is real, both roots lie between p(0) and p(ymin). It should be noted that the roots would coincide with ^min if p(ymin) — t = 0.

The equation for the height functions/! and/2, which is given by

ά2ί ay ^2 + lp(y)-Qf=Q (48)

now can be approximated by

^ + (K2 + v)0 = O. (49)

This is the equation for the parabolic cylinder function. It is now necessary to select the parameter v so that {\ζ2 + v) has its single minimum occurring simultaneously with p(y) — t. Taking a hint from the form of the WKB solutions to these two equations, the substitution

[\p{y)-t]'Aày=[ [K2 + v],/2dC Jy* J+2.VV

(50)

is suggested. The lower limits in each case are zeroes of the respective integrand. Then v can be chosen such that

f W ) - *]* dy = ί 2W\tt2 + v]* dC = - ίπν (51)

where now the upper limits are fixed. For future use it is convenient to define

S={\p(y)-tr>ây (52)

and

?o = i ["'iPiy) - 0 * ày + i f W ) - fl% ay. (53) -*o Jo

Therefore the equation connecting ζ and y can be conveniently written

S - S 0 = f [iC2 + v]^dC. (54)

Solutions of Eqs. (48) and (49) are now taken to be related by

g (55) · % ■

where, within the asymptotic approximation, (άγ/άζ)ί/2 can be regarded as a constant.

In conventional notation [Whittaker and Watson, 1950] the Weber parabolic cylinder functions* are defined by

(-l)m + i2m/2r(m~n\zm

2-(n/2) -1 oo ^ ^ 1 9 / D»(z)=-TTTe"22/4 Σ r-^- (56)

Γ(-η) m=o ml and they satisfy the equation d2P„(z)

dz2 + (n + i-j)Dn(z) = 0. (57)

On comparing this with Eq. (49) one sees that they are equivalent if z is indentified with ζ e*'"'4 and n + \ with + /'v. (Upper signs are to be taken together.) Independent solutions of Eq. (49) are

5 l ( 0 = Z)_iv_I/2(Ce'·"/4) (58) and

02(O = *>+,,-a(C e"'«'4). (59)

Suitable solutions for the height factors are now written

/ i (y ,0 = ci(v)2(^) /2gt(0 (60)

h(y, 0 = c*(v)2(^) Wo (6i)

,(v) = exp \ ~ - i | | «P [- <0 -\lnv- S0 ) ] (62) where

i πν

c2(v) = e x p | - — + i ]exp[+ iQ-^/«v-S0)] . (63)

* Various new asymptotic forms for parabolic functions have been considered recently by Logan and Mason [1961].

The reasons for choosing these particular forms for ct and c2 are discussed by Fock who makes extensive use of the known properties of parabolic cylinder functions. Without going into the details, it can be verified that, with these forms, the asymptotic representations for/i and/2, for y located well above the inversion point (i.e. Ç large and positive), are

-in/4 m , ) s i^r^e" i ( S"2 S o ) (64)

and e + ur/4

^■«"ooFO1·"""1"· (65)

Then, sufficiently well below the inversion point (i.e. ζ large and negative)

and

e - i n / 4 e+in/4-Λ(*l) s *'(v) m=w e""'(S_2S0)+e""v m=w<e+is (66)

gin/4 e~ n v e~in^

^•■^m^>,',s-"''+m=w-°-'s («) where

(2π)% (2π)* f 0 1 ^)=röT^exprT-,(v-vInv)J

(2π)* f πν φ/ , 1 ^(v) = f(ï3hôexprT + '(v~ Τ

(68)

(69)

The preceding asymptotic formulas may now be substituted into Eq. (21) to yield asymptotic forms for F{t, y, y0, q). To simplify the discussion it is now assumed that q = oo which is also a reasonably good approximation for tropospheric radio propagation for any polarization. Thus, for y0 > y

F s Fit, y, y0) - ±A(y0, t)[/2(y, 0 - J^M* θ] · (70) Also, to be specific

M(h) = M(ftmin) + - ( f t 7 * 7 ) 2 106 (71) a n + /

where /;min is the "inversion height" and / is a scale parameter. Thus

P(y) = P(ymJ +(y ' ^ (72) y -r y i

where

Ηέ)ν

In the case of intra-layer propagation (i.e. y < y0 < ymin), it readily follows from Eqs. (66), (67), and (70) that

Fit, y, y0) s 2i[p(y)- i ] , / < [ i>0>o)- i ]%

re-ÎS(w) _ ^ e + iS(j>o)-ire + iS()') _ e - i S ( y ) j i

1 - Λ where

(73)

Λ = — e-"v-2iSo. (74) Zi(v)

The attenuation function V may then be calculated from

V(x,y,y0)"2(nx)^c-^* £ Ä-e-to- (75) ro=l

where Rm is the residue of the function F(t9 y, y0) at the /w'th pole tm. The latter are roots of the equation

1 - Λ = 0. (76) In the present case

R _ _ r 1 sin S(y) sin S(y0) m " [PÜ0 - tml\p(y0) - U * r_d (1 _ A)] ( }

where

S(y)=f\p(y)-t-}*dy. (78)

7. REFINEMENTS TO THE ASYMPTOTIC APPROXIMATIONS

It should be mentioned that the above asymptotic representations for the height functions ft and f2 and the resulting modal equation, 1 — Λ = 0, are only valid if the parameter ζ is large. Actually the roots tm9 for the case of q = oo, are determined from the zeros of the function/i(0, t), or in terms of parabolic cylinder functions by

ffi(Co) = 0 (79)

where ζ0 is the value of ζ at y = 0. The residues are then calculated from the equation

D i My, tJMyp, * J/2(Q, O , Q m Km = 2 ~ ή · ^°U' [>Ί,„

Now, generally y and ζ are connected by

S(>0-So=f [v + CWdC (81)

and if a new variable u is introduced such that *

C = Zjv sinh u (82) then a simpler relation is

2u + sinh 2u = - [S(>>) - S0]. (83) v

Fock, Weinstein, and Belkina [1958] have compared the roots obtained from the more exact Eq. (79) with those obtained from the asymptotic form given by Eq. (76) for a range of values where the magnitude of ζ is comparable with unity. In this case one might expect some discrepancies due to the inapplicability of the asymptotic forms for/^O, /). Defining Atm by

tm = P(yJ + Afm (84) the comparative results are shown in Table 1 for y = 0 and selected values of the parameter yx + ym]n = Y.

Table 1.

Y

23.11 23.11 48.07

Pole No. m

1 2 1

ί

-0.729 + i 0.095 -0.725 + /0.246 -1.282 + / 0.041

From Eq. (76)

-0.085 - i 0.466 - 0 . 1 2 8 - /1.132 - 0 . 1 0 4 - /0.224

Δ/m From Eq. (79)

- 0 . 1 0 7 - i 0.443 - 0 . 1 2 5 - /1.145 -0 .113- /0 .227

The differences between these sets of values are not great and, for many practical purposes, they can be neglected.

The asymptotic formulas given by Eqs. (76), (77), etc. also become invalid for modes of very low attenuation (i.e. imaginary part of tm very rmall). To demonstrate this fact and to obtain an improved asymptotic form, a representation for the height functions in terms of Airy functions is obtained. Again, this follows very closely the work of Fock [1950].

The equation for the height functions fx(y, t) and/2(y, 0 is again written

^ + [ p ( y ) - 0 / = 0 (85)

* The required root of Eq. (82) is selected from the condition that, if / < p(ji), the quantity ζ should be a negative real number.

where it is to be remembered that p(y) has a minimum (at y = j>min) and increases on both sides. For y < ymìn, the maximum value is p(0) and for y > ymin> P(y) eventually becomes equal to y. It can be recalled that t is real and lies between p(ymin) and p(0), then p(y) — t becomes zero for real values °f y = >Ί and y2. In the interval yl < y < y29 the quantity [p(y) — /] is negative, and outside the interval, it is positive. It should be pointed out that, in an actual case, t is not real. However, for modes of low attenuation it is almost real.

To obtain a suitable representation of the height function in terms of Airy functions, the two new variables ξχ and ξ2 are defined by

f Jy

[ί-ρΟ,)]*φ, = Κ£ι)Η (86) )yi

and

Î yi lt-p(y)y>dy = M2)* (87) )y

for the range of y x < y < y2. Then, if the parameter

S-KW% + KW% (88) is large, one can write down the alternate "extended WKB" forms

and

/ = [iéjfìùΑίΑΛξι) + ΒΑξ2)ί (90)

in terms of Airy functions u and v. Since these two expressions for/are to be equivalent, relations between the constants Au Bu A2, B2 can be found by comparing the asymptotic expansions for large values of ξ1 and ξ2. Thus

A2 = iBiQ-s

and B2 = 2At es.

Additional relations between the constants can be obtained by comparing the asymptotic forms of/for large negative ξί with the appropriate known asymptotic form of fi{y, t) and f2(y, t). Thus, finally,

/ i(y,0«2c1(v)c-«Vi(y,0 (91) f2(y>t) = 2c2(v)e-«y2(y9t) (92)

where

^•Mr^]Vi)-ïea"Mii )] (93)

and

Hy> 0 = [rrJöö]4 [ν(ξι) + ί β 2 π ν "^>] · <94> The quantities v, c^v), and c2(v) are defined by Eqs. (51), (62), and (63), respectively.

The modes are thus found from the roots of the equation

ΨιΦ, O = 0, or

|0«1)-Ie2«Mi1)] =0. (95)

The residues are now obtained from

D 21 e-2*Vtp?, Οψ^ο, Οψ2(0, tm) Km = n =; · v>6) β'Ή,.,.

It may be shown that when ξχ is large and negative, the residue formula given above goes over to Eq. (73). However, if p(y) — / is small (i.e. ξί is small or positive), the height factors cannot be calculated from (66) or (67). Nevertheless, the values of tm or Atm practically coincide whether they are calculated from the asymptotic form (Eq. 76), the parabolic cylinder form (Eq. 73) or from the Airy form (Eq. 95). Thus, the simple asymptotic form can apparently be used for the calculation of Atm for all cases of practical interest.

8. A FEW QUANTITATIVE RESULTS FOR TROPOSPHERIC DUCTING

An adequate numerical treatment of propagation within a tropospheric duct is very difficult becasue of the large number of parameters involved. However, to demonstrate some of the important features, Fock, Weinstein and Belkina [1958] have presented their results in dimensionless form for a selected range of the parameters. These results are briefly summarized here.

As before, parabolic variation of p(y\ as given by Eq. (72), is considered. In this case,

P(}'min) = 2>>min + yi (97) and

P ( 0 ) - K ^ i n ) = — " . (98) yi


Three examples are taken corresponding to the parameters given in Table 2.

Table 2

Profile

1 2 3

Y = ymin + yi

109.20 52.5 25.24

JVmin

10.40 5 2.40

/>0>min)

119.60 57.5 27.6

P(0) - piymirò

1.095 0.526 0.252

The form oî(py) — p(ymìr) is illustrated in Fig. 1 where y is the ordinate.

2 3 4

p ( y ) - p ( y m i n )

FIG. 1. The form of the three profiles.

Using Eq. (76) for determining the poles the complex values of M« [=tm — /K-Vmin)] a r e giyen in Table 3 for the three profiles of p{y).

Table 3.

Profile

1

2

3

Y

109.20

52.5

25.24

Pole No. m

1 2 3 1 2 3 1 2 3

Mm

+0.3541 - / 0.064 -0.0062-/0.0150 -0.1618-/0.1669 -0.0497-/0.0492 -0.2272 - / 0.3689 -0.3529-/0.7653 -0.1484-/0.2621 - 0 . 2 6 3 5 - / 0.8820 -0.3146-/1.5143

Then using the equation

Κ = 2(πχ)1/2β"ίπ/4 £ Rmt IXtm (99)


the quantity 20 log10F is plotted as a function of x in Fig. 2 for y = y0 = ^min/5 (solid curves) and y = y0 = y^Jl (dashed curves). Although only three terms were used to compute V, the terms for m = 4, 5, ... etc. contribute an insignificant amount for these examples. It is interesting to note that the attenuation for profile No. 1 is almost negligible and has the characteristic of a cylindrical wave since | V\ varies approximately as xv*. This corresponds to an actual field strength which varies as (distance)" v\ Thus, a tropospheric duct of this kind actually leads to fields above the free space case (i.e. \V\ > 1). The other two profiles do not exhibit this enhanced propagation as is also evidenced in Table 3 where Atm has a larger imaginary part.

FIG. 2. Attenuation function \V\ vs. the distance factor x for various conditions.

The undulating pattern evidenced in the | V\ vs. x curves for profile 1 is due to the interference between the first two modes (i.e. m = 1 and 2). This is not seen in the curves for the other two profiles since the relative attenuation of the second modes is so great that it cannot compete with the first-order modes.

It should be pointed out that if y0 is somewhat above the inversion height ymin9 the highly trapped modes are hardly excited at all. Some evidence of this trend is indicated in Fig. 2 where the field is stronger for the lower value of y0 and y. To demonstrate the extent of the ducting in the three cases shown, the corresponding values of \V\ are indicated in Fig. 2 in the absence of refraction. The typical case shown corresponds to y = y0 = 1.

9. REDUCTION TO THE PHASE INTEGRAL FORM

To permit further discussion of the modes in tropospheric ducts it is desirable to reduce the mode equation

1 - Λ = 0


to an even simpler form. As seen above, the highly trapped waves correspond to / lying between /?(ymin) and p(0). Then, the quantity

1 f ΛΓ

nJyi P(y)-]dy (100)

i s essentially negative. Furthermore, as (— v) becomes large, Xi(v), defined by Eq. (68), approaches unity. Then, making use of the identity

S0 = St + inv/2 (101)

the mode equation reduces to the form

lp(y)-trAdy = (m-i)n (102)

which is equivalent to the phase integral formula first developed by Eckersley [1932] and used by Booker and Walkinshaw [1946]. The corresponding formula for normal propagation in a linear atmosphere (i.e. no trapping) is obtained by setting p(y) = y and y1 = t. Thus

i: [y - i]* dy = (m- ±)π (103)

which yields i- = ß(m-i)Ä]%e-r-/3 . (104)

This corresponds to the formula obtained by Millington [1939] and Vvedensky [1935-37] in their studies of ground wave propagation.

A rather simple way to obtain the phase integral form given above is to equate the sum of the normal wave admittances to zero at the surface y = 0. Since q = 0, the downward wave admittance is zero and, consequently, the upper wave admittance is also zero. The latter condition is equivalent to setting the reflection coefficient equal to — 1 or exp(nr). Thus

i expi-2i J lp(y) - i] ày\ = e1'* = e"i2,r(M-,/2) (105)

where the left-hand side is the reflection coefficient obtained by a simple application of the phase integral method according to the prescription given in Chapter IV. It is evident that Eq. (105) is equivalent to Eq. (102).

The phase integral form of the mode equation is useful to describe many of the broad features of tropospheric propagation in ducts. A useful parameter which can be easily derived is the "critical wavelength" Am. Thus, defining

y ft z = - i - = — (106) /min "min

and M , PU) - K^min) = 4 M(?t) - M(hmiB)

9KZ) KO) - ίϋΌϋπ) M(0)-M(fcmin) <" }

it is apparent that 0(0) = 4 and 0(1) = 0. (108)

Now, a new variable τ is introduced by

τ = 4 ' " y v (109) p(0) - p(ymin)

Then, if zx is a root of g(z) - τ = 0, Eq. (102) takes the form

£'[000 - τ]* dt = 9 ^ (no) where

G = ^ f^ [p (0 ) -p (y m i n ) ]

= ^ f t 10"6[M(0) - M(/tmin)]. (Ill)

Now the critical wavelength for a given M profile with inversion corresponds to the condition where t p(0) or τ = 1. Thus

fW^dx-i^^-/» (112) Jo Um

where GTO = G]A=Am.

Then, using Eq. (Ill), it is found that ß x IO"3

λ™ = m-}. ftm-2[M(0) - M(fcmJ]%. (113)

In most cases β can be replaced by unity and the preceding formula becomes identical to one obtained by Bremmer [1949, p. 258].

The physical meaning to attach to km is that for wavelength λ > Àm9 the attenuation becomes appreciable. However, the transition from highly trapped to weakly trapped is not sharp and the parameter does not really have a precise meaning.

10. THE MODIFIED INDEX OF REFRACTION METHOD

Equation (8) for the function Ψ was obtained from the spherical earth theory after making certain approximations. The simplifications were attained by invoking the fact that the radius of curvature was large compared to other

significant dimensions of the problem. Although it was not explicitly mentioned, this is equivalent to replacing the spherical problem by an equivalent horizontally stratified problem. In this case the dielectric constant KQi) as a function of height could be replaced by the modified value KmoaQi) where

Kmoa(h) - KQi) + j . (114)

In a rather penetrating analysis, Pekeris [1946] has shown that this so-called earth flattening approximation can be used for distances up to one-half the radius of the earth and the error in the value of the field will not exceed 2 per cent at any frequency. Furthermore, the method can be used for any height z of the transmitter or receiver provided these satisfy

ΛΛ3/2

Formulating tropospheric ducting problems in terms of a flattened earth with a modified index profile has the virtue of great simplicity In this case the field of a given mode is proportional to φ where, in cylindrical coordinates (/>, Φ, z),

t=-P(p)Q(z) (115)

on the assumption of azimuthal symmetry (i e. 8/8φ = 0). Then since φ satisfies

[V2 + k*Kmoa(2)W = 0 (116)

it readily follows that P and Q must be solutions of

d2P dP dp2 dp

and d2ö

p-AÂ + *z + pvs*r = ° (117)

dz2 + k*[Kmoa(z) - S*]Q = 0 (118)

where S is a separation parameter. Since each mode will be described in the above fashion it is desirable to attach a subscript m to P9 Q, and S. Thus, for mode m,

Pm = AmH<$(kSmP) (119)

where Am is a constant. Here H^ is the usual Hankel function of the second kind which is chosen to insure outgoing and damped waves as p -> oo. Also, to be consistent in our conventions, Im Sm < 0. For all practical situations in the troposphere |fc5mp| > 1 and

ρ«=Μ^ΓβΗω-'· (120) AA

Thus, the radial dependence of the field is characterized by the complex propagation constant ikSm. To within the far field approximation, k{Sm—\)p may be identified with xtm where x=(kaj2)1W and tm is the solution of Eq. (36). Furthermore, Q is approximately proportional to the height function f(y, t) discussed before. To make the correspondence complete, the p in the "flattened" earth should be identified with the great circle distance αθ in the spherical problem. This fact is in accordance with the cylindrical wave representations established in Chapter VI for v.l.f. mode theory.

There have been many investigations which deal with tropospheric mode theory from the earth-flattening viewpoint. Apart from the virtue of simplicity, the method is economical in the sense that the influences of curvature and profile are combined. To illustrate the application, the bilinear model is considered briefly. This is based on the work of Furry, Freehafer, and others [Kerr, 1951].

In the bilinear model

K&oa(z) =l+p(z-d) for 0 < z < d (121a)

= 1 + q(z - d) for z > d (121b)

and thus A'mod is unity at z = d. Thus, Eq. (118) for the height functions becomes

^|f* + (G + Bm)Qm = 0 (122)

where

Z = z/H with H=(qk2)-^ (123)

G = (/c/<7)2/3 (Kmoa - 1) = 0lqH)(Kmoa - 1) (124)

Sm = (1 - qHBm)1/2 = 1 - qHBmß (125)

The latter equation relates Furry's eigenvalues Bm to the more conventional Sm. It should be pointed out that k is the wave number at z = d rather than the surface value.

The function G may also be conveniently written

G = Z-g, Z>g (126)

G = s\Z-g\ Z<g (127)

where

g = dl H and s3 = p/q.

The form of the function G is indicated in Fig. 3 for 3 values of s. Negative values of s correspond to the presence of a waveguide, s = 0 corresponds to a "normal" atmosphere, and positive values of s would correspond to an

"anti-waveguide ". For the linear dependence of G with height it is immediately apparent that Eq. (122) can be reduced to Airy's differential equation

(£-') w(t) = 0.

Thus, for Z > g Qm(Z) = amwi (—Z + g — Bm)

(128)

(129)

when am is a constant w± is the required form of the Airy function appropriate for outgoing waves at Z = oo On the other hand, in the region 0 < Z < g, it follows that

Qm(Z) = bmwi(—sZ + sg — s-2Bm) + cmw2(—sZ + sg — s-2Bm) (130)

where bm and cm are constants. The Airy functions notation used here conforms with that in Chapter VII. It differs from that used by Furry in a number of respects.

G G

FIG. 3. Three examples of the bilinear profile.

The solution can now be found by imposing the following conditions

(1) Qm and dQm/dZ are continuous at Z = g, (2) Qm = 0 at Z = 0.

The latter condition corresponds to a vanishing electric field on the earth's surface in the case of horizontal polarization. This is an excellent approximation of tropospheric propagation.

The two conditions yield three linear equations connecting the coefficients dm, bm, and cm. In order that they lead to a non-trivial solution, the determinant of the coefficients must vanish. Therefore, it readily follows that

H>I(—Bm)

— W^—Bm)

0

-Wii-Bmls2)

SW^—Bm/s2)

Wi(sg — Bmls2)

-W2(—Bmls2)

W'l-Bmls2)

W2(sg — Bmls2)

- 0 . (131)

Solutions of this equation yield the discrete complex values Bm.

The individual modes vary with distance according to (l/p)V2exp[-/(/:-^(p/L))] (132)

where L = 2(A^2)-1/3. The attenuation constant of the modes is then proportional to Im Bm which is designated Cm by Furry. Numerical values of Cm for the first normal mode (i.e. m = 1) have been obtained in a complicated calculation by Furry from a modal equation equivalent to (131). Using these results, the normalized attenuation constant C\ is plotted in Fig. 4 as a function of the normalized thickness g of the waveguide. Various values of s are

< 3

2.0

0.5l·-

2 3 4 5 6 7

GUiDE THICKNESS, g

FIG. 4. Attenuation constant of dominant mode for bilinear profiles.

shown. The case s = 1 corresponds to the normal or linear atmosphere. For values of s less than unity, the attenuation is decreased.

The corresponding amplitude of the first normal mode is shown in Fig. 5 where | Ui\ expressed in decibels (i.e. 20 logio | Ui\) is plotted as a function of

60

2!

5"

-60

1 1

s = - l

~

^ ^

1 I

1

\ · / — y s/\<&/

^Zj&^

1

FIG. 5. Amplitude in decibels for the dominant mode as a function of height.


normalized height Z. A number of values of g are shown but s is fixed at —1 for all four curves. Here the smallest value of g is associated with very weak ducting, whereas the larger values are associated with strong ducting. As expected, low attenuation and strong ducting are somewhat synonymous.

The total field can be computed by summing over the modes. Thus, the magnitude of the attenuation function V can be written

| V\ = 2(πΡ/Ζ,)ΐ/21 2 exp (iBmPIL)Qm(Zo)Qm(Z) | (133) m

where Zo = zo/H in terms of the height zo of the source. Here the height functions have been normalized such that

| o"ßi(Z)dZ = 0. (134)

If only one mode makes an appreciable contribution

| V\ ~ 2(TTP/L)I/2 e-ci* | βι(Ζ)βι(Ζ0)|. (135) A number of other special profiles have been considered by Hartree,

Booker and Freehafer. These investigations are described adequately in the book by Kerr [1951].

REFERENCES

BOOKER, H. G., and WALKINSHAW, W. (1946) The mode theory of tropospheric refraction and its relation to wave-guides and diffraction (in Meteorological Factors in Radio-wave Propagation), Physical Society, London.

BREMMER, H. (1949) Terrestrial Radio Waves, p. 258, Elsevier, New York. ECKERSLEY, T. L. (1932) Radio transmission problems treated by phase integral methods,

Proc. Roy. Soc. 136, 499. FOCK, V. A. (1950) Theory of radiowave propagation in an inhomogeneous atmosphere

for a raised source, Bull. Acad. Sci. U.S.S.R.—Physics Series, 14, 70-94. FOCK, V. A., WEINSTEIN, L. A., and BELKINA, M. G. (1956) On radio wave propagation

near the horizon with superrefraction, Radio Eng. and Elect. 1, 575-92. FOCK, V. A., WEINSTEIN, L. A., and BELKINA, M. G. (1958) Radiowave propagation in

surface tropospheric ducts, Radio Eng. and Elect. 3, 1411-29. KERR, D. E. (editor) (1951) Propagation of short radio waves, M.I.T. Radiation Laboratory

Series, pp. 140-68, McGraw-Hill, New York. LOGAN, N. A., and MASON, R. L. (1961) Asymptotic expansions for functions occurring

in the theory of diffraction by parabolic cylinders, Lockheed Missiles and Space Company. MILLINGTON, G. (1939) The diffraction of wireless waves round the earth, Phil. Mag.,

S. 1, 27, No. 184, 517-42. (A summary of the diffraction analysis, with a comparison between the various methods.)

PEKERIS, C. L. (1946) Accuracy of the earth flattening approximation in the theory of microwave propagation, Phys. Rev. 70, 518.

VVEDENSKY, B. (1935, 1936 and 1937) The diffractive propagation of radio waves, Tech. Phys. U.S.S.R. 2, 624-39; 3, 915-25; 4, 579-91.

WHITTAKER, E. T., and WATSON, G. N. (1950) A Course of Modern Analysis, Cambridge University Press.

Influence of the Lower Ionosphere on Propagation of VLF Waves to Great Distances]

JAMES R. WAIT

Abstract. Theoretical attenuation and phase characteristics at VLF are presented for a number of idealized models of the lower ionosphere. The results indicate the limitations of the sharply bounded model.

1. INTRODUCTION

Despite the great number of papers dealing with VLF radio propagation, there appears to be a need to consolidate some of the recent theoretical data. It is the purpose of this paper to fulfil this need. The intent is to present results, in convenient graphical form, which illustrate the expected value of the attenuation and phase velocity of the dominant modes in the frequency range from 8 to 30 kc/s. Some of the effects considered are due to the earth curvature, ground conductivity, and finally the gradient of the effective conductivity of the lower ionosphere.

No attempt will be made here to describe the mathematical aspects of the subject in any detail. However, certain basic formulas will be introduced to permit comparisons with previous work [Wait, 1962].

The basic theoretical model is a smooth spherical earth of homogeneous conductivity surrounded by a concentric (spherically) stratified ionosphere. Initially the ionosphere is regarded as a sharply bounded isotropie medium. For this discussion the influence of the terrestrial magnetic field is not considered. The non-sharpness or diffuseness of the lower edge of the ionosphere is treated by using an exponential isotropie model which has the virtue of simplicity.

2. BASIC MODAL EQUATION

To permit a straightforward treatment, it is assumed that the level h above the ground may be characterized by an ionospheric reflection coefficient Ri. Furthermore, it is convenient to make the substitution

Ri = - e x p (aC), (1) where C" is the cosine of the angle of incidence at the reference level in the

t / . Res. NBS 67D (Radio Prop.), No. 4, July-Aug. 1963.

367


ionosphere. While the a is generally a (complex) function of C", it is found that under most practical situations a is approximately a constant for C" ranging from 0 up to about 0.3. This, of course, encompasses the important angles in long distance propagation of VLF radio waves.

The relevant modal equation for the low order (important) TM modes may now be written [Wait, 1962]

2 / 2/Λ3

where C2 = (C)2

2A\3/2 / „ „ 2/i\1/2 + / a C 2 + — ■(-+?)" /log

H^(0 w\(t) ( 4 * 1 - 1 ) ^ = 0 , (2)

and

k = 27r/wavelength, a = radius of the earth, n = mode number, t - -(kal2)*l*C2.

This form of the mode equation is valid for a ground surface which is

(a)

16 20 24 28 FREQUENCY, k c / s

FIG. la. Attenuation rate of the first mode for a perfectly conducting earth and a sharply bounded ionosphere.

Propagation of VLF Waves to Great Distances 369

8 12 16 20 24 FREQUENCY, k c / s

FIG. lb. Attenuation rate of the second mode for a perfectly conducting earth and a sharply bounded ionosphere.

effectively a perfect conductor. The logarithmic term involves the derivatives of the Airy functions of argument /.

Under the somewhat unrealistic assumptions that the ionosphere is a sharply bounded ionized medium and the terrestrial magnetic field may be neglected, it is known that [Wait, 1962]

a ~ — 2 (;Π,+Ξ)· (3)

where ωΓ = ω^/ν. Here ωο = the (angular) plasma frequency,

and v = the effective collision frequency. The attenuation An, of the nth mode, is then given by


An = —Im kSn nepers/meter,

where Sn = (1 — Cffi = 1 — CjJ/2. In more convenient units

An £ Im fcq/2 x 8.68 X IO3 in db/1000 km, (4) when

k = Ιπ/κφβΟΟ,

where fui s is the frequency in kilocycles per second. The phase velocity vn of the nth mode, referred to the velocity of light in

vacuo, is then obtained from

^ - l S R e ( Q / 2 ) . (5)

The complex values of Cn are the roots of the modal equation quoted above. It is considered to be convenient to number the modes in the VLF range so that mode number 1 is the mode of least attenuation. Higher order modes correspond to modes of progressively greater attenuation.

3. ATTENUATION FOR THE SHARPLY BOUNDED MODEL

Using (2) and (4), the attenuation rates A\ and A% for the first two modes of lowest attenuation are shown in Figs, la and lb, respectively. The height of the reflecting layer varies from 60 to 100 km in 5 or 10 km intervals. The ground conductivity σ9 is here taken to be oo while œr has the value 2 X 105. The latter is typical of daytime conditions. It is interesting to note that the attenuation A\ for the first mode has a broad minimum when plotted as a function of frequency. It is also noted that at the lower frequencies the attenuation A\ increases rather significantly for the lower reflecting heights.

In general it is indicated that the attenuation A2 of the second mode is somewhat greater than A±. It is important to observe that A2 is very dependent on the height parameter h.

The influence of finite ground conductivity can be determined by solving a slightly more complicated form of the modal equation. To within a good approximation it has the same form as (2) if the log term is replaced by

Κ ( 0 - qw2(t) 8 K W - iwi(0.

where q = —iQca/iyZgl 120π,

and where Zg is the surface impedance of the ground. Choosing the ground conductivity ag to have a number of typical values, A± was calculated from the modified form of the mode equation. It appears that for propagation over sea water (i.e., ag ~ 4000 millimhos/m) the value of σ9 is effectively infinite. However, over moderately or poorly conducting land, the effect of finite ground conductivity cannot be ignored as indicated clearly in Fig. lc.

o o 2 4 T3

< 3

3 Z Lu h-

«1 1 1

" m - \ V|

-

Γ

l·

1 1

— i r

( c )

I 1 1 1 1 1 1

h--70 km; ωΓ = 2XI05

n = 1

^ * ^ 0 " = I MILLIMHO/METER

^ ^ ^ " - ^ 2 ^ " \ ^

^^^^^— ^ — " S > ^ ^ ^ ^ I ^ ^ ~ " s_ ^ ^ , """""" > v c!U "

CO

1 ! 1 1 1 1 1

4

H

H

^

-j H

- i

-

12 26 16 20 24 FREQUENCY, kc/s

FIG. le. Attenuation rate of the first mode for an imperfectly conducting earth and a sharply bounded ionosphere.

0.004

0.002

> 1 o

>-" O Q

< X

0.000

-0.002 h

-0.004 h

-0.006

-0,008

{ \ \ '

\ \ \

- \ \

> -

--

-

1

V \ ' '

(a)

1 ( 1 1 1 !

aq = 00; GUr = 2 X IO5

n = i

V ^ v ^ s ^ = 6 0 k m

^^^ν^^^Ζ^^^^^^^^^

^ Ο^ " / ^ " ^ ^

^ ^ ^ ^ ^ ^ ι ^ ^ - ^ ,

1 1 1 I l ^ « k .

-

-

-

--

-- ^

-

12 28 16 20 24 FREQUENCY, kc/s

FIG. 2a. Phase velocity of the first mode for a perfectly conducting earth and a sharply bounded ionosphere.

4. PHASE VELOCITY FOR THE SHARPLY BOUNDED MODEL

The values of the phase velocity of the modes 1 and 2 for the same conditions as in section 3 are shown in Figs. 2a5 2b, and 2c. As expected, the

>-" o o

> UJ co < X

0.010

0.008

0.006

0.004

0.002

0.000

-0,002

-

-

-

--_ _

-

i ' \ \ '\ \ \ \ ' \ ' '

0^= 00; Gür= 2XI05 \ \ n = 2 \ N

(b)

1 ! 1 1 1 !

, l \ 1 1 1

v \ \h=60km

\ \ \>5 \

\ X X° X \ \ \75 X s

\ \ \° \ \ \ v \ .

\ N ^ ß X i xooò\,

-

-

J

A ή q

j

x j

12 28 16 20 24 FREQUENCY, kc /s

FIG. 2b. Phase velocity of the second mode for a perfectly conducting earth and a sharply bounded ionosphere.

o o Ld >

< X Q-

0.004

0.002 h

0.000

-0.002

-0.004

- \ u

- \

-

-

1

(c)

1 1

1

^ - σ , -

1

1 1 1 1 1 1

h = 70km ; OJr= 2 X I 0 5

n = i

00 MILLIMHOS/METER

^— 10

1 1 1 1 l " " Γ -

-

-

-

-

16 20 24 FREQUENCY, k c / s

FIG. 2C. Phase velocity of the first mode for an imperfectly conducting earth and a sharply bounded ionosphere.

phase velocities differ only slightly from c, the velocity of light in vacuo. Nevertheless, this difference is very significant. Jt is interesting to note that the phase velocity v± of the first mode may be actually equal to c for frequencies around 15 kc/s when the reflecting heights are around 65 km. For frequencies above this, the phase velocity is less than c and the mode becomes a "slow wave". As indicated by Fig. 2c, the ground conductivity appears to have only a very small influence on the phase velocity of VLF radio waves.

5. A SIMPLIFIED FORM OF THE MODAL EQUATION

For certain extensions and other applications, it is important to ascertain if any simplifications can be made to (2). The key step is to recognize that when t is near zero, the factor / log [w'2(t)lw[{t)] may be replaced by π/3.

o 3 Ld > LU C/) < X CL

0,006

0.004

0.002

0.000

-0.002

-0.004

-0.006

-0.008

-0.010

H 1 r

COMPARISON OF EXACT ( ) AND APPROXIMATE ( ) METHODS OF SOLUTION

I

16 20 24

FREQUENCY, k c / s

28

FIG. 3. Comparison between accurate or "exact" modal and an approximate form used for simplified calculations.

This is valid when \{kaßy C\ is small compared with unity. At the same time, terms such as (C2 + 2A/#)* are replaced by

(2hlaf (l + ^ ,

which is justified when C2 is somewhat less than hja. Employing these approximations, the modal equation (2) reduces to a simple algebraic equation in C2. Explicitly, the solution may be written

_ [12* - 5] (TT/6) - (2to/3) (2hla)W - za(2A/a)* ka(2hlaf + (m/2) (2A/a)-* ' W

To give some idea of the goodness of this simple formula, the phase velocity vi for the first mode is shown plotted in Fig. 3 for a = 0. It is quite evident that for most purposes, the approximate method of solution is adequate.

6. EXPONENTIAL LAYER

To illustrate the influence of a non-sharp boundary at the lower edge of the ionosphere, it is assumed that the parameter wr varies exponentially with height z. Thus

tor = tor exp [ß(z — A)], (7)

where ώτ is the value of œr at the reference height z = A. For a typical D-layer profile, the exponential form is a surprisingly good fit if β is about 0.5 km- 1. For daytime, A is approximately 70 km while, for nighttime conditions, A is about 90 km. For these calculations ώτ is assigned the value 2.5 x 105.

The reflection coefficient for such an exponential layer may also be expressed in the form—exp (aC) when referred to the reference height z = A. Furthermore, a is again nearly a constant for a wide range of angles of incidence. A numerical treatment of the exponential layer for plane wave incidence has already been given in some detail [Wait and Walters, 1963]. For present purposes, the complex values of a previously obtained are used in the simplified modal equation (6) given above.

For presentation of results, the attenuation and phase velocity are presented as differences to the corresponding results for the sharply bounded model. This technique has two advantages: (1) it makes use of the more accurate attenuation and phase data using the sharply bounded model, and (2) it illustrates the influence of a non-sharp boundary in a direct manner.

The attenuation increment Δ An, which is plotted in Figs. 4a, b, c, and d, is defined by

ΔΑη = Α<ξ}-Αη9 (8) where A{£ is the resultant attenuation rate for the nth mode when the ionosphere has an exponential profile, while An is the attenuation rate of the sharply bounded model with œr = 2 X IO5.


IO 15 20 25 30

FREQUENCY, kc/s

FIGS. 4a, b, c, and d. The increase or increment of the attenuation for a diffuse ionosphere over that of a sharply bounded ionosphere. (The effective con

ductivity of the diffuse ionosphere varies according to exp [ß(z—h)].)

Τ Τ Γ Τ 7 Τ Τ Τ Γ 1 Γ Ϊ Τ Γ Π Ί Τ Π Π" I I I I II I I I I

<n 5 h -

t 8 l·-

25 30 10 15

FREQUENCY, kc/s

FIGS. 5a, b, c, and d. The increase or increment of the phase velocity for a diffuse ionosphere over that of a sharply bounded ionosphere.


The attenuation increment Δ Ai for the first mode is shown in Figs. 4a and b for h values of 70 and 90 km. The corresponding attenuation increments Δ A2 for mode 2 are shown in Figs. 4c and d. It is interesting to observe that, for the larger values of ß corresponding to a relatively abrupt boundary, Δ An is relatively small and does not depend significantly on frequency. For smaller values of ß, corresponding to a diffuse boundary, the attenuation rate may be significantly increased. This is particularly apparent at the higher frequencies.

The phase velocity increment N is defined in a similar fashion. Explicitly,

where v^ is the phase velocity of the nth mode for an exponential ionosphere while vn is the corresponding phase velocity for the sharply bounded model as indicated in Fig. 5. It is apparent that Nn is a small quantity for the range of frequencies indicated. In general, it may be observed that a diffuse ionosphere (i.e., small ß values) is associated with higher values of phase velocity. Physically, this is a consequence of the lowering of the effective height of reflection. It may be noted that the effect is particularly noticeable for the higher range of (VLF) frequencies.

The actual ionosphere is expected to encompass the whole range of jö values indicated in Figs. 4 and 5. Thus, the phase perturbations are a sensitive indicator of the relative diffuseness of the lower ionosphere.

The reader should be cautioned that the results in the present paper make no allowance for the terrestrial magnetic field. This factor will change the attenuation rates to some extent but the conclusions regarding the gradualness of the lower ionosphere are still valid.

The author thanks K. P. Spies and Mrs. L. C. Walters for carrying out the original calculations. The work was supported by the Advanced Research Projects Agency, Washington, D.C., on ARPA Order No. 183-62.

7. REFERENCES

WAIT, J. R., and WALTERS, L. C. (1963) Reflection of VLF radio waves from an inhomo-geneous ionosphere, Pt. I, / . Res. NBS 67D (Radio Prop.), No. 3, 361.

WAIT, J. R. (1962) 1st Edition (see main text of present volume).

ADDITIONAL REFERENCE

GALEJS, J. (1963) Terrestrial ELF propagation in the presence of an isotropie ionosphere with an exponential conductivity-height profile, Proceedings of the International Con

ference on the Ionosphere, Chapman and Hall, Ltd.

Height-Gain for VLF Radio Waves]

JAMES R. WAIT and KENNETH P. SPIES

Abstract. The height dependence of the field strength of VLF radio waves is considered. Using previously developed theory, the height-gain function of the first two modes is calculated in terms of Airy functions of complex argument. It is indicated, for frequencies of the order of 25 kc/s, that the height-gain function reaches a maximum value at a height of the order of 40 km when the reflecting layer is about 70 km. The form of the height-gain function is also shown to be dependent on the finite conductivity of the ground. An experimental curve for 18.0 kc/s based on a rocket measurement shows some agreement with the theory.

1. INTRODUCTION

Despite the extensive literature [Budden, 1962; Wait, 1962; etc.] on the propagation of VLF radio waves hardly any attention has been paid to the influence of varying the heights of the transmitting or receiving antennas. Although the theory is usually developed for arbitrary heights of the source and observer, the explicit calculations are usually carried out only when both heights are set equal to zero. Because of the long wavelength involved it is expected that the heights of the terminal points would need to be very great before any significant change is felt. However, with the use of high altitude rockets, field strengths can be measured right up to the ionosphere and, thus, the calculations are more than just of academic interest.

It is the purpose of this paper to describe the results of some computations which enable the height-gain function to be plotted for the range from the ground up to the reflecting layers. To simplify the calculations the ionosphere is regarded as a sharply bounded medium. The excitation factors of the various modes are also discussed since they are closely related to the height-gain functions. The significance of the excitation factor itself has been discussed recently [Wait, 1962] where some explicit results were given and the curves of the excitation factor given here extend these somewhat.

2. FORMULATION AND STATEMENT OF FORMULAS

At the risk of appearing repetitious, the formulation of the problem and the final results are briefly stated here. The source is a vertical electric dipole

t / . Res. NBS 67D (Radio Prop.), No. 2, March-April 1963.

379

located on or above the surface of a smooth spherical earth of radius a, conductivity ag, and dielectric constant eg. Spherical coordinates (r, 0, φ) are chosen with the dipole located at r = a + zo and Θ = 0. The ionosphere is represented as an isotropie reflecting shell located at r — a + h. The electrical properties of this shell are not specified except to say that the tangential electric and magnetic fields are related at the level r = a + h by a surface impedance Z.

For harmonic time dependence, the radial component of the resultant electric field is written

£ r ~ m . m , Ve*«* (1) α(θ sin θγ ν '

apart from a constant factor. An expression for V was derived previously [Wait, 1961] as the sum of waveguide modes. It may be written conveniently in the following form

v = 4(^1* e-fir/4 y e-<*nGn(y)Gn(y)An9 (2)

where x = (kajiye, y = (2/kaykzo, and y = (l/ka)* k(r — a). The other factors in this equation are discussed below.

The complex values tn are solutions of the equation

1 - A(t)B(t) = 0, (3) where

A(t) = _ ïn^-y^ + AjzBl] (4) ΛΚ}) [w&t - yo) + qiW2(t - yo)\ ' W

?< = -i{kal2fZjr,0, vo = 120ττ,

9 s -/(Ära/2)* f - ^ r ^ - ì and j 0 = (2/Âra)»A7i.

The functions wi(/) and w^{t) are Airy integral functions and the primes indicate a derivative with respect to the argument. Methods used to obtain numerical values of tn are described in a technical note [Spies and Wait, 1961].

The functions Gn are height-gain functions and they are normalized to unity for y or yo equal to zero. Explicitly,

GnW==f(tZö) (6)

where /('«, y) = wi(tn -y) + A(tn)w2(tn - y). (7)

Height-Gain for VLF Radio Waves 381

The function An is a modal excitation factor [Wait, 1961]. Here it is normalized such that, in the limit of zero curvature and perfect ground conductivity, it becomes unity for all modes. Under this condition

An - 2 |fe -q)~ ^ _ ^ + ^ ^ _ ^ J . (8)

In many applications at VLF the ionosphere may be described in terms of an effective conductivity parameter œr. On neglecting the terrestrial magnetic field and assuming that ω < v where v is the collision frequency, it follows that cor = w^jv where ωο is the (angular) plasma frequency. Then, to within a good approximation, the surface impedance is given by [Wait, 1962]

Ì = (>+~) (-)'· being essentially independent of angle of incidence (or mode number) for grazing modes.

3. EXCITATION FACTOR

The excitation factor An determines the efficiency of launching of a given mode in the earth ionosphere waveguide. Using (8) and the known values of tn [Spies and Wait, 1961], calculations of An were carried out. The resulting curves of Λι and Λ2 are shown in Figs, la and lb for σ9 = oo corresponding to perfectly conducting ground. To simplify matters even further, q% is taken to be 00, which corresponds to a perfectly reflecting ionosphere with a local reflection coefficient of — 1. It is immediately evident from these curves that the excitation of the least attenuated mode becomes very weak at the higher frequencies. However, for the range of frequencies and heights shown, the second mode is being well excited. In fact, there may be a 2 db increase over the excitation amplitude for a flat earth.

For the cases shown in Figs, la and lb, Λ^ is real. Actually, finite ground conductivity or losses in the ionospheric reflecting layer will modify An slightly and cause it to become complex. To illustrate this phenomenon, Λι is calculated for various values of σ9 for an œr value of 2 x 105 which is considered to be typical for daytime conditions. The results are shown in Figs. 2a and 2b. It is immediately evident, on comparing the curve for σ9 = oo with the corresponding curve for h = 70 km in Fig. la, that the influence of the finite value of a>r is quite small. However, the effect of finite ground conductivity is quite noticeable. It is rather interesting to see that the excitation factor is actually increased as the ground conductivity is decreased. This fact is not incompatible with the behavior of the height-gain functions as discussed overleaf.


o

0

0

-5

-.10

-15

-20

-25

-λ(\

-

-

"

_

" I I 1

(a)

1 1 1

1 1 1 1

^ " — — . - ^ h = 6 0 km

\ \9°

\ioo

1 1 1 1 \

1 1 I 0" = oo g

q . = oo

n = 1

1 1 \ l

-J J

-J —\

—\

1

\ J

■A

-j

-8

-10

Ί Γ

(b)

Ί 1 1 Γ

h = 70 km

10 12 14 16 18 20 22 24 26 28 30 FREQUENCY , k c / S

FIGS, la and lb. The excitation factor for the first and second order modes for an earth-ionosphere waveguide with perfectly reflecting boundaries. In

this case Δη is real·

T3

o

O evi

2

o

-2

-4

-6

-8

HO

-12

-14

-Ifi

^ :

-

-

-

- 1 —

(ay

1

—i 1

h=70 km ω Γ * 2 χ Ι 0 5

n= 1

1 1 -

T

J** '

1

Ί 1 1

= 1 MILLI M HO/ M E T E R -

< / \ V 20

\ V \ >r1

1 1 1 J

o ÜJ

< X Q.

CM

10

0

-10

-20

-30

-40

-

=—

-

1

(b)

1

I 1 I I 1

h=70km ωΓ = 2χΙΟ? n=l

0 ^ = 0 0 ^ . ■—

c^=20 Mil 1 IMHO/METEB

IO

"-^--—£-^

^ S , 2

^ S J ^ * ^

1 ! ,. 1 , , I , . ,_,! ,

I |

^ ^ ^

-

-

-

-

^ ^

V

. 1 14 16 18 20 22 24 26 28 30

FREQUENCY , k c / S

FIGS. 2a and 2b. The (complex) excitation factor for the first-order mode for imperfectly reflecting boundaries.


4. HEIGHT-GAIN FUNCTIONS

The height-gain factors Gn(y) and Gn(y) have the same functional form and, therefore, only one of these quantities need be considered. The factors Gn(y) may be computed either directly from (6) or use may be made of power series developments [Wait, 1962]. For the results given here, the former method was employed.

The results are shown in Figs. 3a to 6b with all relevant parameters being indicated on the curves. In Figs. 3a, 3b, 4a, and 4b, Gn(y) is real since both

2.2

18

1.4

1.0

0.6

0.2

-

—

Ί h = n =

q.=

v

(a)

1 1 1 1 70 km 1 CD CO /

1 1 1 1

1 l i — L I 1 > ^ = 3 0 k c / s \

25 >

f=iokc/s>v\v

1 1 1 1 1 1

1 1 J

—I -j

\ 1

\ 1

\ \ \~"

i i ^ l

20 30 40 50 H E I G H T , z , k m

70

FIGS. 3a and 3b. The height-gain functions for the first and second order modes for an earth-ionosphere waveguide with perfectly reflecting boundaries. In this case Gn(y) is real. In these curves the reflecting height is fixed at 70 km.


0 10 20 30 40 50 60 70 80 90 100 HEIGHT, z , k m

FIGS. 4a and 4b. The height-gain functions for the first and second order modes for an earth-ionosphere waveguide with perfectly reflecting boundaries.

In these curves the frequency is fixed at 20 kc/s.

the ground and the ionosphere are perfectly reflecting. However, when either wr or ag is finite, Gn(y) is complex as indicated in Figs. 5a, 5b, 6a, and 6b.

The sets of curves in Figs. 3a and 3b are for a fixed height h of 70 km and, because q% = oo, the value of GJj) must go to zero at z = 70 km for all the frequencies. A similar behavior is evident for Figs. 4a and 4b but here the ionosphere heights are taken to be 60, 70, 80, 90, and 100 km in turn. When cür is finite it can be seen from Figs. 5 and 6 that Gn(y) is finite at the upper boundary and the phase is of the order of —90°. Such a behavior is compatible with the boundary condition at the sharply bounded ionized layer.

It is seen from the curves (e.g., Figs. 3a, 4a, 5a, and 6a) that the field

2.2

1.8

_ L4

>% 2 1.0

0.6

0.2

---

— -

n * h»

(yrx

^*"»· - -^^

EXP

_ l _

1 1 1

1 70 km 2x IO5

00

(LOMAX)

1 1 1

1 1 1 1 I 1 1 f «30 kc/s

25 kc/s

20kc/s \

- ^ - * * » ^ 5 k c / s ^ \ ^

(a)

1 1 1 1 1 1 1

1 1

-J

- i

\ Ί

\ A

\ \ 1

\ \ \ ι

x 1 1 * 1 I

Li. O

UJ CO < X a.

IO*

0

-20*h

~ -40«

-60«

-80*

-100·

-120'

-

--i-L

l·

h

ί T 1 1 !

n= 1

h = 70 km

ωΓ= 2 x i o 5

crg = co

1 1 1 1 1

I 1

(b)

1 1

1 1 1 1

25 k c / s ^ 20 k c / s ^

15 kc/s ^

l 1 1 1

1 1

f = 30kc/àl

Ov\ 1 S \ Yvi 1

Wvi i

1 1 1 0 IO 20 30 40 50

HEIGHT, Z , k m

60

FIGS. 5a and 5b. The amplitude and phase of the height-gain function for an imperfectly reflecting upper layer and a perfectly conducting lower boundary. The dashed curve in (a) is based on a rocket measurement at 18.0 kc/s during

daytime for a path over sea water.


associated with the first order mode rises to a maximum at a level below the reflecting layer. In this sense the mode is analogous to Lord Rayleigh's whispering gallery (acoustical) mode which he observed on the inside of the dome of St. Paul's cathedral [Rayleigh, 1910; Budden, 1962]. When dealing with the earth-ionosphere waveguide it is appropriate to call this the earth-detached mode since it depends only very slightly on ground conductivity

FIGS. 6a and 6b. The amplitude and phase of the height-gain function for an imperfectly reflecting upper layer and a finitely conducting lower layer

(typical of propagation over land).


[Wait, 1962; Spies and Wait, 1961]. It may be observed that the form of the height variation is generally quite insensitive to the electrical characteristics of the reflecting boundaries.

The effect of finite ground conductivity is seen by comparing Figs. 5a and 5b for ag = oo with Figs. 6a and 6b for σ9 = 5 x 10~3. It is evident that the finite conductivity of the lower boundary modifies the amplitude of Gn(y) only slightly although the phase is noticeably affected. This behavior is consistent with the series development for height-gain functions given previously [Wait, 1962]. In particular, it was indicated that, for low heights, the finite conductivity tends to reduce |GwCv)| below unity and the phase of Gn(y) is positive by a small amount. For propagation over sea water it is permissible to regard ag as being infinite.

As mentioned in the introduction, the height-gain functions should describe the variation of the field measured in a rocket as it proceeds from the ground up to the ionosphere. Such an experiment has actually been carried out by Lomax [1961] at Eglin Air Force Base in Florida. He employed the 18.0 kc/s transmission from station NBA in the Canal Zone, Panama. At this distance, in the daytime, it is expected the first-order mode predominates. A sketch of Lomax's results for the vertical electrical field is shown in Fig. 5a. It follows the theoretical curve for 15 kc/s quite closely. Over the range of heights from 2.5 to 60 km Lomax estimates that the signal strength as plotted is accurate to within 2 db. The slight increase at very low heights has been attributed to the exhaust plume which has the effect of increasing the effective length of the receiving antenna. At great heights there were also some departures between the calculated and experimental curves, but here the experimental data is not considered to be reliable.


The curves given in this paper, although based on an idealized model, should provide preliminary information concerning the behavior of the VLF field at large heights in the atmosphere. Contrary to commonly accepted ideas the field may increase by a significant amount as the transmitting or receiving antennas are raised. The effect should be particularly noticeable at the upper end of the VLF band at long ranges.

It might be mentioned that the results in this paper, for the height-gain functions, refer specifically to the vertical electric field. However, to a very good approximation they also hold for the horizontal magnetic field component. On the other hand, the horizontal electric field component which was not discussed in this paper actually is proportional to the derivative with respect to height of the function Gn(y). Thus, we can expect the horizontal electric field to be very small on the ground and at low heights. However, at heights above 30 km the horizontal field may be quite large.


I thank Mrs. L. C. Walters and Mrs. C. M. Jackson for their assistance with the computations, Mr. V. Pfannenstiel who drafted the illustrations and, finally, Mrs. Eileen Brackett for help in preparing the manuscript.

The work has been supported by the Advanced Research Projects Agency, Washington, D.C., Order No. 183-62.

6. REFERENCES

BUDDEN, K. G. (1962) The Waveguide Mode Theory of Wave Propagation, Prentice-Hall, New York.

LOMAX, J. B. (1961) Measurement of VLF transmission characteristics of the ionosphere with instrumented Nike-Cajun Rockets, Final Report from Stanford Research Institute on Contract NOw 60-0405 to the U.S. Navy.

RAYLEIGH, LORD (1910) The problem of the whispering gallery, Phil. Mag. 20, 1001. [See also 27, 100 (1914).]

SPIES, K. R, and WAIT, J. R. (1961) Mode calculations for VLF propagation in the earth-ionosphere waveguide, NBS Tech. Note No. 114.

WAIT, J. R. (1961) A new approach to the mode theory of VLF propagation, / . Res. NBS 65D (Radio Prop.), No. 1, 37-46.

WAIT, J. R. (1962) Refers to 1st. Edition (see main text of present volume).

VLF Mode Problem for an Anisotropie Curved Ionosphere]

JAMES R. WAIT

Abstract. The influence of earth curvature in the theory of reflection from the ionosphere is considered. By choosing a rather idealized model, the significance of usual earth-flattening procedures can be displayed quite readily. To simplify the analysis, it is assumed that the earth's magnetic field is vertical everywhere. It is shown that the curved ionosphere may be represented by homogeneous planar slabs, provided the local value of layer curvature is used. The results in the present paper are compared with some corresponding expressions obtained by Johler and Berry [1962].

1. INTRODUCTION

Boundary value problems involving magneto plasmas, such as the ionosphere, are a great deal more complicated than the corresponding problems for isotropie media. For this reason a number of gross simplifications are often made in order to achieve tractability. For example, in most of the existing literature on radio propagation it is assumed that the ionosphere may be regarded as a planar stratified medium. In fact, Budden's exhaustive treatise [Budden, 1961] is restricted entirely to this case. Also, in prior work on VLF mode theory it has been somewhat tacitly assumed that the ionosphere may be regarded, in a local sense, as a planar stratified medium. Justifications for this step are based on arguments valid only for isotropie media. Earth curvature is accounted for only in the nonionized waveguide region beneath the base of the ionosphere [Wait, 1961a, 1962; Katzin and Koo, 1962; Budden, 1962]. Since the anisotropie properties of the ionosphere play an important role in VLF mode theory, it seems that this aspect of the subject warrants some scrutiny, and this is the main purpose of the present paper.

The situation considered here is rather idealized. The ionosphere is regarded as a spherically stratified plasma with a superimposed radial magnetic field of constant strength. Such a model has been considered by van der Wijck [1946], Bremmer [1949], and more recently by Krasnushkin [1961]. If this restriction, on the magnetic field, were not made the problem would become intractable unless some other simplifying assumptions were made such as a purely transverse field [Wait, 1961b].

Î / . Res. NBS 67D (Radio Prop.), No. 3, May-June 1963.

391 CC

2. FORMULATION

Choosing a spherical coordinate system (0, φ, r), the base of the ionosphere can be regarded as the spherical surface r = c. For an ionospheric-type plasma, the dielectric constant (e) has the form of a tensor. The relation between the displacement vector D and the electric field E is then

D = ( , ) E ,

which may be written more explicitly in matrix form äs

De °Φ Dr

— «11

— «12 0

«12 «22 0

0 ' 0 «33

Εβ Εφ Er

0)

(2)

The elements of the dielectric tensor may be a function of r but constant with respect to Θ and φ. They depend on the strength of the earth's magnetic field and on the electron density, ion density, and frequency of collisions between them. The case usually considered is that in which the electromagnetic forces influence only the electrons and the motion of the ions is neglected. In this case [Bremmer, 1949]

£22

£12

£33

j(v + ίω) ωΙΙω ω\ + (y + iœf

1

f- 0 + /ω)2

{y + /ω)ω

(3a)

(3b)

(3c)

where ωο is the (electron) plasma frequency, ωη is the (electron) gyrofrequency, and v is the effective collision frequency (for electrons). The generalizations of these formulae to include the influence of heavy ions and the effect of an energy-dependent collision frequency have been discussed elsewhere [Sen and Wyler, 1960; Johler, 1962; Wait, 1962]. For purposes of the present analysis, it is sufficient to point out that, even under these general conditions, the dielectric tensor has the form given by (2), provided the magnetic field is vertical. Therefore, the subsequent theory will not be restricted just to an electron plasma with constant collision frequency.

Maxwell's equations,

curl E = — /μωΗ,

curl H = /(e)cüE,

where (e) is a tensor, may be written explicitly in the form

1 d ~~ ~rdr (rE<^ = ~ίμωΗ<»

(4)

(5)

(6)

VLF Mode Problem for an Anisotropie Curved Ionosphere 393

\ [ I <'**> - w] = -*"°Η* (7)

1 F„ (sin 0 £,) = -$««>#„ (8)

r sin 0 So

- - j r (ΓΗΨ) = iw[euEe + citEJ, (9)

r är ^rH^ ~ ~dÌ \ = 'ω^~ ei*E° + €2ζΕΦ^ (10)

In the above it has been assumed that 3/3φ = 0. It is now a simple matter to express the r and Θ components of the field in

terms of the φ component. To facilitate this, two auxiliary functions M and N are introduced which are defined by

ΕΦ~ r 3Θ ' ( 1 2 )

ΗΦ = 7 W · {13)

Then, without difficulty, it follows that 1 d I dM\

1 d / 8N\ ίί3'ωΕ"=^τΓθ8θ[ύηθ-8θ)' (15)

1 d2M ίμωΗθ =~rdrtë> (16)

1 d2N ieizco dM 1€ηωΕ» = - 7 dm - ~T ~m ■ (17)

Furthermore, it is found that M and N satisfy the two coupled equations

€33Γ är m * + *" \Vo) r2N + ' w " Jr (m Mj

+ s ïn^è( S Î n 0 ^) = 0 , (18)

+ sînlâl(SÎn0âM)=0 (19)

«11 ar

where k2 = eo/χω2. In the case of isotropie media, M and N are proportional to Debye potentials. Equations (18) and (19) are applicable in regions external to the source.

3. SOLUTION

A solution of (18) and (19) is now sought in the form

N = F(r)Pv(-cos 0), (20)

M=G(r)Pv(-cos0), (21)

where Pv is the Legendre function which has the required asymptotic behavior and has no singularities except at Θ = 0. It is a solution of the equation

The radial functions F and G must then satisfy

É33 d / 1 dF\ _L VI € 3 3 7 7 . · d (€12 A <V + ^ 77 Π n<X\ dr fc Tr) + k2 To F + / ω € 3 3 dr fc G) " ^ ^ " * = °> ( 2 3 )

d2G &2 Γ €j2" dr 2 e0 I en

€19 dF v(v -4-1) G - ίμω — — - -^-f1J G = 0. (24)

^ en dr r2

In the case of isotropie media (i.e., €i2 = 0), the equations become uncoupled. As a further check, the medium may be taken to be homogeneous and isotropie (i.e., en = *22 = €33 = € where € is a constant). Then Fand G satisfy the same differential equation given by

Solutions are linear combinations of the functions Zh^\(Z) and ZA<2) (Z), where Z = fc(e/€0)1/2r and where A*1* and A<2) are spherical Hankel functions. Most proponents [Budden, 1962; Katzin and Koo, 1962] of earth-flattening procedures start with an equation equivalent to (25). Then they develop the idea that the term

may be replaced by

where €eff(r) is the effective dielectric constant for the equivalent planar region. Unfortunately, in the anisotropie case the solutions of the coupled equations cannot be expressed in terms of spherical Hankel functions even if

the medium is homogeneous. Thus, the earth flattening procedures are open to question when dealing with magnetoplasma media.

To solve the coupled wave equations for the general case, resort must be made to numerical techniques. To simplify the procedure it is suggested that the ionosphere above the surface r = c = ro be broken into a number of concentric homogeneous regions which are bounded by the surfaces ro, ri, r% . . . rm-\, rm . . . r^ . For example, within the mth region, the elements of the dielectric tensor may be replaced by constants. Thus, for rm-\ < r < rm,

m = *22 = €m> € 3 3 = *m a n c* € 1 2 = *S™>

which corresponds to a dielectric tensor having the form

€m

— ig m 0

igm €m 0

0 0 €m

00

for the mth region. The coupled equations now have the form

(26)

and

where

ΛΛ~ +k2 — Fm- -j- — X2Fm = 0

d2Gn

dr2 + k2 * (l - 4) G. + μ^τη âFm

dr A2Gm = 0, €m

X2 = v{v + 1)

(27)

(28)

In most cases the quantity λ2 can be regarded as a constant within the layer. Thus, it is replaced by

λ2 = XI = v(v + 1)

(29)

for the mth layer. Solutions of the coupled differential equations may now be found in the form

Fm=pmc-Umr (30) and

Gm = qme-Umr (31)

Substituting these into (27) and (28) leads to the two algebraic equations

and

um Pm + k2 — pm — X^pm + qm = 0 (32)

-pugmUmPm . 9 , , 9 *m + Wmtfm + &2 —

^m eo ( · - * ) * · -A^m = 0. (33)

In order for a solution to exist, the determinant of the coefficients pm and qm must vanish. This condition can be written compactly in the form

where

k2z2

< + « i [si + si] + sX - s\ - f - = 0. (34)

s*, = k*^-Xl. (36)

There are four solutions of this equation ; two representing outgoing waves and two representing incoming waves. These are denoted ±um and ±u'm. The equation is equivalent to the Booker quartic for plane waves in anisotropie media [Budden, 1961].

The general solution in the rath layer may then be written

km — Pm£ and

m +pme m +Pme m +P m e »' W

- > < - - > Gm = qmQ m + qmQ m + qme m + qmz m ' ^ö)

when the /?'s are arbitrary constants. The q's are related to corresponding/?^ through (32) or (33). Therefore, for convenience in what follows,

qm = d(Um)Pm, qm = CL(—Um)Pm, qm = « ( O / 7 ™ ' q'm = a(—Um)qm,

where the function a(w) is known. The imaginary part of the ums is chosen to be positive and, thus, the arrows -> and <- represent outgoing and ingoing waves, respectively. In the outermost region, where m = M, only outgoing

waves are permissible and, thus, pM = pM = qM = qM = 0. Since the tangential field components are to be continuous at the interfaces, it follows, from (12), (13), (16), and (17) that four continuity conditions are required at the surface r = rm- These are

Fm = Fm+U (39)

Gm = Gm+1, (40)

dGm dGm+i âr dr (41)

i [dT - gmœGm] = ^L· [ d T ^ - S»+i"C*+i] · (42)

Therefore, four independent relations between the coefficients are available at each interface from m = 1 to m = M.

To obtain the complete solution of the problem it is necessary to consider the fields in the free space or waveguide region a < r < ro. Here the fields are obtained from

1 dMo ΕΦ=Ί--8Θ> <4 3>

i / , = - — ° (44) "Φ r 8Θ ' l >

1 a / . 8Mo\ r^rTe 8Θ [Slne~Wy ~Ψ<»ΗΤ = - ä - j - ä τ , sin θ — , (45)

1 d I dNo\

ί€οωΕ"=7^Γθ8θ[ύηθ-8θ)' W

ιμωΗβ = ', ~m ' (47)

l€o"E° = ~ 7 ärä* <48) which is a special case of equations (12) to (17) for isotropie media. In this case

No = F0(r)P„(-cos Θ), (49) and

Mo = Go(r)Pp(-cose), (50) where Go and Fo satisfy

Go &+*-*?* f o = 0 . (5.)

Therefore, solutions in the region a < r < ro are of the form

l^i = /lA(J) (Arr) + /2A<2» (*r) (52) and

η ^ Γ = giA<v» (*r) + gzh™ (kr). (53)

The boundary conditions at the level r = ro are

Fo = Fi (54)

Go = Gi, (55)

dGo dGi d7~ = dT ' (56)


1 dFo 1 fdFi I . . . .

For all practical purposes, the ground may be characterized by a surface admittance Yg and a surface impedance Zg. Thus

Ηφ = Υ9Εφ (58) and

Εφ = -Ζ9Ηφ, (59)

at r = a. In terms of Go and Fo these may be written

i^YgGo=^r, (60) and

ieowZgFo = ^— (61)

at r = a. In the case of a homogeneous ground of conductivity σ9 and dielectric constant eg,

and

Z. = ,ο (-^-)* fl - - ^ - 1 * . (63) In most actual situations the square bracket factors in these expressions may be replaced by unity, and, furthermore, displacement currents in the ground are usually of minor significance. Thus

Zg% 1ΙΥβΖ(βμωΙσ9)Κ (64)

The boundary conditions as exemplified by (39), (40), (41), (42), (60), and (61) yield 4M + 2 algebraic equations connecting 4M + 2 unknown coefficients. Setting the determinant of these coefficients equal to zero leads to an explicit equation to solve for the eigenvalues or modes v. To illustrate the form of this modal equation, the case of a homogeneous ionosphere is considered. Thus M = 1 and the boundary conditions yield 6 linear equations

-> -> in the coefficients/1,/2, gu g2,p19 and/?r The determinant of these coefficients is written out explicitly in Table 1. Setting this monster equal to zero yields a rather involved transcendental equation for the determination of the modes v. The size of the determinant grows linearly as the number of slabs is increased (i.e., M > 1). In the case of any number of slabs, the determinant equation for the mode can be written in matrix form as follows:

deJ["A °W* i * i i f W * W _ p οη

Tabl

e 1.

D

eter

min

ant

of th

e co

effic

ients

for

the

case

of

hom

ogen

eous

ion

osph

ere

[ΐ€οω

ΖΛ,(1

) (ka

) [i€

oa>Z

ghvM

(ka)

0

0

-kh v

{iy

(ka)]

-khy

W

(ka)]

0 0

[ίμω

ΥΧ^

(ka)

[ip

vYgh

J® (k

a)

-kh v

uy(k

a)]

-kh v

{2y (k

a)]

*fl

hjv

(kc)

0 0

- Λ

>>' (

kc)

€0

À„<2 >

(kc)

0 0

- h v

vy (

kc)

€0

0

A>>

(kc

)

kW»'

(kc)

0

0

A>>

(kc

)

khj»

y (k

c) 0

— Q

-UlC

-a(U

i)Q-u xc

iiia(

u^e-

u ic

1 r

ei

w]e

-vic

_ e-

MxC

-oii

/iO

e-"!

'0

Uia

(Ui)Q

-u ic

1 r

, ,

/x

€1

wje

-"^

o O cr

3

NOTE

: In

this

det

erm

inan

t Âv«>

(Z)

= ZÄ

V«>

(Z),

Ä V«)

'(Z)

= -^

[ZA

;«) (

Z)].

whe

re i

= 1,

2.

dZ

where the first two factors are matrix reflection coefficients. Explicitly,

rw xV» (x) - iZM Kv - [in' *Λ«> (x) - iZghoj x - ** K ]

and _ pn'*&»»(*)-/IVqol

Λ Α - [in' JCA«> (x) - i YgVo\ x - ka ' l j

where In' is the logarithmic derivative defined by

dF(x)/ds l n F ( x ) = ~Ύ(χΤ ■ m

Rv and Rn can be interpreted as ground-reflection coefficients for vertically and horizontally polarized waves, respectively. The explicit form of „i^, JÄIJ, etc., for the present problem could be written out in full. However, there is no need to do this as they have the same form as the reflection coefficient matrix for a plane stratified ionosphere. This aspect of the subject has been discussed recently in some detail [Wait, 1963]. To make direct use of numerical data already available for planar reflection coefficients, it is necessary to invoke the Debye approximations for the Hankel functions hyM {kc) and A„<2) (kc) and their derivatives. The restriction for this step is

(kaßfC» 1, where C is the (complex) cosine of the local angle of incidence at the ionosphere. In terms of v and kc it is not difficult to show that

,, Γ1_(» + ιη» ' ~ L ike? \ ■

In the theoretical model discussed in this paper, the earth's magnetic field was taken to be vertical everywhere. A rigorous treatment of the earth-ionosphere waveguide with a non-vertical magnetic field does not seem to be possible. However, an approximate approach to the subject, which utilizes the concept of surface impedances, also leads to a modal equation of the form of (65) above [Wait and Spies, 1960; Wait, 1961a, 1963].

4. FINAL DISCUSSION

The particular problem discussed in this paper has also been considered by Johler and Berry [1962]. However, their solution for the anisotropie ionosphere does not agree with the one given in this paper and with a previous formulation of the author [Wait, 1961a] using a similar approach. It appears that the origin of the discrepancy is their assumption that the Hertz potentials IIe and IIm in the ionosphere are not coupled by the boundary conditions.*

* It is also of interest to note that their general solution does not reduce directly to the well-known isotropie solution given by their equation (73) when the earth's magnetic field is removed.


This coupling is evidenced by the forms of (18) and (19) in the present paper. Furthermore, their Hertz potentials IIe and IIm, which are related to the Debye potentials TV and M, are not solutions of uncoupled wave equations in the ionosphere.

The coupling of ordinary and extraordinary waves by the boundary conditions appears to be an essential point in the analysis. Physically this is important because the transmitted waves in the ionosphere are almost circularly polarized even though the waves in the space between the earth and the ionosphere are nearly linearly polarized.

5. REFERENCES

BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier Publishing Co., New York and Amsterdam.

BUDDEN, K. G. (1961) Radio Waves in the Ionosphere, Cambridge University Press. BUDDEN, K. G. (1962) The Wave-guide Mode Theory of Wave Propagation, Prentice-Hall,

Englewood Cliffs, N.J. JOHLER, J. R., and BERRY, L. A. (1962) Propagation of terrestrial radio waves of long

wavelength, / . Res. NBS 66D (Radio Prop.), No. 6, 737-72. KATZIN, M., and Koo, B. Y.-C. (1962) Studies in ionospheric propagation, Pt. 1, The

exact earth-flattening procedure in ionospheric propagation problems, Final Report on Contract AF 19(604)-7233 for U.S. Air Force Cambridge Research Laboratories.

KRASNUSHKIN, P. E. (1961) Propagation of long and very long radio waves around the earth and the C, D, and E layers of the ionosphere from the standpoint of information theory, Doklady Akademii Nauk, SSSR 139, 67-70.

SEN, H. K., and WYLER, A. A. (1960) On the generalization of the Appleton-Hartree magneto ionic formulas, / . Geophys. Res. 65, 3931-50.

VAN DER WIJCK, C. TH. F. (1946) Thesis, University of Delft, Holland. WAIT, J. R. (1961a) A new approach to the mode theory of VLF propagation, / . Res. NBS

65D (Radio Prop.), No. 1, 37-46. WAIT, J. R. (1961b) Some boundary value problems involving plasma media, / . Res. NBS

65B (Math, and Math. Phys.), No. 2, 137-150. WAIT, J. R. (1963) The mode theory of VLF radio propagation for a spherical earth and a

concentric anisotropie ionosphere, Can. J. Phys. 41, No. 2, 299-315. WAIT, J. R., and SPIES, K. (1960) Influence of earth curvature and the terrestrial magnetic

field on VLF propagation, / . Geophys. Res. 65, No. 8, 2325-31. JOHLER, J. R. (1962) On radio wave reflections at a continuously stratified plasma with

collisions proportional to energy and magnetic induction, Proc. Conference on Ionosphere, Physical Society London (in press).

WAIT, J . R. (1962) Refers to 1st. Edition, see main text of this book.

ADDITIONAL REFERENCES

BUDDEN, K. G., and CLEMMOW, P. C. (1957) Coupled forms of the differential equations governing radio propagation in the ionosphere, 2, Proc. Cambridge Phil. Soc. 53, 669-82.

FERRARO, A. J. (1962) Multi-slab concept applied to radio-wave propagation in the ionosphere, and its limit to the continuous ionosphere, / . Geophys. Res. 67, 3817-22.

FERRARO, A. J., and GIBBONS, J. J. (1959) Polarization computations by means of the multi-slab approximation, / . Atmos. and Terrest. Phys. 16, 136-44.

JOHLER, J. R. (1962) Propagation of the low-frequency radio signal, Proc. IRE 50,404-427. CROMBIE, D. D. (1961) Reflection from a sharply bounded ionosphere for VLF propagation

perpendicular to the magnetic meridian, / . Res. NBS 65D (Radio Prop.), No. 5, 455-63.

Reflection of VLF Radio Waves from an Inhomogeneous Ionosphere. Part L

Exponentially Varying Isotropìe Model] JAMES R. WAIT and LILLIE C. WALTERS

Abstract. The oblique reflection of VLF radio waves from a continuously stratified ionized medium is considered. The profile of the effective conductivity is taken to be of an exponential form. This is a fair representation for the actual D layer of the ionosphere. It is shown that the gradient of the conductivity change has a marked effect on the reflection characteristics.

1. INTRODUCTION

In the study of VLF radio wave propagation it is often assumed that the ionosphere can be regarded as a sharply bounded medium on its underside. This step may be considered reasonable in view of the long wavelength and the relatively rapid change of the electron density of the D region. Furthermore, experimental data are often in accord with calculations based on this model. The general agreement is particularly good at highly oblique incidence provided that appropriate corrections are made for earth curvature. Nevertheless, there are many occasions when the sharply bounded model appears to be inadequate. Therefore, it is worthwhile to consider a more realistic model of the lower ionosphere.

On examining much of the recent literature on the characteristics of the lower D layer, it appears that the effective dielectric constant of the medium can be well approximated by an exponential function. Then, to within this approximation, the relative permittivity may be written in the form

tf(z) = * o ( l - i £ e x p j f e V (1)

where Ko is a reference permittivity, and L and ß are constants. The level z = 0 may be defined as the reference level and thus

K(0) = K 0 ( l - i ^ , (2)

which is a familiar form. t The work in this paper was supported by the Advanced Research Projects Agency,

Washington 25, D.C., under ARPA Order No. 183-62. (/. Res. NBS 67D (Radio Prop.), No. 3, May-June 1963.)

403


Under the assumptions that the angular frequency ω is much less than both the collision frequency v and the plasma frequency ω0, it is known that [Wait, 1962]

#o = l a n d L = —- , (3) ωο

provided that the earth's magnetic field can be neglected. The latter assumption is strictly valid only if v is somewhat greater than the gyrofrequency ωΗ. In actual cases, however, the isotropie assumption is useful even when v is of the order of ωπ, provided the magnetic field is not transverse to the direction of propagation [Crombie, 1961; Johler and Harper, 1962]. For the purposes of the present paper, the influence of the terrestrial magnetic field is not considered.

The constant ß in the exponent of the equation for K{z) is a measure of the sharpness of the gradient. For example, when β = I km-1, it means that the ratio ωΐ/ν or N/v increases by a factor 2.71 for each km of vertical height. The best available information on expected values of β can be found in the recent work on pulse cross modulation [Barrington et al, 1962] and from recent rocket measurements [Kane, 1962]. From these results it appears that β is of the order of | km - 1 for quiet daytime conditions although it may diifer by a factor of two or more at certain times. In the present study it appears to be desirable to allow β to vary from 0.3 to 3.0 to encompass most cases of interest.

2. BACKGROUND THEORY

The propagation of electromagnetic waves in a medium whose permittivity e(r) varies in an exponential manner has been studied extensively. The usual case considered is for horizontal polarization where the electric vector is parallel to the stratification. For example, if the electric vector has only a y component Ey, it is a simple matter to show that [Wait, 1962]

[£-2 + k\K{£) - S*)] Ey = 0, (4)

when the field varies in the x direction according to exp (—ikSx), where S is a dimensionless constant. Here k = ω\Ζ(€0μο) is the free-space wave number and eo and μο are the permittivity and permeability of free space, respectively.

At a sufficiently large negative value of z, the relative permittivity K(z) becomes unity, and thus, Ey satisfies

( £ + k2c2) Ev = °> (5) where C2 = 1 — S2. The general solution of this equation has the form

Ey = Eo(Q~ikCz + Ahe+i*cz)e-iksz9 (6)

Reflection of VLF Radio Waves from an Inhomogeneous Ionosphere. Part I 405

when Eo and Rh are constants. Recognizing that the solution of (4) must reduce to (5), for z tending to — oo, it may be shown that [Wait, 1962]

(k\2v 1 <—>' (7) -iLY° (γο)\

where v0 = 2ikC/ß. The term EoQ~ikCzQ~îkSx may be regarded as an incident wave whose

direction of propagation makes an angle Θ with the z axis where C = cos Θ or S = sin Θ. Then EoRc+ikCze~'ikSx can be interpreted as a reflected wave and R can be defined as the reflection coefficient of the exponential layer. It should be stressed that this reflection coefficient, while referred to the level z = 0, is only valid in the free space region at large negative values of z.

The reflection coefficient Rh, given by (7), may be conveniently written in the form

/ 2Τ72 \ ÄÄ = exp I - wj C\ exp /Φ, (8)

where

where λο = Ιπ/k is the free-space wavelength. Thus, the amplitude of the reflection coefficient is given by the very simple form

w = εχρ ( - λ^ cy (9)

which is independent of L. On the other hand, the phase factor Φ is relatively complicated since it involves the factorial function of imaginary argument. Nevertheless, (8), for the complex value of Rn can be used to obtain quantitative results for exponential-type layers.

3. EXTENSION TO VERTICAL POLARIZATION

In the VLF radio problem one is mainly interested in vertical polarization such that the magnetic vector is parallel to the stratification. For example, if the magnetic field has only a y component, Hy, and assuming again that the fields vary in the Λ: direction according to exp (—ikSx), it follows that [Wait, 1962]

Td2 1 dK(z) d l o / 1

Intrinsically, this is a more complicated form than (4) for horizontal polarization.

In the region of large negative z the (10) reduces to the elementary form

(£* + k*c)H* = 0> (11)

Hy = 0. (10)


and thus Hy = H^~ikCz + R^^zy-ikSx^ ( i 2)

where Ho is a constant and Rv is, by definition, the reflection coefficient for vertical polarization.

Unfortunately, for the exponentially varying permittivity of the form defined by (1), it does not appear to be possible to obtain a closed form expression for Rv. For this reason, the required quantitative results were obtained by a numerical method. Essentially, the procedure is to replace the continuous K(z) profile by a finite number of steps. The situation is illustrated in Fig. 1 where the function

1 exp βζ, (13)

is shown plotted versus z along with its step approximation. Thus, between the limits z = — zo and z = + T, the medium is divided into M homogeneous layers of width hi, hi, . . ., hm, . . ., AM-I, AM. The value of L{z) in each of these slabs is then replaced by a constant value Lm. The method is really equivalent to the usual method [Budden, 1961] of numerically integrating the basic differential equations.

FIG. 1. The step approximation to an exponential profile.

The problem now boils down to finding the wave impedance Z\ at z = —zo in terms of the properties of all the individual slabs. Here Z± = ExjHy which is to be evaluated at z = —ZQ. The reflection coefficient Rv, which is referred to the level z = 0, is then found from

C—ΖιΙηο HS 4 (14)

for z = —zo, and where 770 = VOW^o) = 120 π. Here zo is chosen sufficiently large that l/L(z) may be regarded as zero. (Strictly speaking, zo -> 00.)

4. ITERATIVE PROCESS

From the theory of wave propagation in stratified media [e.g., Wait, 1962], it is known that Z\ can be obtained by a series of iterative processes. Thus

where

2 i =

02 =

Qm =

Zi = K!Qh

K2,i Q2 + tanh D± 1 + K^i Ô2tanhDi *

^3,2 Qs + tanh D2

1 + ^3,2 03 tanh D2 '

Km+i,m Qm+i + tanh Dn

1 + Km+ifm Qm+i tanh Dm '

(15)

(16)

(17)

(18)

and so on. The various factors are defined below. The parameter Dm which is a measure of the thickness of an individual slab is defined by

D„ Nr, ( < - ! ) ' ·

(19)

where Nm is the refractive index of the rath slab given byf

Nn - ( - * = ) '

The quantity Km+i, m is the ratio of the wave impedances of layer ra + 1 and layer ra. Thus

Km+1 K-m+1, m — Kn (20)

where

Km = ^0 Λ _ * V

is the wave impedance in the mth layer. Alternately,

Ai m+1, m —

(' - r~) \ Lm+lJ

C2 Lm+1

C2 1

Lm

(21)

Finally, because the quantities Qm are given in terms of Qm+i it is necessary

f The square roots are defined such that Dm = ( n m\ Nm if Nm->oo, while

Dm = ( î ^ ^ j c i f ΛΓ*-*1. Furthermore, Nm^ (1/Lm)*e-**/4 if |W™| > >1 andW™ = 1 if Lm->00.

to know the initial value QM+I at z = T. For example,

n — KM+I,MQM+I + tanh DM . _ . ^M - 1 + ^M+i,M[tanh DM]QM+I ' ( ]

and similarly, QM-\ is expressed in terms of QM· The process is continued until g i is obtained.

In some problems of this kind it is not necessary to know precisely the value of QM+I. However, the economy of the calculations is greatly improved if a good starting value of QM+I is known. Recognizing that the wave impedance at z = T is given by

ZM+I = QM+I, (23)

and utilizing the condition |JVM+I| > > 1, it follows from previous work [Wait, 1962] that

V(i) Y NM+I Ö UM+l = "r ~ 2= (24)

λο " Μ + 1 j8j

where ^o and K\ are modified Bessel functions. In a practical sequence of calculations it is necessary to choose T large

enough that the final result for Rv is insensitive to further changes in T. For example, if yS = 1 km - 1 and λ0 = 15 km, it was found that T = 4 km was sufficiently large to obtain four-figure accuracy in Rv. Furthermore, for some values of β and λ0, it was found that JlO approaches a constant as zo was increased to 20 km. In general, for smaller values of β it was necessary to increase T and ZQ to larger values in order to achieve stability of the final results. The width of the steps, hm, must also be chosen sufficiently small to achieve an adequate simulation of the smooth profile. Generally, it was found that \Dm\ < 10-2 was a satisfactory criterion.

5. DISCUSSION OF REFLECTION COEFFICIENT CALCULATIONS

Using the basic definition of the reflection coefficient Rv, given by (14), the amplitude and phase of Rv were calculated for a wide range of conditions. While, strictly speaking, Rv is defined as the limit for z = — zo, where zo -> oo, it is of interest to first demonstrate how Rv approaches this limit. An example of such calculations is shown in Figs. 2a and 2b. Here C = 0.1, corresponding to highly oblique incidence, and λο = 15 km, o r / = 20 kc/s. Various values of β are shown on the curves. It is immediately apparent that for sufficiently negative values of z (i.e., sufficiently far below the layer), both the amplitude and phase of Rv approach a limit.

FIG. 2. The reflection coefficient as a function of height for an exponential profile and a fixed angle of incidence.

Actually, the curves in Figs. 2a and 2b have more than just a mathematical interest. Physically, the jR„, corresponding to z = —zo where zo is finite, is the reflection coefficient for a permittivity profile of the form

K(z) = Ko Γΐ - -L e*l for z > - z0

= 0 for z < — zo. (25) Therefore, these curves describe reflection from a sharply bounded ionosphere which behaves exponentially above its lower edge. Of course, as zo tends to oo,


the discontinuous profile becomes a continuous exponential. It is rather interesting to observe that as the discontinuity is moved from above the reference level (z = 0) to below, the reflection coefficient passes through a minimum. This is related to a Brewster angle phenomenon since to the right of this minimum in Fig. 2a the reflection is metallic-like while, to the left, the reflection is dielectric-like.

G 0.05 UJ —I

UJ 0.04

0.0!

■ VERTICAL POLARIZATION HORIZONTAL POLARIZATION

î T f J J I L 0.2 0.4 0.6

C 0.8

FIG. 3. The amplitude of the reflection coefficients for vertical and horizontal polarization as a function of C which is the cosine of the angle of incidence.


For the remainder of the present paper, attention will be confined to the limiting value of Rv far below the reference level z = 0 (i.e., z ^ — zo = — oo). On the other hand, it should be emphasized that Rv is always referred to the level 2 = 0.

The magnitude of the reflection coefficients Rv and Rh are shown plotted in Fig. 3 for vertical and horizontal polarization, respectively. The values of \RV\ were obtained from (14) using the multislab model described. The values of |JRÄ| were calculated directly from (9) and, on this log-linear scale they are merely straight lines. As an important check, the value of \Rn\ was also obtained from a multislab model using the same procedure as described for vertical polarization. Within four-figure accuracy, the values of \Rh\ obtained by the two methods were the same.

A number of significant features are evident in Fig. 3. In the first place, \RV\ exhibits a Brewster angle phenomenon, provided ß is sufficiently large. Here the reflection process is dielectric-like at grazing angles and is more or less metallic-like at normal incidence. Of course, \Rh\ does not exhibit this phenomenon. However, for very small values of ß, corresponding to a relatively slowly varying medium, the curves for \Rh\ and \RV\ become rather close to one another. This is consistent with the optical behavior of waves in an inhomogeneous medium. In fact, by a direct application of the phase integral method of Eckersley, the equation for \Rh\ is found also to be applicable to \RV\. The use of the phase integral method for such applications is necessarily restricted to slowly varying media [Wait, 1962]. It is apparent, that for VLF radio waves, when ß is of the order of 0.5, the phase integral method is inapplicable to vertical polarization.

Another interesting feature of Fig. 3 is the near linear dependence of all the curves for small values of C Fortunately, it is just these values which are important in the long-distance propagation of VLF radio waves. The linearity of the phase curves for Rv are indicated in Fig. 4 for the same conditions. Here, it is evident that they all approach —180° at grazing incidence. The phase curves for Rh exhibit a similar property but they are not shown here since they have only an academic interest at VLF.

The variation of the reflection coefficient Rv as a function of the gradient parameter ß is illustrated in Figs. 5a and 5b at oblique incidence. It is rather remarkable that \RV\ is relatively insensitive to ß if it is in the range from 0.7 to 3.0. Furthermore, it appears that \RV\ has a broad maximum for ß approximately equal to 1.2.

The general behavior of the amplitude and phase of the reflection coefficients at highly oblique incidence suggests that, if Rv is written in the form

Rv = — exp (aC), the function a should be almost independent of C Writing

α = a i + /(X2,

120

- 60 CO v

ce o Q O h -> h*

O

LxJ CO < X CL

-60 h

120 l·

L L L, h

—

u

r

-

1 1 1 1 1

λ0=!5 L=l/2

1 I I ± I

i ' ' ~r i \~

J?/ ^^ *y^

y r -

_

-

—

Î ! 1 !

~

-

0 0.1 0.2 0.3 0.4 0.5 0-6 0-7 0.8 0.9 1.0 C

FIG. 4. The phase of the reflection coefficient for vertical polarization as a function of C.

0.2

-60

-100

! ! ! I I I I

^ Ο Ο , Η

(o)

t t I f » ' i T t t T i 1 T T T ? Î f I t i 1 r ! t t 0.5 1.0 1.5 2.0 2.5

* -120

iu -140

< -160

-180

i i I [ i i i i 1 i i i i j ι i i i \ ι i i ι i i i i

(b)

T t 1 H t l i 1 i i i i 1 I i i i I t T ? î f t t î t

0.5 1.0 1.5

ß 2.0 2.5 3.0

FIG. 5. The reflection coefficient as a function of the gradient parameter ß for an exponential profile.


where αχ and a2 are real, it is a simple matter to compute the complex coefficient a from the numerical data of Rv. The results are shown in Figs. 6a and 6b where (—ai) and α2, respectively, are plotted as a function of C in the range 0.05 to 0.30. It is apparent that over this range of C, the coefficient a can be regarded essentially as a complex constant.

5.0

4.0

ö 3.0 I

2.0

1.0

(a)

-

1 1 /3 = 0.3 km - 1

0.5

0.7

A 2.0 ^- 1.0

1 1

.._r_

L =

1

1/2,

1

λο =15

!

-\

— - =

-

0.05 0.10 0.15 0.20 0.25 0.30 c

o.u

7.0

6.0

5.0

4.0

3.0

20

1 Λ

(b)

1

£ = 0.5

0.7

1.0

I

1

1

1 1

L = 1/2 , λ 0 =15

I 1

-

~ -η

-—

—

— _

0.05 0.10 0.15 0.20 0.25 0.30

FIG. 6. The real and imaginary parts of the function a defined by Rv= — exp(aC) for λο = 15 and various values of β.


For all the results given in the foregoing figures, it has been assumed that L = \ and λο = 15. Actually, the results for the magnitude of Rv and Rn do not depend on L· In fact, \RV\ and \Rn\ depend on the parameter βλο. Furthermore, the phase of the reflection coefficients is also simply related although the situation is slightly complicated by the choice of the reference level where L is required to have a special value. In fact, it is convenient to choose the reference level (i.e., z = 0) so that

_ ων _ 1 15

In this way, the results can be readily compared as a function of frequency or wavelength. At the reference level z = 0, the effective conductivity parameter ωΓ or ωΐ/ν has the value 2.51 X 105.

For wavelengths other than λο = 15, it is also found that Rv may be approximated by the function — exp(aC) where a is approximately a complex

I I 1 I L _ J 1 f 1 1 I io I I I ! I I I I L _ l I 10 14 18 22 26 30 10 14 18 22 26 30

λ0 IN km λ0 IN km

FIG. 7. The real and imaginary parts of the function a plotted as a function of the wavelength λο for C=0.16 and various values of β.


constant. To illustrate the wavelength dependence, —a± and a2 are plotted as a function of λ0 in Figs. 7a and 7b, respectively. For these curves, C is chosen to be 0.16; the corresponding curves for other values of C in the range 0.10 to 0.20 are almost indistinguishable. The ordinate in Fig. 7a is simply related to the magnitude of the reflection coefficient and, thus, small values of —ai are associated with high reflection coefficients. It is apparent that for small values of β corresponding to a diffuse layer, the reflection coefficient becomes very small for the shorter wavelength. On the other hand, for a rapidly varying layer, corresponding to large values of β, the reflection coefficient decreases with increasing wavelength. For intermediate values of β, the reflection coefficient has a maximum in the wavelength range between 10 and 30 km. For example, when β = 1.0, the optimum wavelength is about 17 km or approximately 18 kc/s.

The curves for a2 in Fig. 7b also have a particular significance. Noting that

arg Rv = — 7Γ + a2C,

it is apparent that a2C is the phase shift resulting from the imperfect reflecting properties of the exponential layer. To attach a physical meaning to this term it is often desirable to imagine the reflection taking place at a height Δζι below the level z = 0. In this case, Δζι is chosen so that the arg of Rv is always — π. Clearly,

a2 = 2&Δζι = 4ττΔζι/λο, or

Δ ζ ι = α2λο/(477).

For example, at λο = 15 km (i.e., 20 kc/s) and β = 1, the effective height of reflection is depressed by approximately 3 km. For smaller values of β, it is seen from Fig. 7b that Δζι may be much greater.

6. FINAL REMARKS

The results, for the coefficients a± and a2, given here may be introduced into the waveguide mode theory. In this way, attenuation rates and phase velocities of the modes may be obtained. This is a valid subject for another paper.

The influence of nonexponential profiles also appears to be a subject worthy of some attention. Using the iterative method described in this paper, it appears that any reasonably smooth profile may be treated in a straightforward manner. Also, with some modification, the technique may be used to evaluate the effect of discontinuities in gradients and local regions of excess ionization.

General conclusions about the influence of the profile of the lower ionosphere on YLF propagation must await completion of extensive and systematic calculations. It is hoped to report such results in the near future. Detailed


comparisons of theory and experiment are also deferred until these more complete computations become available.

We thank Mrs. Carolen Jackson and Mrs. Eileen Brackett for their assistance in the preparation of this paper. In addition, we would like to thank Douglass D. Crombie for his useful suggestions and comments.

7. REFERENCES

BARRINGTON, R., LANDMARK, B., HOLT, O., and THRANE, E. (1962) Experimental studies of the ionospheric D-region, Report No. 44, Norwegian Defence Research Establishment, Kjeller, Norway.

BUDDEN, K. G. (1961) Radio Waves in the Ionosphere, Cambridge University Press. CROMBIE, D. D. (1961) Reflection from a sharply bounded ionosphere for VLF propagation

perpendicular to the magnetic meridian, / . Res. NBS 65D (Radio Prop.), 455-463. JOHLER, J. R., and HARPER, JR, J. D. (1962) Reflection and transmission of radio waves at

a continuously stratified plasma with arbitrary magnetic induction, / . Res. NBS 66D (Radio Prop.), 81-99.

KANE, J. A. (1962) Re-evaluation of ionospheric electron densities and collision frequencies derived from rocket measurements, Chapter 29 in Radio Wave Absorption in the Ionosphere\ Pergamon Press, Oxford.

WAIT, J. R. (1962) Refers to 1st. Ed.; see Chap. 4 of present book.

Reflection of VLF Radio Waves from an Inhomogeneous Ionosphere. Part II.

Perturbed Exponential Model] JAMES R. WAIT and LILLIE C. WALTERS

Abstract. The oblique reflection of radio waves from a continuously stratified ionized medium is considered. In this paper the medium is assumed to be isotropie. The height profile of the effective conductivity is a Gaussian curve superimposed on the (undisturbed) exponential form. The reflection coefficient is shown to be influenced by the vertical location of the Gaussian perturbation. In some cases the magnitude of the reflection coefficient is increased while, in other situations, it is decreased. In nearly all cases, insofar as phase is concerned, the presence of the perturbation corresponds to a loweringof the reflection height.

1. INTRODUCTION

In a previous communication from the present authors [Wait and Walters' 1963], oblique reflection of (VLF) radio waves from a continuously stratified ionized medium was considered. The profile of the effective conductivity was taken to be exponential in form. Actually, this is a fairly good representation of the actual D layer of the ionosphere under daytime conditions. Henceforth, that paper will be referred to simply as (I).

It is the purpose of the present paper to consider profiles which are no longer exponential in form. Since the objective is to gain insight into the mechanism of reflection from perturbed layers, a number of idealizations are made. First, it is assumed, under quiescent conditions, that the ionospheric conductivity varies exponentially with height. Then the idealized perturbation is assumed to have a Gaussian form. Again, for sake of simplicity, the earth's magnetic field is neglected as in (I). This is well justified when considering effects which result from ionization in the lowest ionosphere.

2. DESCRIPTION OF THE PROFILE

The notation follows that used in (I) as closely as possible. Thus the undisturbed profile, as a function of height z, is defined by the conductivity

t The work in this paper was supported by the Advanced Research Projects Agency, Washington 25, D.C., under ARPA Order No. 183-62. (/. Res. NBS 67D (Radio Prop.), No. 5, Sept.-Oct. 1963.)

417


parameter 1/L(z) where

ih) = iexp(ßz)> (1)

and L is a constant, ß is a gradient parameter and z is the height above the reference level z = 0. Under the isotropie assumption, we know that

L = ^ + ^ (2)

in terms of the angular frequency ω, collision frequency v, and plasma fre quency ωο. At VLF, v > ω, and therefore

L ς — where cor == — , (3)

to within a very good approximation. In general, it is seen that \jL{z) is proportional to N{z)jv{z) where N(z) and

v(z) are the electron density and collision frequency regarded as a function of height. The constant ß, in the exponent, is a measure of the sharpness of the gradient. For example, β = 1 km-1 means that the ratio of ωΓ(ζ) or N(z)jv{z) increases by 2.71 for each km of vertical height. From the recent work of Barrington et al. [1962], Kane [1962], and Belrose [1963], it appears that β for an undisturbed ionosphere may be in the range from 0.2 to 0.8. If the level from about 60 km to 70 km is considered, it appears that β = 0.3 typifies many of these daytime D-layer profiles. A detailed study of the influence of changing β is to be found in (I). For this paper, β is chosen to be 0.3.

Having specified our undisturbed profile, we now wish to introduce the perturbation. It is assumed that the collision frequency profile is unchanged whereas the ionization is to be increased by an amount ΔΝ(ζ) where

AN(z) = ANo exp [-5-7 (4)

and ANo, F and D are constants. Clearly, the maximum value of AN(z) is ANo which is located at z = F. Furthermore, the thickness of this layer is 2D which is the vertical distance between the levels where AN(z) drops to ANol*.

In order to estimate correctly the influence of this Gaussian shaped layer, it is necessary to assume something about the collision frequency profile. A careful study of the recent literature indicates that an exponential variation of v(z) with height z is not unreasonable. The form chosen here is

v(z)=v0Qxpi- -zV (5)

where β = 0.3 km-1. Therefore, the resulting conductivity perturbation has


the form àN(z) ΑΝο (β \ Γ (z-Fy-\ -& = ^ e x p (2 ήexp [-(-D-) \ · (6)

VERTICAL DISTANCE Z , km

FIG. 1. The undisturbed and disturbed conductivity profiles used in this paper.

The complete profile, under these idealized conditions is given by

zfe = W) {exp m + A exp (2 z)e x p [-(^1}' (7) where the right-hand side is proportional to the effective conductivity of the medium as a function of height above (or below) the reference level at z = 0. The coefficient A defines the strength of the perturbation. In fact,

A ΔΝο A - No '

where No is the electron density of the undisturbed profile at the reference level z = 0. In this paper, as in (I), L(0) = 7.5/λ where λ is the wavelength in kilometers.

It is admitted that other ways to define a perturbation in the profile may be preferable. Here the electron density anomaly, for a given value of A, does not change with its vertical location F. Consequently, we may anticipate that the influence of this type of perturbation will be diminished at sufficiently low heights because of the increasing collision frequency. However, we shall see the problem is not quite this simple as other factors come into play.

A sketch of the profiles used is given in Fig. 1. The undisturbed profile is the exponential form, while the disturbed profiles have the superimposed Gaussian "bump". The location of the "bump" for five typical profiles is specified by the appropriate value of F.

4»

LVV 160 120

80

■σ 4

0 CE

L·. °

0 LU

< X CL -4

0

-80

-120

_icn

—

- —

_, C

—

! (b

)

λ A D

ß

=J)A

0.20

0.15

O! I

1

-- 15

km

=

2 =

2 km

=

0.3

km'

—.

^

1

1 1

1 s\l

\/

\

i 1

i l\

' l

' ;

\

\ \P

r \

\ \

\ \

\ \

\ \

\ \

\ \

^ ^

\ f\

\ \

1 \

\\ \

\ \

V\"·

1 '

| —

— _ —

j — ,

s^..

=d J —S

0.

20

^0.1

5

0,1

1 ,

0 10

-6

0 -5

0 -40

-3

0 -2

0 F,

km

FIGS

. 2a

and

b. T

he re

flect

ion

coefl

Bcien

t as a

func

tion

of th

e ver

tical

loca

tion,

F, o

f the

Gau

ssia

n pe

rtur

batio

n, fo

r va

rious

ang

les o

f inc

iden

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Reflection of VLF Radio Waves from an Inhomogeneous Ionosphere. Part II 421

3. RESULTS OF CALCULATIONS

The method used to calculate the reflection coefficient R has been described in detail in (I). The quantities considered are the amplitude \R\ and the phase of R for a vertically polarized plane wave incident at an angle whose cosine is C. The reflection coefficient is evaluated in the free space region corresponding to z-> — oo. However, it is important to remember that the phase is referred to the level z = 0.

The plan of the calculations is to vary the value of one parameter while keeping the others constant. To obtain a complete understanding of the various phenomena, an enormous number of calculations is needed. In order to keep the problem within reasonable bounds and to reduce the expense of the computation, only a limited number of cases was considered. These results are shown in graphical form in Figs. 2 to 6. In all cases ß = 0.3 km-1.

In Fig. 2a the amplitude of the reflection coefficient is plotted as a function of F for λ = 15 km ( / = 20 kc/s), A = 2, D = 2 km, and C values varying from 0.05 to 0.4. Small values of C here correspond to angles near grazing. For long distance propagation of VLF radio waves, values of C near 0.1 are most important. For this case, it is interesting to note, when F is near or above zero, that |jR| takes the same value as for the undisturbed profile. As

n i I I I I I I 1 I i, 1 I I - 5 0 - 4 0 · - 3 0 - 2 0 -10 0 10

F3km (FIG. 3a).

220

180

140

100

en ■S60

ce u_ ° 20 LÜ CO < X Q_

-20·

-60

-100

-140

F3km

FIGS. 3a and b. The reflection coefficient, as a function of F, for various widths of the Gaussian perturbation when C=0.2 (i.e., angle of incidence is 78°).

the "bump" or perturbed layer is lowered, the reflection coefficient first increases then decreases. Eventually, as the "bump" is brought down to very low heights, \R\ returns to its undisturbed value. The other curves for highly oblique incidence have a similar behavior. Thus, the "bump" may either improve or degrade the reflection. Presumably, at the lower heights the Gaussian layer is acting as an absorber whereas, at greater heights, it enhances the reflection. At the steeper angles of incidence, the situation becomes more complicated. It is probable that this results from interference between multiple reflected rays between the upper side of the "bump" and the exponentially varying layer. Such an interference phenomenon becomes more


pronounced at steeper incidence because the vertical component of the wavelength is becoming comparable with typical values of F.

The phase of R is shown in Fig. 2b for the same conditions as in Fig. 2a. Again, it is apparent that, when the Gaussian "bump" is such that Fis near 0 or above, the phase of JR attains its undisturbed value. When the angle of incidence is highly oblique the phase undergoes an increase (i.e., decrease of lag) as the "bump" comes down to lower heights. Sufficiently far below the reference level, the phase of R returns to its undisturbed value. It is well to note that as C becomes small (i.e., approaching grazing incidence), the phase of R is approaching —180°.

J i I i 1 i I i 1 I 1 i 1 i 1 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10

F, km

- 6 0 | I I I I \ j i i i [ I I r

F, km

FIGS. 4a and b. The reflection coefficient, as a function of F, for various widths of the Gaussian perturbation when C=0.1 (i.e., angle of incidence is 84°).

EE

-30

-20

F, k

m

FIGS

. 5a

and

b. T

he r

efle

ctio

n co

eflS

cient

as

a f

unct

ion

of 7

% fo

r va

rious

wav

elen

gths

(fro

m 1

0 to

30

km).


For highly oblique incidence the influence of the "bump" is to lower the effective height of reflection for the whole range of F. However, a very interesting phenomenon occurs at steeper incidence. As can be seen in Fig. 2b, when C = 0.2 the phase undergoes a rather rapid change as F varies from about —22 km to —27 km. As C is increased further there is an apparent discontinuity when the phase changes by 360°. Such a change of 2π radians is quite permissible since the ordinate is arbitrary to within any integral number of 2π radians. Thus, the phase curves for C = 0.3 and 0.4 could have been drawn in the range below —160°.

The curves in Fig. 2b, even if they show nothing else, demonstrate that phase shifts in reflection phenomena may have some unusual cycle ambiguities.

The influence of the width of the Gaussian perturbation or "bump" is shown in Fig. 3a for the amplitude \R\ and in Fig. 3b for the phase of R. Here A = 2, λ = 15 km, and C = 0.2. The amplitude curves show that, when D is increased, the overall influence of the layer becomes somewhat greater. There is some tendency for the thinner layers (i.e., smaller D) to be more effective at greater heights. The corresponding phase curves show that the thicker layers always produce a larger phase change. Furthermore, as D exceeds 2 km, a point is reached where the "360° jump" takes place.

The curves in Figs. 4a and b are for the same conditions as Figs. 3a and b except that now C = 0.1, corresponding to nearer grazing incidence. The amplitude curves have a very similar shape. The phase curves are also similar except that the "360° jump" is no longer present.

The wavelength dependence of the reflection coefficient is shown in Figs. 5a and b. For these C = 0.1, A = 2, and D = 2. The wavelengths chosen (10, 15, 20, 25, 30 km) correspond to frequencies of 30, 20, 15, 12, and 10 kc/s. Qualitatively, the curves have a very similar shape. There is some tendency for the shorter wavelengths to be accompanied by more pronounced changes. In all cases the "bump" acts as an absorber at low heights while it enhances the reflection at greater heights.

Finally, in Figs. 6a and b, the influence of A, the relative magnitude of the anomalous electron density, is shown. As expected, the individual curves are similar in shape with the larger values of A corresponding to an increased change over the undisturbed values. It is important to note that the phase anomaly is almost directly proportional to A.

4. FINAL REMARKS

The results given here constitute a small portion of extensive computations dealing with reflection of waves from inhomogeneous media. In subsequent parts to this series other types of profiles will be considered. Also, the applications to the mode theory of VLF propagation are to be described in some detail.


-30 -20 F,km

-30 -20 F,km

FIGS. 6a and b. The reflection coefficient, as a function of F, for various amplitudes A of the perturbation.


The authors wish to thank A. G. Jean and D. D. Crombie for their helpful suggestions during the course of this work.

5. REFERENCES BARRINGTON, R., LANDMARK, B., HOLT, O., and THRANE, E. (1962) Experimental studies

of the ionospheric D-region, Report No. 44, Norwegian Defence Research Establishment, Kjeller, Norway.

BELROSE, J. S. (1963) Low-frequency propagation (unpublished manuscript). KANE, J. A. (1962) Re-evaluation of ionospheric electron densities and collision frequencies

derived from rocket measurements, Chapter 29 in Radio Wave Absorption in the Ionosphere, Pergamon Press, Oxford.

WAIT, J. R., and WALTERS, L. C. (1963) Reflection VFL of radio waves from an inhomogeneous ionosphere. Part I. Exponentially varying isotropie model, / . Res. NBS 67D (Radio Prop.), No. 3 361-7.

Reflection of VLF Radio Waves from an Inhomogeneous Ionosphere. Part III. Exponential

Model with Hyperbolic Transition]

JAMES R. WAIT and LILLIE C. WALTERS

Abstract. This is a continuation of two earlier papers on the subject of reflection of waves from inhomogeneous isotropie media. In this particular paper an exponential conductivity profile is perturbed in such a manner that the conductivity is increased for all heights above a certain level. A hyperbolic tangent transition is employed in order to avoid discontinuities in the conductivity versus height profile.

1. INTRODUCTION

In two earlier communications from the present authors [Wait and Walters, 1963a and 1963b] oblique reflection of (VLF) radio waves from continuously stratified ionized media was considered. In the first of these, denoted (I), the profile of the effective conductivity was taken to be exponential in form. It was pointed out that an isotropie exponential model is a fairly good representation of the D layer of the ionosphere under quiet DAYTIME conditions. In the second paper, denoted (II), an idealized perturbed exponential model was considered. The perturbation consists of a localized increase of electron density which itself has a Gaussian profile. The reflection coefficient was shown to be influenced by the vertical location of this Gaussian perturbation. In the present paper, which is part III of the series, the exponential profile model introduced in (I), is modified by allowing the electron density to be increased for all heights above a specified level. Such a modification to the quiescent ionosphere could result from ionizing radiations associated with a solar flare [Jean and Crary, 1962; Pierce, 1963; Chilton et al, 1963].

For the detailed numerical results given here, it is assumed that the earth's magnetic field may be neglected. This is justified for highly oblique incidence at VLF provided that attention is restricted to effects which result from ioniza-tion in lowest daytime ionosphere [Johler, 1962].

t / . Res. NBS 67D (Radio Prop.), No. 6, Nov.-Dec. 1963. 429

2. DESCRIPTION OF THE PROFILE

The notation follows that used in (I) as closely as possible. Thus, the undisturbed profile, as a function of height z, is defined by the conductivity parameter \jL(z) where

llL(z) = (llL)exp(ßz), (1)

and L is a constant, ß is a gradient parameter, and z is the height above the reference level. Under the isotropie assumption, we know that

_ ω{ν + Ιω)

in terms of the angular frequency ω, collision frequency v, and the plasma frequency o>o. Furthermore, at VLF, v > ω which leads to

L £ ωΙωτ where ωΓ = ωΐ/ν. (3) In general, the conductivity parameter is proportional to N(z)lv{z) where

N(z) and v(z) are the electron density and collision frequency regarded as a function of height. The constant ß, in the exponent, is a measure of the sharpness of the gradient. As in (II), it will be assumed that β = 0.3 km - 1

typifies an undisturbed D-layer profile. We shall now turn our attention to the modification of the exponential

profile. As in (II), it is assumed that the collision frequency profile is unchanged whereas the ionization is to be increased by an amount ΔΝ(ζ) where

ΔΝ(ζ) = ΔΝ0 i t a n h ^ - ^ + l j , (4)

and ΔΝο, F, and D are constants. It is evident that ΔΝ(ζ) is equal to ΔΝο at z = F which may be described as the "lower edge" of the modified layer. When z is somewhat less than .Fit is seen that ΔΝ(ζ) is zero whereas when z is much greater than F9 ΔΝ(ζ) becomes equal to 2ΔΝο. It is clear that the vertical location of the "lower edge" is governed by the value of F. The rapidity of the transition from 0 to 2Δ#ο is controlled by the magnitude of D; the quantity 2D could be described as the "transition thickness".

Following the discussion given in (II), the collision frequency profile is taken to be

v(z) = v0exp(-ßzl2), (5)

where j8 = 0.30 km- 1. The resulting conductivity perturbation has the form

ΔΝ(ζ) ΔΝο exp(£z/2) [ l + t a n h ( ~ ) ] . (6)

V(Z) VQ

The complete conductivity profile is now given by

z<b = z<k{exp m + A exp(2z) [l+tanhC~^)]}' (7)

Reflection of VLF Radio Waves from an Inhomogeneous Ionosphere. Part III 431

< OC

2 > O

a o o

20

16 h

12 h

UNDISTURBED PROFILE-/-X = l5km

(ORf;20kc/s') D = 2km A=2 ß = 0:3 km-1

TYPICAL DISTURBED PROFILES-

(a) ^

^\6V^i JL

-16 -12 - 8 - 4 0 4

VERTICAL· DISTANCE Z IN km

FIG. la

<. S >

j -

r

l·-r~

—r i X = i5km

(ORf =.20kc/s) F =-5 km A = 2 y3=0.3km-'

•(b)

1 1

TYPICAL DISTURBED PROFILES."^

1 1

Ύ J / V

//r / / //

\r/jy UNDISTUR

Ì 3ED

M / Z ^ PROFILE-^

1 - " " ^

-!

-

—

-14 -10 -8 -6 -4 -2

VERTICAL DISTANCE z IN km

FIG. lb

FIGS, la, lb, lc. The profile of the conductivity of the model illustrating the influence of the various parameters.


0 1 ! 1 « r f ^ — T 1 1 1 1 -20 -16 H2 -fi -4 0 4 8 12

VERTICAL DISTANCE z IN km

FIG. lc

where the right-hand side is proportional to the effective conductivity of the medium as a function of height above (or below) the reference level at 2 = 0. The coefficient A defines the strength or magnitude of the perturbation. The constant L(0) is chosen to be equal to 7.5/λ where λ is the wavelength in kilometers.

To enable the reader to grasp the significance of the various possible forms of profiles, some typical examples are illustrated in Figs, la, lb, and lc. In Fig. la, the vertical location of the "lower edge" of the perturbed layer is varied while other factors are kept constant. In lb, the abruptness of the transition at the "lower edge" is varied for fixed values of Fand A. Finally, in lc, the strength of the perturbed layer is varied while keeping F and D constant.

3. RESULTS OF THE CALCULATIONS

The method used in the reflection coefficient calculations is the same as that used in (I) and (II), so this aspect of the subject need not be discussed. As before, the amplitude |JR| and the phase of R for a vertically polarized wave are considered. The reflection coefficient is evaluated in the free-space region corresponding to z -> — oo ; however, it is to be remembered that the phase is referred to the level z = 0.

In Fig. 2a, the amplitude of the reflection coefficient is plotted as a function of F for λ = 15 km (i.e., / = 20 kc/s), A = 2, D = 2 km, and C values

200

120

40

£ -4

0

-120

H60

I

i- (b

) %

Γ*

^^

—£>

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0.05

1

ι V \ \ &

\ \ \

**

Ν.

/ \

Nw

/ *'*

\ ^

\^\

\

\ 1

\ Ί

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\ \

\ \ \

\ \ \

\ \

\ \ ν

\ \

\ \

\ \

\

1^

>

Γ X

-'l5

km

Α =

2 D

=2 k

m

β =

0,3

km"1

\ \ \ \ \

\ \

\ \

. ->—

1

* —\

-δθ

-40

-30

-20

F(k

m)

FIGS

. 2a

and

2b. A

mpl

itude

and

phas

e of

the

ref

lectio

n co

eflSc

ient f

or v

ario

us v

alue

s of

C, t

he c

osin

e of

the

angl

e of

inci

denc

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s th

e ver

tical

di

stanc

e, re

lativ

e to

the

refe

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ight

, to

the

"low

er e

dge"

of t

he p

ertu

rbed

laye

r.

i-LJ

UJ O o

0.6

0.5

0.4

0.3 h-

0.2

0.1

I

l·-w //''/ H ////

s/-'/ \.'^Lyi<y

Y (a)

ι _

ι r

of :/ v,l !/ o / ;/ li/

ff// ff// / 91/ / / / / / //' // /// #/ F

J/

1 L

Φ^\ ^/ .-W

if/ V\ '/ / V\ / / j^-y\ / / o*^\% x ^ \ x

/ / V__ / /

λ =15 km C =0.2 A =2 β =0.3 km"'

i l L

^vs>*.

—

—

—

-60 -50 -40 -30 -20

F(km)

-0

Q: o

UJ en < •x CL

160

120 p ^

40 -

-40 h

-120

—

—

i

"*"■*«*

(b)

!,...

i—

"v^v V ^

. . . , . [

1 1 !

λ =l5km C =0.2 A =2

, 0 = 0 . 3 km"1

vCvsN

A\ \ \ \

1

—

ΥΛ\Ν °Λ '

1 1 1

<yfr Ί

ΙΊ -60 -50 -40 -30 -20 -10 10

F (km) FIGS. 3a and 3b. Amplitude and phase of the reflection coefficient for various values of D, the "half-thickness" of the layer transition for C=0.2.

C8

0.4

Ί Γ

_L _L -60 -50 -40 -30 -20

F ( k m )

-20 p

~ -40 h-

B "60 h

u- -80 h-

? -loo

-120

-p-

-

i

j 1 i ! j

X = i5km (b) — . C = O.I ^ ^ A = 2

^ . β =0 .3 km1

V \

\^ \^ Y ^

xv\\ v \ \ A \ \ \ V \ \ \ \ — °\ΛΛ \>>v\\ \ 0 \ \ \ v

\ K \ \<?

Ö 0 \ \ \ \

V 3 \ \ X - ~ 1 1 ! 1 ^ ^ h =

-40 -30 -20 -10 0

F ( k m )

FIGS. 4a and 4b. Amplitude and phase of the reflection coefficient for various values of D, the "half-thickness" of the layer transition for C==0.1.


£T

1 -

O U-

O O 2 o h-

ÜJ

u. Id tr.

U-0

0.7

0.6

0.5

0.4

0.3

0-2

0.1

Λ

:

-

j^

1 { Λ^

y /

y

(σ)

1 1 !

^ " " Χ ^

\ .

-j

—

C = 0.l ^_ Α = 2 j D = 2km y3 = 0.3km_I ^_

ι ι 1 -50 -40 -30 -20

F ( k m ) .

120

- 40 h-

•±- -40 l·—

? -80 h -

-120

-160,

—

—

—

—·

1

-~^ /5^

_ 2 0 ^

__25__ _ 30

(b)

!

.4

1 l

1 1

1 1 1

c =0.1 A=2 D =2 km ß =0.3 km-1

1 f ^ ^

,__

—

—-

*—-

*—I

-40 -30 -20

F ( K m )

FIGS. 5a and 5b. Amplitude and phase of the reflection coefficient for various values of λ, the wavelength.


F ( k m )

-60 -50 -40 -30 -20 -10 0 F ( k m )

FIGS. 6a and 6b. Amplitude and phase of the reflection coefficient for various values of A, the magnitude of the perturbation.


varying from 0.05 to 0.4. Small values of C correspond to angles near grazing; these are important for long distance propagation of VLF radio waves. It is evident that when Fis somewhat above zero, \R\ assumes the value for an unperturbed exponential layer. As the "lower edge" of the perturbed layer moves down (i.e., F becoming negative), the tendency is for \R\ to increase somewhat. As the perturbed layer is moved farther down (i.e., becoming increasingly negative), \R\ tends to be reduced. The behavior illustrated here is not dissimilar to the corresponding case of a Gaussian "bump," described in (II), as it moves down to low heights. In both cases the perturbation enhances the reflection at greater heights but degrades it at lower heights.

The phase of R is shown in Fig. 2b for the same conditions as in Fig. 2a. It is apparent that when the "lower edge" is somewhat above F = 0, the phase attains its undisturbed value. As F becomes negative the phase increases and, for grazing angles, this increase is monotonie in nature. However, for the steeper angles of incidence there is some evidence of oscillations, which are presumably due to certain interference phenomena.

The dependence of the "transition thickness" on the reflection coefficient is shown in Figs. 3a and 3b for C = 0.2. These show, as one might expect, that a sharp transition is associated usually with an increased reflection coefficient. As the transition is stretched out, the tendency is for the maximum at F ^ —15 km to disappear. It is particularly interesting to note that the phase is not overly sensitive to changes of D.

The influence of varying D for a highly grazing situation is shown in Figs. 4a and 4b. These show behavior which is qualitatively the same as Figs. 3a and 3b.

The wavelength dependence of the reflection coefficient is shown in Figs. 5a and 5b for λ = 10, 15, 20, 25, and 30 km corresponding to frequencies of 30, 20, 15, 12, and 10 kc/s. As expected, the curves have similar shapes to each other and the tendency is for the shorter wavelengths to be associated with more pronounced changes.

Finally, in Figs. 6a and 6b, the influence of varying the strength A of the perturbed layer is indicated. Generally, as A is increased, the changes are more pronounced. It is interesting to note that at the very low heights the magnitude of the reflection coefficient is very sensitive to small changes of A. This behavior is similar to that found in the troposphere when considering the reflection of VHF radio waves from gradient changes of the refractive index [Wait, 1962].


In the present series of three papers, the reflection of plane waves from an inhomogeneous isotropie medium has been considered. Attention has been confined to horizontally stratified layers whose effective conductivity varies


smoothly with height. Special attention is given to vertically polarized waves at highly oblique incidence and at very low frequencies because these are important in long distance propagation.

In the first paper, the conductivity profile is taken to be exponential in form. Although this is a simple model, it is quite convenient to employ for quantitative estimates of the amplitude and phase changes as a function of gradient, angle of incidence, and frequency. The application of the results of (I) to mode theory has already been carried out [Wait, 1963].

In the second paper of this series, denoted (II), the exponential profile is allowed to have a perturbation in the form of a Gaussian curve. In the third paper (i.e , the present one), the perturbation is in the form of a hyperbolic tangent. Basically, the difference between these two classes of profiles is that in (II) the perturbation is localized whereas in (III) it is spread out over a range of heights. It is probable that the results in (II) are more applicable to ionization produced by the hard X-rays emitted from a space nuclear burst. On the other hand, the results in (III) would be more appropriate to the ionization produced by a solar flare.

The direct application of the results in (II) and (III) to mode theory computations and further analysis including the earth's magnetic field is reserved for future communications In the meantime, it is hoped that the present results should add to the knowledge of wave propagation in inhomogeneous media.

We thank A. G. Jean for many useful suggestions concerning this work. The work has been supported by the Advanced Research Projects Agency, Washington, D.C., under ARPA Order No. 183-62.

5. REFERENCES

CHILTON, C. J., STEELE, F. K., and NORTON, R. B. (1963) VLF phase observations of solar flare-produced ionization in the D region of the ionosphere, / . Geophys. Res. (to be published).

JEAN, A. G., and CRARY, J. H. (1962) VLF phase observations on the ionospheric effects of the solar flare of Sept. 28, 1961, / . Geophys. Res. 67, 4903.

JOHLER, J. R. (1962) Radio wave reflections at a continuously stratified plasma with collisions proportional to energy and arbitrary magnetic induction, Proc. London Conference on the Ionosphere (distributed by Chapman and Hall, London, 1963).

PIERCE, E. T. (1963) Stanford Research Institute, private communications. WAIT, J. R. (1963) Influence of the lower atmosphere on propagation of VLF waves to

great distances, / . Res. NBS 67D (Radio Prop.), No. 4, 375-381. WAIT, J. R., and WALTERS, L. C. (1963a) Reflection of VLF radio waves from an in-

homogeneous ionosphere. Part I. Exponentially varying isotropie model, / . Res. NBS 67D (Radio Prop.), No. 3, 361-367.

WAIT, J. R., and WALTERS, L. C. (1963b) Reflection of VLF radio waves from an inhomogeneous ionosphere. Part II. Perturbed exponential model, / . Res. NBS 67D (Radio Prop.), No. 5, 509-523.

FF

Some Remarks on Mode and Ray Theories of VLF Radio Propagation]

JAMES R. WAIT

Abstract. Some of the assumptions used in treatment of the mode theory of VLF radio propagation are discussed briefly. The connections with geometrical-optical theories are also pointed out.

In several recent papers the author has discussed waveguide theory of radio propagation for a smooth spherical earth and a concentric anisotropie ionosphere [Wait, 1960, 1963a, 1963b]. To facilitate the solution of this intrinsically difficult problem, certain assumptions were made. In view of great interest in this problem, it appears to be worthwhile to discuss briefly the nature of these assumptions since they have been alluded to in a series of three papers [Berry, 1963; Johler and Berry, 1963; Johler, 1963].

The source is taken to be a vertical electric dipole located at Θ = 0 and r = b of a spherical coordinate system. The earth's surface is located at r = a and the reference reflecting layer is at r = c. The medium between a and c is regarded as free space.

Within certain limitations, which have been discussed extensively [Wait, 1962], the boundary conditions at r = a and r = c may be described in terms of surface impedances. In the general case, the surface impedances are functions of the (complex) angle of incidence. However, for treating VLF propagation in the earth-ionosphere waveguide, it is justified to regard the elements of the surface impedance matrices as constants. As a result, lateral-type waves are neglected. The latter correspond to waveguide modes propagating within the earth and the ionosphere. In the limiting case of a flat earth model, they become part of the continuous spectrum, and they are usually described as the branch-cut waves. A detailed investigation of these lateral waves for a planar model has been carried out by Anderson [1962]. As expected [Wait, 1960], they do not contribute significantly to the total field.

Another consequence of using impedance boundary conditions is that waves "reflected" at the center of the earth are neglected. In essence, this is

Î Radio Sci. J. Res. NBS 68D, No. 1, Jan. 1964.

441

accomplished by Bremmer [1949, 1958] when he replaces the spherical Bessel functions jn{kgr) by hn

{1) (kgr), where kg is the propagation constant of the earth. In view of the large magnitude of the imaginary part of kgr, it turns out that this is an excellent approximation.

Thus, in one fell swoop, the lateral waves and the waves reflected from the center of the earth are dispensed with by the introduction of impedance boundary conditions. Furthermore, in the application of the Watson transform, the desired even property of the integrand is assured [Wait, 1960, 1962].

In much of the theoretical development of this subject, the spherical wave functions hj^ (x) and hv^ (JC), of general argument x, are ultimately approximated in terms of Airy functions (or Hankel functions of one third). Thus, if \v — JC| < < Jt2/3, it is permissible to write

xhvW (x) £ - i(xl2)V*W2(t)9 and

*V 2 ) (x) = + /(*/2)1/6wi(0, where

/ = (v + i - X) (2/x)1/3,

and W2(t) and w±{t) are Airy functions. For example

W2(t) = (n)*[Bi(t) + iAi{t)l

mit) = (TT)*[Ä(0 - iAi(t)]9

in terms of Airy functions Ai(t) and Bi(t) defined by Miller [1946]. These Airy function representations are highly accurate for the VLF mode

problem as has been demonstrated by Wait [1960] and Spies and Wait [1961]. Using such approximations, the modal equation takes on a reasonably tractable form, and numerical results for attenuation and phase velocity may be computed in an economical manner. Despite this, some further approximations are often warranted if simplicity and physical insight are to be gained. For this reason the Debye approximations for the spherical wave functions are found to be very useful [Wait, 1962]. In the present context, they correspond to utilizing the leading term in the asymptotic expansion of the Airy functions valid for large negative t.

The Debye approximated mode equation is accurate when \Cn{kaj2y\ and \C'n(kal2)h\ > 2, where Cn and Cn are the cosine of the (complex) angle of incidence for the earth and the ionosphere, respectively!. Because (fc«/2)4, at VLF, is the order of 20 and C'n is never smaller than about 0.15, the latter condition is never seriously violated. However, the quantity \Cn(kal2)h\ may become very small at VLF for important modes at frequencies greater than 10 kc/s.

Another approximation occasionally used corresponds to the assumption of a flat earth. Such a model has been used extensively by Budden [1962].

t In this context, we must have ReCn and ReCn' > 0

Some Remarks on Mode and Ray Theories of VLF Radio Propagation 443

The resulting simplicity permits a careful study of the nature and the properties of the modes. Actually, the flat-earth mode equation is obtained as a limiting case of the spherical-earth equations when both [ReCn(kal2y] and |C£(a/A)| > > 1. Extensive comparisons between flat-earth and spherical-earth computations have been reported previously [Wait and Spies, 1960; Wait, 1962].

In certain cases, when the convergence of the sum of waveguide modes is poor, an alternative representation is desirable. Such a representation may be found from geometrical optics wherein the total field is considered to be a sum of rays or hops. The ray or hop series has been obtained directly from the contour-integral representation by Bremmer [1949, 1958]. This result is derived by expanding the integrand, which is of the form [1 — JRgRi]-1, into a geometrical progression. Individual terms are then evaluated asymptotically to yield the classical geometrical-optical formulas. The method has been extended by Wait [1961] to permit the results to be used in the vicinity of the geometrical horizon of the various rays (i.e., hops). A key point in the development is the ordering of terms so that Rg and R$ are defined as (spherical) reflection coefficients connecting "downgoing" and "upgoing" wave functions. The latter have the forms A„(1) (kr) and hv^ (kr), respectively, in the earth-ionosphere region. If another choice of wave functions is used (e.g., jv(kr) and A,/1* (kr)), the individual integrals in the expansion should not be identified with geometrical-optical hops in the usual sense. The distinction between these methods of expansion is not trivial. In the first method, the only significant poles are those of the ground wave type, whereas, in the second method, additional poles appear which are closely related to the usual waveguide modes. Thus, if care is not exercised in the choice of wave functions, the individual integrals in the expansion will correspond to hybrid waves which have both guided-wave and ground-wave components. While the resultant field as computed should be identical, the utility of such hybrid expansions could be questioned.

For further discussion of these various points, the reader might wish to consult the referenced literature.

REFERENCES

ANDERSON, W. L. (1962) Fields of electric dipoles in sea water—the earth-air-ionosphere problem, / . Res. NBS 66D (Radio Prop.), No. 1, 63-72. [Correction in 67D, 63-65, Jan.-Feb. 1963.]

BERRY, L. A. (1963) Some remarks on the Watson transformation and mode theory, Digest of Papers, Symposium on the Ionospheric Propagation of VLF Radio Waves, pp. 39-42, Boulder, Colo., Aug. 12-14.

BREMMER, H. (1949) Terrestrial Radio Waves. Elsevier Pubi. Co., New York and Amsterdam. BREMMER, H. (1958) Propagation of electromagnetic waves, Handbuch der Physik 16, 243. BUDDEN, K. G. (1962) The Waveguide Mode Theory of Wave Propagation, Prentice-Hall,

Englewood Cliffs, N.J.


JOHLER, J. R. (1963) Concerning geometrical-optical propagation theory for long waves propagated between the ionosphere and the ground, Digest of Papers, Symposium on the Ionospheric Propagation of VLF Radio Waves, p. 51, Boulder, Colo., Aug. 12-14.

JOHLER, J. R., and BERRY, L. A. (1963) On LF/VLF/ELF terrestrial radio wave fields, Digest of Papers, Symposium on the Ionospheric Propagation of VLF Radio Waves, pp. 43-50, Boulder, Colo., Aug, 12-14.

MILLER, J. C. P. (1946) The Airy Integral, Cambridge University Press. SPIES, K. P., and WAIT, J. R. (1961) Mode calculations for VLF propagation in the earth-

ionosphere waveguide, NBS Tech. Note No. 114 (PB-161615). WAIT, J. R. (1960) Terrestrial propagation of very-low frequency radio waves, / . Res. NBS

64D, No. 2, 153-204. WAIT, J. R. (1961) A diffraction theory for LF skywave propagation, / . Geophys. Res. 66,

No. 6, 1713-24. WAIT, J. R. (1963a) Concerning solutions of the VLF mode problem for an anisotropie

curved ionosphere, / . Res. NBS 67D (Radio Prop.), No. 3, 297-302. WAIT, J. R. (1963b) The mode theory of VLF radio propagation for a spherical earth

and a concentric anisotropie ionosphere, Can. J. Phys. 41, 299-315. WAIT, J. R., and SPIES, K. P. (1960) Influence of earth curvature and the terrestrial magnetic

field on VLF propagation, / . Geophys. Res. 65, 2325-31. WAIT, J. R. (1962) refers to 1st. Ed.; see Chap. 5 and 6 of present book.

Two-Dimensional Treatment of Mode Theory of the Propagation of VLF Radio Waves]

JAMES R. WAIT

Abstract. This paper is partly of a tutorial nature. The intended purpose is to exploit the essential two-dimensional nature of wave propagation in the earth-ionosphere waveguide. It is shown that, without resorting to erudite arguments in the complex plane, the usual working formulas of VLF mode theory may be derived directly from orthogonality considerations. Furthermore, the physical insight gained by the present development immediately suggests how the formulas may be generalized to an earth-ionosphere waveguide of nonuniform width.

1. INTRODUCTION

In the theory of terrestrial radio propagation, the earth is represented by a spherical body and the atmosphere is usually idealized by concentric spherical layers. In this sense the problem is formulated as a three-dimensional one. However, in subsequent developments and in the reduction to useful formulas, approximations are usually made which display the inherent two-dimensional character of the problem. Typical of these approximations are the replacement of the spherical wave functions by Airy functions and the Legendre functions by leading terms in their asymptotic expansions.

It is the purpose of the present paper to develop the final formulas directly from a two-dimensional formulation. This should help dispel any doubts concerning the validity of the approximated three-dimensional solutions. A demonstration such as this is timely since a two-dimensional laboratory model has been used recently to study VLF propagation in a spherical waveguide. \ At the same time we shall discuss the excitation factors, height-gain functions [Spies and Wait, 1963], and mode conversion coefficients [Wait, 1962a] for the two-dimensional model. For the purposes of this paper the ionosphere is assumed to be equivalent to an isotropie medium.

t Radio Sci. J. Res. NBS 68D, No. 1, Jan. 1964. { Microwave model experiments being conducted by the author's colleague E. Bahar

445


2. PRIMARY FIELDS

In formulating the radio problem in spherical coordinates, it is customary to choose the source as a radial electric dipole or a radial magnetic dipole. By an appropriate superposition, the fields of an arbitrary source may be determined. In a two-dimensional model it is more convenient to work with line sources. For example, in place of a radial electric dipole on a spherical surface it is convenient to employ a line dipole source. To introduce the subject the fields, of such a line dipole located in free space, are derived from basic principles.

The line source consists of a uniform distribution of vertical electric dipoles. The situation is illustrated in Fig. 1 where the source current density is / amps per meter, for a strip of width d/, along the x axis. Because the source current has only a z component, the resulting Hertz vector also has only a z component, Uz. Clearly, the Hertz vector of the distribution is obtained by integrating the contributions from the distribution of vertical electric dipoles; thus [Wait, 1959]

llz = f+0° Idi Q~ikr

4777 €οω r dx' 0)

to y, *)

FIG. 1. Line dipole source and coordinate system employed for the discussion of primary fields.

where k = 27r/wavelength, r=[(x- xy + y2 + ζψ2,

and e0 = 8.854 x IO"12.

The integration with respect to x' may be readily carried out, to give [Wait, 1959]

Two-Dimensional Treatment of Mode Theory 447

Idi Πζ = ^η— Ko(ikp), (2)

where p = (y2 + z2)1'2 and Ko is a modified Bessel function of order zero. The resulting magnetic field, which has only an x component, is obtained from

Η χ - μο* dy ' ( 3 )

where μο — 4π x IO-7. In the far zone, where kp >71 , this reduces to

IaHikW H^- 2 0 ^ e " * e s i n e · (4)

where sin a = yjp. It is of interest to observe that the radiation pattern characterized by sin a, is identical to the pattern of an isolated vertical dipole. However, in the case of a line dipole source, the radiation field is a cylindrical wave whose amplitude varies inversely as the square root of distance. This is to be contrasted with the radiation field of a point electric dipole which is a spherical wave with an inverse distance variation of the amplitude. An obvious point of similarity between the line dipole and the point dipole is the common phase factor exp (—ikp), occurring in the radiation fields.

The preceding rather elementary derivation is based on a prior knowledge of the expression for the Hertz vector for a point dipole. It is instructive to obtain the same final formula for Hx by using a transform method.

The starting point is to recognize that Hx must satisfy the wave equation

/ a 2 a2 \ (& + &i + η H* = °- (5)

Solutions are of the form ex (±/λζ) exp (±uy) where u = (λ2 — k2)1^2 and λ is a parameter. A general solution is constructed by writing

p + oo Hx= F(A)e-A**e^dA, (6)

J —00

where F(X) is an unknown function and where the contour of integration is along the whole real axis of λ. The radical is to be chosen in u such that Re u > 0. Thus, the minus sign in the exponent is to be employed when y > 0 and the positive sign is chosen when y < 0. Difficulties with branch points on the contour are avoided by allowing k to have a very small but finite negative imaginary part (i.e., —Im k > 0).

The function F(X) is obtained by noting that Hx must become numerically equal to —Iß at z = 0 and as y tends to zero from positive values. Thus

Hx]y=o+ ^ - { δ(ζ) d/, (7a)


where δ(ζ) is the unit impulse function at z — 0. It should also be noted that

Hx]y=o- ~ + | δ(ζ) dl. (7b)

Employing the spectral representation of the unit impulse function, i.e.,

1 f+0° S(z)= r - e-*A*dA, (8)

277 J-oo it follows from (6) and (7a) that

/ d /

nv = -Thus

W) = - 4 - . (9)

H x ~ ~ An

+ 00

e - ^ e - ^ d A , (10) — 00

for y > 0. This integral in (10) may be evaluated by noting that

tfol/M = 2 + 00

e ^ ' ^ e - ^ w ^ d A , (11)

which may be differentiated under the integral sign with respect to y. As a result,

iJfc/d/ Hx= - —— Kiiikp) sin a, (12)

where sin a = j;//o and A i is the modified Bessel function of order one. In the far zone, where kp > 1, it follows that

{iky 1*1 ài ~2(2πρ)

which is identical to (4) as it should be.

Hx - - ~^—^72 e-^e sin a, (13)

3. FIELDS IN THE CONCENTRIC CYLINDRICAL WAVEGUIDE

To come to grips with the general problem, a rather simple model is chosen. The waveguide region consists of two concentric cylindrical surfaces of radii a and a + h, respectively, as indicated in Fig. 2a. In terms of a cylindrical coordinate system (A*, 0, z) these bounding surfaces are defined by r = a and r = a + h where the tangential fields are assumed to satisfy impedance-type boundary conditions The field variations in the z direction are taken to be zero so that the modes in the structure may be either TM (transverse magnetic) or TE (transverse electric). Only the TM modes will be considered here since their analog in VLF radio propagation is of greatest practical interest. The analysis for the TE modes is almost identical.

For the TM modes in such a cylindrical region, the magnetic field has only an axial or z component which is denoted H (without writing the subscript z). From Maxwell's equations, it is seen that the electric field has only r and Θ components.

LINE DIFOLE ΙΐΟ,.γ) SOURCE . *>\

FIG. 2a. Concentric cylindrical model and conventional coordinate system (r, 0).

The magnetic field in some aperture plane Θ = 0O, a + h > r > a is now assumed to be known and designated by i/(0o, r), being independent of z. The boundary conditions for H(ß, r) are

ΕΘ = -ZgH atr = a, (14a)

ΕΘ = ZiH atr = a + h, (14b)

where Zi and Zg are the respective surface impedances. Within the region a + h > r > a, the magnetic field must satisfy the wave equation

/Id 3 1 a2 \

Solutions are any linear combinations of the form

H<*Kkr) where v is a complex quantity independent of the coordinates. H(l] and H{1 are Hankeo functions, of order i>, of the first and second kind, respectively.

Solutions which are periodic require that v is an integer. However, this is an unnecessary restriction since individual modes need not be periodic. Instead, the value of v will be determined from the boundary conditions. Furthermore, without loss of generality, it is specified that Re v > 0, and attention is restricted to solutions which behave as e~ive. This will correspond to waves which are attenuated in the positive Θ direction. The waves propagating in the negative Θ direction are identical in form.

If attention is restricted to large radii of curvature such that both ka> > 1 and hja«\ are satisfied, the Hankel functions may be approximated by

their Airy function representation. There is a vast amount of literature on this particular subject and here only the final form is quoted. Thus [Wait, 1962b]

Hf W = ^(fy* *i(t-y)9 (16)

where

-£Γ<- ko),

/ 2 \ ! / 3

and wi is an Airy function. Similarly, i / 2 \ 1 / 3

OBSERVER ___ / ^^<<C6 j

^ s ^ . .

FIG. 2b. Concentric cylindrical model with the "natural" coordinates (x, y).

in terms of the Airy function w%. In terms of more conventional notation

Wl(t) = πιΙ2[Βΐ(ί) - iAi{t)l (18) and

W2(t) = «i/a[A(0 + iAi(t)l (19) where the Airy functions Ai and Bi have been defined and tabulated by Miller [1946]. The functions wi and w% satisfy

d2 w{t) di2 - tw(t) = 0, (20)

which is known as Stokes' (or Airy's) equation. To facilitate the subsequent discussion, certain dimensionless parameters

are introduced which simplify the notation. These are * = (kaßfM, xo = (kafflWo, y = [2/(Jfca)F»Jfc(r - a),

yo = [2l(ka))Wkh, q = -i(kal2)^Z;lV0, qt = -i(kal2yvzeMo, and ψ> = (μοΙ^ο)1'2 = 120ir.

The parameters x and y, while describing horizontal and vertical distances, should not be confused with Cartesian coordinates. The quantities q and q% describe conveniently the electrical properties of the bounding walls of the waveguide. The situation is illustrated in Fig. 2b.

The (approximate) solutions of (15) are now written as linear combinations of

wi(* — y) Q-iXt

W2(f — y).

The discrete values of t (i.e., the eigenvalues), denoted by tn, are determine. from the boundary conditions. Thus, the resultant field has the form

H = H(x, y)= Σ *»Φ(ίη, y) e - ^ - V n e-**e<*-V, (21) n = 1 , 2 , 3 . . .

where

Φ(ί», y) = wi(tn -y) + A(tn)w2(tn - y), (22) where bn and A(tn) are undetermined coefficients. It is evident from (20) that Φ satifies

(^2-t + y)*>(t,y) = o- (23)

The boundary conditions given explicitly by (14a) and (14b) may be written in the form

d Η Φ ( ^ ) 1 +?Φ(/,0) = 0, (24) ay ]y=o

and

£φ('^)1 -qMt,yo) = 0. (25) uy J y=y0

The latter two equations determine both the eigenvalues tn and the coefficient A(tn). Using (22) and (25), it is seen that

wj(i ~yo) + qtwiit — yo) A{t) W(t - yo) + qiw2(t - yo)\ ' ( 2 6 )

where the prime indicates a derivative with respect to the argument t — yo of the Airy function. If (24) is also to be satisfied, it follows that

A(tn)B(tn) = 1 = e-œnn, ( 2 7 )

where

B(t) - - Ht) - qWl(t)\' ( 2 8 >


2 Φπι - (tm - y)®m = 0, ΦΜ = Φ ( ^ , >>)· (30)

Within the approximations used, the modal equation (27) for this cylindrical model is identical to the one developed for a concentric spherical model [Wait, 1961].

It should be mentioned at this stage that the surface r = a + h can be regarded as a reference surface where the ratio of the tangential fields is specified. In the general case, Z* or q% may be a function of the eingenvalues tn. However, for VLF it is a good approximation to regard Zi or qt as constants. This is equivalent to stating that the surface impedance does not depend on the angle of incidence. In a similar manner Zg or q can be regarded as constants.

The orthogonality properties of the modes are now studied. We consider two sets of values, tn and tm, which satisfy the boundary equations (14a) and (14b). However, for any value of y these must also satisfy

d2

j~2 ®n - (tn - γ)Φη = 0, Φη = φ(ίΛ, y), (29)

and d2

dy'

After multiplying the first of these equations by Om and the second by Φη, they are subtracted from one another. Both sides of the resulting equations are then integrated with respect to y over the range 0 to yo. This results in

d d ~]y0 fv° Φη dy ° m ~~ Φγη dy Φ Ί = ^tn ~ tm>> ΦηΦγη dy' ^

In view of the boundary conditions on Φ^ and Om at y = 0 and y = yo, the left-hand side of the preceding equation is zero. Thus, the integral on the right also vanishes if tn is not equal to tm. Therefore, we have the important result

Î yO

Φ(ί*ι, yMtn, y) dy = 0 if m Φ n. (32)

It now follows that, if both sides of (21) are multiplied by Φ(ί™, yo) and integrated from 0 to yo,

Îyo

H(xo, y)®(tn, y) dy

»»=755 (33) J Mtn,yWdy

The normalizing integral

Î2/0 Wtn,y)?dy, (34)

is now expressed in a more convenient form. This is accomplished by noting that

( W « , y)f dy = -(tn - y)mn, y)f + [Φ'(/», j ) F (35)

This can be proved by differentiating both sides with respect to y and making use of (29). Then using the definition of Φ(ίη, y) in terms of Airy functions, it follows that

Φ(ί«, 0) = - -T7- 2l , (36) ^2lW — qwzytn)

and

Φ(ίη' y0) = - w'ifn - yo) + W » ^ o ) ' ( 3 7 )

where use is also made of the Wronskian condition

wi(t)w'2(t) - w[(t)w2(î) = - 2 / , (38)

which is valid for any value oft. Finally, on making use of (35), (36), and (37) along with (14a) and (14b), it is found that

N = __ 4('"-<?2) , j ^ - y o - g ? ) f 3 9 i K ( / . ) - qW2(tn)f ^ [W(tn2 - Jo) + qtW&n ~ Jo)]2 K '

This can be regarded as a fairly important result. Equation (33) for the coefficient bn can be written in the convenient form

H(xo,yM*n,y)dy

where A ^ ° L 2Λ (* 2Λ / ^ f a ) - qw2(tn) VI'1 .(41)

We are now in the position to obtain the fields resulting from a line dipole source at height z above the earth's surface. Thus, the aperture field is specified by

H(r, 0) = δ(ζ - z)(//2) d/, (42)

where z = r — a and where / and d/ have the same meaning as used previously. In terms of the natural coordinates, the assumed field in the aperture plane is given by

H(X090) = k^j Sty-y)-dl, (43)

A / 2 W 3 A where ^ = 1 ^ - 1 kz. Using this result and (40), it follows that

^ = I»W^W\to) k' (44)

The resultant field for a dipole line source, at 0o = 0, may now be written compactly in the form

H(x9y) = HM(x,y)V, (45) where

ν=-±Γ *~ίπ/* 2 ^ix\AnGn(y)Gn(y\ (46) yo „=-- 1,2,3...

Idltik)* IdlQin^ / 2 \ *

""<*· » - 2(ö)i · ** - W (έ) * ·"". («7)

and Φ(*η, 0) '

G»O0 = ^77-7^ · (49) Φ(*η, 0)

The quantity //<°>, defined above, is numerically equal to the broad-side field of the line dipole source in free space at a distance equal to ad (e.g., compare with (4) for α = π/2). The quantity Gn(y), described as a height-gain function, is the ratio of the field of mode n at height y to the field of the same mode at y = 0.

The coefficient An, which is defined by (41), is called the "excitation factor". The physical significance of this factor may be evident from the identity

An = YJfWnJyW^y ' (50)

which may be verified by comparing the right-hand sides of (33) and (40). As may be seen, height-gain functions Gn(y), which increase appreciably with height, correspond to small excitation factors.

The expression for V given above is identical in form to the corresponding result derived for a spherical model under the assumption that hja <ξ 1 and ka > 1. In both cases, V may be described as the ratio of the actual field to a "primary field". Furthermore, the height-gain function Gn(y) and the excitation factor An also have identical definitions in the cylindrical and spherical models.

4. EXTENSION TO WAVEGUIDES OF NONUNIFORM WIDTH

The development of the theory in this paper lends itself quite readily to certain generalizations. A case of some interest is when the height of the upper boundary varies in the direction of propagation. In the two-dimensional model considered here, the (normalized) height yo(x') is considered to be a function of x\ the (normalized) distance. As indicated in Fig. 3, yo(xf) varies, in a smooth manner, from y0(0) above the source to yo(x) above the observer. The (normalized) surface impedances of the walls are also regarded as a slowly varying function of x'.

On physical grounds it can be expected that a waveguide with slowly varying properties will not differ fundamentally from a waveguide of constant cross section. In other words, at a distance x\ the structure of the modes is characteristic of a uniform waveguide of constant width which is equal to

^ χ ' )

, = y ^

FIG. 3. Waveguide of nonuniform width.

yo(x') and with constant wall impedances equal to q(x') and qi(x'). To obtain the overall behavior of the field one is led to integrate over the whole range of x' from 0 to x. Thus, the appropriate form of the (complex) phase term is

—/ tn(x') dx' exp — / tn(x') dx' exp (—ika6),

where x = (ka/iy^d, x' = {kaj2)l^d\ and tn(x') is the slowly varying eigenvalue. From what we have said above, it follows that tn(x') is a solution of

A[tn(x')]B[tn(x')] = exp (-ΐ2πη), (51)

where A[tn] and B[tn] have the same definitions as (26) and (28), where now yo, q, and q% are functions of x'.

For the slowly varying waveguide, it is also necessary to employ the height-gain functions determined by the local width. Thus, Gn(y) is to be replaced by the function

n , , , <&[tn(x'), y] , _ Gn(X>y)=<!>[tn(x'),0]> ( 5 2 )

GG

which is an obvious generalization of (48). The resultant expression for the field is thus given by

H(x9y) = HM(pc9y)V, (53)

where

V = [JSJC)]* e _ Ì" / 4 2 6 X P [ " ' l o tn(x')àx] ^nGn(0,y)Gn(x,y). (54) w = l ,2 ,3 , · · ·

The "excitation factor" is now obtained from

~ = [yo(Q)yo(.x)]112 . . . , Λ η ( ÇyO(O) ÇyO{x) ~] 1/2 p S a )

2JI [G n (0 j )Pd^ [Gn{x9y)fày\

which is an obvious generalization of (50). It is then evident that

An = [ A ^ A ^ F 2 , (55b)

where Λ{% is defined by (41) if yo9 q, and q% are replaced by jo(0), #(0), and qt(0), respectively. Similarly, the definition of Λ{$ has the form of (41) if the corresponding quantities yo(pc)9 q(x), and qi(x) are employed.

It might be mentioned that if the curvature of the walls is sufficiently small and q% is essentially zero, An will become equal to unity as in the guide of constant width [Wait, 1962b]. Thus, in this limiting case, it may be seen from (54) that the fields of the individual modes vary approximately as I/VLFOCX)], apart from the exponential factors. This behavior is consistent with conservation of power since the outward power flow is proportional to ljyo(x) or inversely to the width of the waveguide.

The form of (54) is suggestive of a WKB approximation for propagation in a horizontally stratified medium with slowly varying properties. Implicit in the development is the assumption that conversion of. modes from one order to another may be ignored. Thus, the properties of the waveguides must vary only slightly in a distance equal to one wavelength.

5. WAVEGUIDE WITH LOCALIZED OBSTRUCTION

If the width of the waveguide changes suddenly, it is apparent that the results in the previous section are not valid. In this case, it is suggested that a Kirchhoff-Huygen's approach is useful [Wait, 1962c].

It is now imagined that the incident field results from an equivalent line magnetic source at x = 0 (i.e., Θ = 0). Thus,

H{x, y) = 2 am<b(tm9 y) Q~ixtm for x < x0 (56)

m

where am is a coefficient which does not depend on x or y. It is assumed that the aperture plane x = xo is obstructed in such a manner that the effective

FIG. 4. The concentric cylindrical waveguide with a localized obstruction in the aperture plane x = x0.

aperture is a slit extending from y = yi to y% (i.e., r — a = z\ to 22). The situation is illustrated in Fig. 4. Thus, within the Kirchhoif approximation,

H(xo, y) = Σ am<b(tm, y) e-*xfm for yi < y < y2 = 0 for 0 < y < yi = 0 for j2 < y < yo· (57)

In other words, we are assuming that the field within the aperture of the slit has the same value as if the slit were not present. It is known from a study of the rigorous solutions of diffraction by slits that this is an excellent approximation [Born and Wolf, 1959] provided the width of the slit is greater than about a wavelength.

The field in the region x > xo can now be expressed in the form

H(x, y) = Σ Σ Α^Φϋη, y) e ^ - * A e-**0'm (58) m n

where

ÎÎ/2

Q(tm,y)<l>(tn9y)dy "n " yo [Φ(ί»,0)ρ am' m

We see clearly that the incident mode of order m excites modes of order n where m and n are positive integers.

It is convenient to write Airn) = [PM + Q(m)]am (gQ)

where, for m φ η,

and

where

Gn{y) = ΦΟΪ^) (63)

and

Gm0;) = Φ Ο ^ · (64)

In obtaining the above forms for i^m) and g^w) use has been made of the orthogonality condition given by (32).

When m = n, we have AC,) = [/»<»> + ßWje, (65)

where

pw = i _ ?A» fî" Jo

[COOP dj (66) 0

and 2Λ Cyl

Qln) = i- - r [^ωΐ2^ (67) ■JO JÎ/O

where use has been made of (39). The integrals over the range 0 to y± in the preceding equations can be

regarded as the influence of the obstacle on the ground, whereas the integrals over j>2 to jo are related to the protuberance at the ionosphere. To evaluate these integrals it is desirable to expand the G functions as a power in y.

Since

Gn(0) = 1 (68)

[dGn(y)ldy]y=o = -q (69) and

d2 G(y) d j 2

it is not difficult to show that

(tn - y)G(y) for any y, (70)

Gn(y) = 1 - gy + - γ - — g - — y3 + . . . . (71)

Thus,

Gn(y)Gm(y) = 1 - 2qy + (tn + tm + 2q2) ^ - (1 + 2tnq + 2tmq) ~ + . . .

(72)

and the expansion for [Gn(y)]2 is obtained by simply replacing tm by tn in the preceding result.

The integrations for the P integrals are now readily carried out. They yield

qyi + (tn + tm + 2q2) p(ro)_ _ ?Δ» yo yi ^ n

where

gm, n —

(1 + 2tnq + 2tmq) ^

<b(tm, 0) _ w'2(tn) — qw2(tn) Φ(ίη, 0) "" w'2(tm)—qw2(tm)

and

pin) _ 1 2An yo

y\ — qyi + (tn + q2) - l (1 + 4tnq) yi

0{r

3 v* , » w 1 2 -

The g integrals are evaluated in a very similar manner. Thus

2Λ« (m) JO

,(yo - J2)3

+ (in + tm + 2q^^-~6:----- - (1 + 2fn# + 2/™?*)* " Γ Λ " ^ + ·

O'o - yi) - q\(yo - yzf (yo - yzY

12

and

Q{nn) = 1 2An

yo

+ (t

[Gn(yo)T (yo - y2) - qKyo - yzf

9λ (yo - J2)3 Λ (yo - J2)4

tf2) 3 (l + 4ίΛ#) — γ ^ — +

(73)

(74)

(75)

(76)

(77)

Due to the typically large values of qi at VLF, the preceding series for the Q functions are probably not useful. It would be better to work directly with equations (62) and (67).

6. DISCUSSION OF FORMULAS

Some of the previous results are now discussed briefly. For purposes of illustration, it is assumed that the ionosphere is a sharply bounded medium whose effective conductivity is e0cur where eo = 8.854 x 10~12. Under this condition extensive numerical values of the coefficients tn satisfying equation (27) are available [Spies and Wait, 1961]. Using these values, the various quantities entering into the formulas for the modal coefficients can be evaluated in a straightforward manner.

It is seen that the modal coefficients, given by equations (73), (75), (76), and (77) all contain the factor An- This factor is a modal excitation factor and it is a measure of the efficiency of excitation of a given mode from a line or dipole source [Wait, 1961, 1962b]. In the present context it is normalized so

that it approaches unity for perfect ground conductivity (q = 0) and a flat earth (a = oo). In general it is a complex quantity. To illustrate its behavior wr is set equal to 2 x 105 and h is taken as 70 km. Furthermore, the ground is assumed to be perfectly conducting. Under these conditions An for « = 1 has the following complex values for the frequencies indicated :

Ai = 0.95 |1ίΤ (10 kc/s), 0.79 [ItT (15 kc/s),

0.59 [4.6° (20 kc/s), 0.37 |7.4° (25 kc/s),

0.19 [10.6° (30 kc/s). (78)

This mode corresponds to the mode of least attenuation. It is characterized by an excitation factor which decreases approximately as the inverse of the frequency. Under the same conditions An, for n greater than 1, is roughly unity over the same frequency range [Wait, 1962b].

The modal coefficient P^ defined by (75), in the case of n = 1, can be written

P[l) = 1 - 2Ai j . (79)

Here, h\\h is the ratio of the heights of the obstacle on the ground to the height of the ionosphere. This quantity would never be greater than about 0.05 and thus the modification of the first mode by even an extremely high mountain range would be small. This is particularly the case at the upper end of the VLF band where the excitation factor is small.

The relative conversion of the field from an incident mode of order 1 to a mode of order 2 is obtained from the factor P^w) defined by (73) for m = 1 and n = 2. Approximately, this can be written

P<D = - 2A2gi, 2(Ai/A). (80) The complex quantity g±, 2 is defined by equation (74) for m = 1, n = 2, and q = 0. For the same conditions, its magnitude for the frequencies indicated, are given as follows

|gi, 2I - 1.72 (10 kc/s), 1.47 (15 kc/s), 1.08 (20 kc/s), 0.70 (25 kc/s), and 0.41 (30 kc/s). (81)

Since |Λ2| is of the order of unity, it is thus apparent that the conversion to higher modes may be significant.

The influence of the protuberance on the upper boundary is described by equations (76) and (77). The situation is similar to that of the ground obstacle except that the factors Gn(yo) appear. Actually, these are the ratio of the field just below the ionosphere reflecting layer to the field at the ground.

Of particular interest is the possibility that, as a consequence of an ionosphere irregularity, a mode of order 1 may be excited by an incident mode of

order 2. The magnitude of this first-order mode relative to amplitude of the second-order mode is obtained from equation (76) with m = 2 and n = 1. Thus, approximately

ß<2> = -2Aig2, Ι<320Ό)<Π0Ό) i^-J—)· (82)

The numerical magnitude of the excitation factor Λι has already been discussed. As noted, it may be quite small for frequencies of the order of 25 kc/s. However, in certain cases, the height-gain function Gi(y), is somewhat greater than unity. Thus, the conversion to the lower-order mode may be quite significant. This point can also be demonstrated directly from equation (62) which, in this special case, has the form

2Λι Cy0

Q\2) = - - 1 ft, i G2(y)G1(y)dy. (83) JO Jy2

Modes of the "whispering gallery" type [Budden and Martin, 1962], also known as "earth-detached modes" [Wait, 1962b], are associated with a low excitation efficiency (i.e., Λι is small). However, the height-gain function G±(y) for a "whispering gallery mode" is an increasing function of height. Thus, the product of the integral over y% to yo and the excitation factor Λι may be of appreciable magnitude.

Further work on this subject awaits the completion of the extensive tabulations of numerical values of the coefficients tn and the height-gain functions Gn(y) for range of the parameters. Also, the complications resulting from the earth's magnetic field are incorporated following the approach introduced in a previous paper [Wait, 1963].

I thank Prof. S. Maley and Mr. K. P. Spies for their helpful remarks, and Mrs. Eileen Brackett for her assistance in preparing the manuscript.

7. REFERENCES

BORN, MAX and WOLF, EMIL (1959) Principles of Optics, Pergamon Press. BUDDEN, K. G., and MARTIN, H. G. (1962) The ionosphere as a whispering gallery, Proc.

Roy. Soc. Ser. A, 265, 554-569. MILLER, J. C. P. (1946) The Airy Integral, giving Tables of Solutions of the Differential

Equation y" — xy. Cambridge University Press. SPIES, K. P., and WAIT, J. R. (1961) Mode calculations for VLF propagation in the earth-

ionosphere waveguide, NBS Technical Note No. 114. WAIT, J. R. (1959) Electromagnetic Radiation from Cylindrical Structures, Pergamon Press. WAIT, J. R. (1961) A new approach to the mode theory of VLF propagation, / . Res. NBS

65D (Radio Prop.), No. 1, 37-46. WAIT, J. R. (1962a) An analysis of VLF mode propagation for a variable ionosphere height,

/ . Res. NBS 66D (Radio Prop.), No. 4, 453-461. WAIT, J. R. (1926) Refers to 1st edition, see main text of present volume. WAIT, J. R. (1962c) Mode conversion in the earth-ionosphere waveguide, NBS Technical

Note No. 151.


WAIT, J. R. (1963) The mode theory of VLF radio propagation for a spherical earth and a concentric anisotropie ionosphere, Can. J. Phys. 41, 299-315 and 819.

WAIT, J. R., and SPIES, K. P. (1963) Height-gain for VLF radio waves, / . Res. NBS 67D (Radio Prop.), No. 2, 183-187.

Reflection of Electromagnetic Waves from a Lossy Magnetoplasma t

JAMES R. WAIT AND LILLIE C. WALTERS

Abstract. A method is outlined for calculating the reflection coefficient from a horizontally stratified ionized medium. The profiles of electron density and the collision frequencies are both taken to be exponential functions. The d-c magnetic field is taken to be horizontal and transverse to the direction of propagation. The specific results described are applicable to the oblique reflection of VLF radio waves in the D layer of the ionosphere for propagation along the magnetic equator. It is confirmed that the reflection coefficient is nonreciprocal in both amplitude and phase. For a wide range of the parameters the magnitude of the reflection coefficient is greater for west-to-east propagation than for east-to-west propagation.

1. INTRODUCTION

The lower ionosphere is primarily responsible for the propagation of VLF radio waves to great distances. In theoretical treatments of this problem it is often assumed that the lower edge of the ionosphere may be represented by a sharply bounded and homogeneous ionized medium. Actually, such a model was used by G. N. Watson [1919] over 40 years ago. Applications and refinements of such a model have been discussed frequently in the recent literature [e.g., Budden, 1962; Wait, 1962; Johler, 1963]. One such refinement is the inclusion of the earth's magnetic field in the analysis. This, of course, renders the medium anisotropie. If the vertical inhomogeneity (or horizontal stratification) of the ionosphere is also considered simultaneously, the situation becomes very complicated indeed. An extensive study of analytical methods to treat such problems has been carried out by K. G. Budden and his colleagues at Cambridge University. Much of this work is summarized in a monumental text [Budden, 1961] which will be the standard work on the subject for some time. Budden makes extensive use of "full wave" methods which may be described as a frontal assault on the differential equations satisfied by the field components in the medium.

In this paper we shall consider a special case of a horizontally stratified and anisotropie ionosphere. Specifically, the earth's magnetic field is assumed

Î Radio Sci. J. Res. NBS 68D, No. 1, Jan. 1964.

463


to be purely transverse to the direction of propagation. Strictly speaking, this is applicable only to the situation when the path of propagation is along the magnetic equator. However, the characteristics in this special case prevail at other latitudes if the transverse component of the field is appreciable. At least this is borne out by a numerical study of the sharply bounded ionosphere for an arbitrary magnetic dip angle [Johler, 1961]. In any case, the resulting simplicity of the differential equations for the limiting case of a purely transverse magnetic field encourages one to consider this situation in more detail. In particular, it is desirable to investigate the influence of gradient of both the electron density and collision frequency. In much of the previous work on this subject the collision frequency has been assumed constant.

In a study of the recent literature [e.g., Belrose, 1963] it is found that both the electron density N(z) and the collision frequency v(z) vary approximately in an exponential manner with height z. For example, in the undisturbed daytime ionosphere we may assume that

N(z) = No exp (bz), (1)

and

v(z) = VQ exp (~az), (2)

where a and b are positive constants and z is some specified level in the ionosphere. From a study of the experimental data [Belrose, 1963], it appears that, if the reference level is 70 km above the earth's surface, No ~ 102

electrons/cm3 and vo ~ 107 sec-1. The gradient parameters are then expected to be given approximately by b £ 0.15 km - 1 (±0.1) and a ^ 0 . 1 5 km - 1

(±0.02). The quoted values of these constants must be considered tentative and certainly subject to change. Furthermore, it must be understood that significant departures from the exponential shape are to be expected under disturbed conditions.

2. FORMULATION

The situation is shown explicitly in Fig. la. A vertically polarized plane wave is incident at angle Θ on to a horizontally stratified ionosphere. The z axis is taken to be positive in the upward direction. At the reference level z = 0, the electron density and the collision frequency have values designated by No and vo, respectively. The "scale height" which is equal to l/b or I/O, as indicated in Fig. lb is of the order of 6 km for both of these profiles. As mentioned above, these are typical of the daytime D layer for both the N and v profiles.

The lower ionosphere, which is idealized here as a stratified ionized medium, may be regarded as an electron plasma. The (angular) electron plasma frequency ωο is thus given by

Reflection of Electromagnetic Waves from a Lossy Magnetoplasma 465

z = 0

FIG. la. Illustrating oblique reflection of VLF radio waves from an inhomo-geneous (horizontally stratified) ionosphere with a transverse d-c magnetic field.

cog = 3.18 x 109 x N, (3)

where N is the electron density in electrons per cubic centimeter and ωο has dimensions of radians per second. The dielectric properties of such a (cold) plasma may be described in terms of a tensor dielectric constant (e). Thus, the displacement vector D and the electric field E are connected by

D = (e)E. (4) Choosing the d-c magnetic field to be along the axial direction,f the tensor has the form

w = -iq 0

0 (5) Uq e

[0 0

where harmonic time dependence according to exp (+/W) is assumed. The coefficients e', e", and q are given by [Wait, 1962]

e' i(y + ίω)ω%Ιω *0 ω| + (y + /ω)2 '

q CÜTOJQICO

ο

ω | + (ν + Ι'ω)2 '

! _ H (y + ίω)ω

(6)

(7)

(8)

where €o = 8.85 χ 10~12 is the dielectric constant of free space and ωτ is the (angular) gyro frequency.

Maxwell's equations, which are applicable to this situation, are

t In what follows, the x axis is taken to be in the axial direction. Thus eq. (4) is given by (Dy\ [Ey\


i(e)œE = curl H, (9)

-ίμωΗ= curl E, (10)

where μ is the magnetic permeability of the plasma. The first equation above may be written

iœE = (e-i) curl H, (11)

N(z) or GL>O(Z)

N(z) = N0 exp(bz)

z/(z) = vQ exp(-az)

FIG. lb. Sketch of the profiles of electron density N(z) and the collision frequency υ(ζ).

where

^(e- 1 ) =

M -IK 0

iK M 0

0 eoiV J

with

and

M= γ-, e € 0

K

(O2 - ?2 '

-qeo

(12)

(13)

(14) ( O 2 - ? 2 *

For convenience in what follows, we now choose the gyro axis to be in the

x direction (i.e., the d-c magnetic field is taken to be parallel to the x direction). As a result djdx = 0 since the incident wave is specified to be transverse to the d-c magnetic field. It is now a straightfoward matter to show that

/ a 2 a2 k2 \ b n + TTä + —z)Hx = 0, (15) \dy2 dz2

MMJ

+ ^ A W = 0, (16) €0/

)wEz — iK

— ΐμοωΗν

— ίμοωΗζ =

dHx

dz

= M

= -M

-M

ΒΕχ dz'

dEx

By'

dHx

dy'

W + e^2 + ûj where k = (βομο)έω = 2π/λ and M = μο/μ. Here λ is the free space wavelength and μο is the magnetic permeability of free space.

From Maxwell's equations, the other field components may be obtained from Hx and Ex. For example,

dHx dHx U0œEy=M^f + i K ^ , (17)

(18)

(19)

(20)

It is immediately evident that the general problem splits into two parts. The total field may be regarded as the superposition of two partial fields; one is characterized by E = (Ex, 0, 0) and the other by H = (Hx, 0, 0). These may be called TE (transverse electric) and TM (transverse magnetic) waves, respectively. For a vertically polarized incident wave, the first partial field (i.e., the TE waves) is not excited, so further attention is restricted to the TM waves.f

3. PRELIMINARY PROBLEM

As a simple preliminary problem we shall consider the ionosphere to be sharply bounded and homogeneous. Thus, for z > 0, the dielectric properties are to be characterized by the tensor (e) which does not vary with z. The region z < 0 is free space.

The incident wave is of the form

H™ = ho Q~ikCz e-**^, (21)

where ho is a constant and where C = (1 — S2)1^2 = cos Θ in terms of the angle of incidence Θ. Then the reflected wave must be of the form

t Actually, the TE waves in this case are not influenced by the d-c magnetic field and the medium is effectively isotropie.

Hx =h0e+ikCze-ikSvJl, (22) where R, by definition, is the reflection coefficient. For the region z > 0,

Hx=f(z)exp(-ikSy), (23) where, by virtue of (15),/(z) satisfies

[£+*&-")]*>-·· <24) Therefore,

Hx = Ao^expi- ik\—τ - sA zlexp (-ikSy), (25)

where Γ, by definition, is a transmission coefficient. The unknown functions R and T are found by applying the boundary

conditions which require the continuity of Ey and Hx at z = 0. This readily leads to

and

where

2C T = C + Δ· i 2 6 )

C - Δ Ä = = C + A· ( 2 7 )

r i i i / 2

Δ = M — r - S 2 + iKS. (28) LMM J

A UQ

If M = — = 1, which is the usual case,

Γ 1 + iL 11/2

Δ = (1 + /L)2 - y2 (29)

where (v -f- ΐω)ω , ω;τω

L = r-— and y = —- . wo o

The latter result was given by Barber and Crombie [1959] who employed a somewhat more involved derivation.

4. THE GENERAL PROBLEM

We shall now return to our originally stated problem. The continuous profiles of N(z) and v(z) are replaced by a very large, but finite, number

of steps. In other words, the inhomogeneous medium is replaced by a stack of thin homogeneous layers. For purposes of discussion, there shall be P such layers while a typical layer is the/?th layer. Thus, p ranges from 1 to P through integral values. Somewhere at a sufficiently negative value of z, the medium may be regarded as free space. This level is denoted z = —ZQ.

Within the /?th layer the fields may be regarded as a superposition of upgoing and downgoing waves. These are characterized by the functions exp (—ißpkz) and exp (+ißpkz), respectively, where

Γ 1 11/2

*-tiÄ"s ,j · <30)

The characteristic impedances associated with these wave types are

*t = - | ί (31) and

£* = %, (32)

where the + sign denotes upgoing and the — sign denotes downgoing waves. From (17) and (18), it is seen that

K+ = ηο(Μρβρ + iKpS), (33)

and

Kp - Vo(Mpßp - iKpS), (34)

where

ηο = (μο/€ο)1/2 = 120π ohms. The problem may now be solved by an application of nonuniform trans

mission line theory [Schelkunoff, 1943; Wait, 1962]. Thus, the reflection coefficient which is referred to the lower edge of the bottom slab is given by

K+ - Zi C - Δ *° = J Ç f + Z Î = C + Δ' ( 3 5 )

where Δ = Z1/770 and where Zi is the input impedance at the bottom of layer number 1. Now, Z\ may be expressed in terms of Z2 which, in turn, may be expressed in terms of Z3. The process is continued until the topmost layer is reached where Zv = K+ is assumed known. The details of this derivation are given elsewhere [Wait, 1962].

The required number of layers is best determined by studying the stability of the solution as the number is increased. Because of the relatively long wavelength involved and because of the finite losses in the medium, the solution converges nicely as the number of layers is increased. For the cases


discussed here, the step size was never greater than 0.1 km and this appeared to be smaller than necessary.

5. PRESENTATION OF RESULTS

The final results of the numerical calculations are presented in such a fashion that the phase of the reflection coefficient R is referred to the level 2 = 0. Thus, by definition,

R = [E0 exp (/2fcCzoV>oo· (36) Physically, this means that the observer at z = —zo is sufficiently far below the ionosphere that the medium may be regarded as free space. In practice zo is chosen to be large enough that the phase of R does not vary with further changes in ZQ. For the case here, zo was of the order of 40 km. This particular normalization of the phase has been used on previous occasions [Wait and Walters, 1963].

Following the usage in previous papers [Wait and Walters, 1963], the quantity ωτ = ω^(0)/ν(0) is specified. In particular, œ/œr = \ at 15 kc/s, or ωΓ = 6π x IO4 sec-1. The "effective" conductivity σβ of the medium at this level z = 0, is then given by ae = €Qœr~ 1.7 X 10~6 mhos/meter. With exponential-type profiles the fixing of the parameter of ωτ is not an essential restriction. It is a simple matter to shift the reference level from z = 0 to any other value if desired.

The parameters of the problem are thus λ, C, b, a, and ωτ/νο- In order to display the relative influence of these quantities, it is desirable to plot the amplitude and phase of R as a function of ωτ/νο from —3 to + 3 for a range of values of λ, C, b, and a. It should be noted that λ is in km, C is dimension-less, while b and a have dimensions of km- 1. Consequently, the scale length in the present problem is the kilometer. By changing this scale the results may also have significance at higher frequencies.

In Figs. 2a and 2b the magnitude of the reflection coefficient [R] and the phase of R are plotted as a function οίωτ/νο. Negative values of the abscissa correspond to propagation from west to east along the magnetic equator. The cosine angle of incidence is fixed at 0.1. Thus, the angle is highly oblique, being only 5.7° from grazing. For long-distance propagation of VLF radio waves, such highly oblique conditions prevail.f For the curves in Figs. 2a and 2b, the collision profile is chosen so that the collision parameter a = 0.15 km - 1 and the wavelength λ = 15 km correspond to a frequency of 20 kc/s. For these curves, the electron density parameter b takes the values 0.1, 0.2, and 0.5 km-1. It is evident that for ωΤ\ν§ ~ 0, the steep gradient of electron density is associated with maximum amplitudes of reflection. However, when

t In fact, the attenuation of the dominant mode in the earth-ionosphere waveguide at VLF is approximately proportional to 1 — \R\ for highly oblique incidence [Wait, 1962],

ωτ/νο is finite, this may no longer be the case. In fact, the asymmetry of the curves about ωτ/νο = 0 is a measure of nonreciprocity in the reflection process. As indicated, the reflection coefficient for propagation from west to east is greater than for propagation from east to west. There is also some nonreciprocity in the phase curves but it is not great.

In Figs. 3a and 3b, a set of curves show the influence of varying the collision frequency parameter while keeping the electron density parameter fixed at b = 0.15 km"1. For these curves, as before, C = 0.1 and λ = 15 km. It is evident that the steeper gradient of the collision frequency corresponds to larger reflection coefficients.

0.9

0.7 H

IRI

0.5

0.3

-100*

-120«

^

-

-

1 1

O.I

1 1

1 i 1 λ = 15 km

• a = 0.15 km"1

C = o.l

1 1 1

-

-

, -

0

τΛο

UJ to < -140«

-160*

-180«

0

)·

)°

y

1«

-

—

1 i λ = 15 km a = 0.15 km"1

C= O.l

1

1 ι

fr = n. ι km"' _

0.2.

n ^

1 1

~i 1

-

1

HH

ωτΛο

FIGS. 2a and 2b. Reflection coefficient curves illustrating the dependence on electron density profile.

In Figs. 4a and 4b, \R\ and the phase of R are shown as a function of ωτ/νο for various frequencies in the range from 6 to 60 kc/s. For these curves, C = 0.1, b = 0.15, and a = 0.15. The tendency is for the reflection coefficient to be diminished at the higher frequencies. In this case, the medium is acting like a good absorber rather than a reflector.

IRI

u.o

QJ

a6

0.5

0.4

1 1

a = 0.2 km"1

0.15 " ^ "

O.I

- 0.05

1 1

1 1 1 λ = 15 km

b = 0.15 km"'

^ ^ ^ . ^ ^ C = 0.l

1 1 1

-

-\

-2 τΛο

LJJ

< -120° x

-140°h

-160°

1 1 λ = 15 km

_ b - 0.15 km"1

C = o. i ^ ^ ^ ~

Γ | I

1 1 1

a = 0.05 km"1

n i

"'* -—

1 1 1

—

-3 0 ωτΛο

FIGS. 3a and 3b. Reflection coefficient curves illustrating the dependence on collision frequency profile.

In Figs. 5a and 5b, \R\ and the phase of R are plotted for different values of the angle of incidence. For these curves, λ = 15 km, b = 0.15 km-1, and a = 0.15 km-1. In general, it may be seen that the reflection coefficient is diminished for the steeper angle of incidence. It is rather interesting to note

that the asymmetry (or nonreciprocity) in the phase curves is more pronounced at the steeper angles.

Finally, in Figs. 6a and 6b, \R\ and the phase of M are plotted for various values of the collision frequency parameter a for λ = 15 km and C = 0.1. These curves differ from Figs. 3a and 3b in that here the parameter β = b + a is fixed rather than just a. In other words, the profile of Njv as a function of z is fixed while the gradient of v is changed. In the isotropie case, where ωτ = 0, it is interesting to note that R is determined only by the gradient of Njv. However, for a finite gyrofrequency, the situation is changed

IRI 0.6

0.4 -

0.2

1 1 1

20km (i5kc/s) "

10km (30kc/s)

5km (60kc/s)

i 1 a = o.iskm"1

b = o.iskm"1

^ ^ ^ ^ C = o.i ~~

~ ~ ~ ~ ~ ~ ^ " ^ 1 1 1 ~~\ " Γ

0

-40«

-80«

-120«

- I 6 0 ° h -

-200«

1 ' ' X=5km (eokc/s)_ .

T jokm (3Qkc/sj .

2Qkm(i^kc/s)

• p ^ i<m (6kc/s)

J 1 1

1 1

a = 0.15km'1

b = 0.15 km"' C = O.I

1 1

-

-

—

0

^τΛο

FIGS. 4a and 4b. Reflection coefficient curves illustrating the dependence on wavelength.

significantly. In general, the nonreciprocity is accentuated when a is diminished. For example, if v were assumed to be a constant, the dependence of the gyrofrequency is much greater than for a collision frequency which varies


with height. In much of the earlier work [e.g., Budden, 1955] on full wave solutions in ionospheric radio waves, it is often assumed that v can be regarded as a constant. Clearly, such an assumption may lead to very misleading results.

0.8

0.7

a6

0.5

IRI"

0.4

0.3

0.2

v~3 -2 - Γ 0 1 2 3 ωτΛο

0

- 4 0 e

kl

I -80·

-120°

H60°

- 3 - 2 - 1 0 I 2 3 ωτΛο

FIGS. 5a and 5b. Reflection coefficient curves illustrating the dependence on angle of incidence.

6. DISCUSSION AND CONCLUDING REMARKS

The numerical results given here should provide some insight into the nature of reflection from an inhomogeneous ionized medium. The nature of

λ = 15 km b = 0.15 km"1

C = o.i (~84°ι α = 0.15 km"1


the dependence on N, v, and ωτ is quite complicated. Nevertheless, it appears that the sharper gradients of electron density are usually associated with higher reflection coefficients. The dependence on the collision frequency profile is not so clear-cut.

In nearly every case it may be seen that the presence of the transverse magnetic field is to cause the reflection coefficient |jR| to be nonreciprocal. Furthermore, for a wide range of the parameters, \R\ is greater for west-to-east propagation than for east-to-west propagation. This is in accord with experimental data of Round et al. [1925] who observed that, for propagation over distances of the order of 6000 km, signals from VLF transmitters to the west are received more strongly than from those to the east. This observation has also been confirmed by Crombie [1958] in a series of field strength measurements in New Zealand and by Taylor [1960], who analyzed the

ωτΛο

FIGS. 6a and 6b. Reflection coefficient curves illustrating the dependence on collision frequency profile when the Nv profile is fixed.


waveforms of atmospherics. For some inexplicable reason, Budden [1955] deduces from a full wave solution that the directional dependence is just opposite to this. Although his model is not the same as the one considered here, it is difficult for this writer to accept the validity of his results in this regard. However, it is possible that, because of the complexity of the various phenomena, a reversed trend may emerge for certain special conditions, particularly for the nighttime ionosphere [Rhoads et al, 1963]. It is also worth mentioning that Budden [1955] has some qualms concerning the accuracy of his numerical data at small values of C.

7. REFERENCES

BARBER, N. F., and CROMBIE, D. D. (1959) VLF reflections from the ionosphere in the presence of a transverse magnetic field, / . Atmos. and Terr est. Phys. 16, 37.

BELROSE, J. S. (1963) Present knowledge of the lowest ionosphere, a chapter in Radio Wave Propagation (ed. by W. T. Blackband), Pergamon Press, Oxford.

BUDDEN, K. G. (1955) The solution of the differential equations governing the reflexion of long radio waves from the ionosphere II, Phil. Trans. Roy. Soc. (London), Series A, 248, 45-72.

BUDDEN, K. G. (1961) Radio Waves in the Ionosphere, Cambridge University Press. BUDDEN, K. G. (1962) The Waveguide Mode Theory of Wave Propagation, Prentice-Hall,

Englewood Cliffs, N.J. CROMBIE, D. D. (1958) Differences between east-west and west-east propagation of VLF

signals over long distances, / . Atmos. and Terrest. Phys. 12, 110-17. JOHLER, J. R. (1961) Magneto-ionic propagation phenomena in low- and very-low-radio-

frequency waves reflected by the ionosphere, / . Res. NBS 65D (Radio Prop.), 53-61. JOHLER, J. R. (1963) Radio wave reflections at a continuously stratified plasma with colli

sions proportional to energy and arbitrary magnetic induction, Proc. International Conference on the Ionosphere, 436-445, Chapman & Hall Ltd., London.

RHOADS, F. J., GARNER, W. E., and ROGERSON, J. E. (1963) Some experimental evidence of direction effects on VLF propagation (private communication).

ROUND, H. J. T., ECKERSLEY, T. L., TREMELLEN, K., and LUNNON, F. C. (1925) Report on measurements made on signal strength at great distances during 1922 and 1923 by an expedition sent to Australia, J.I.E.E. 63, 933-1011.

SCHELKUNOFF, S. A. (1943) Electromagnetic Waves, Van Nostrand, New York. TAYLOR, W. L. (1960) VLF attenuation for east-west and west-east daytime propagation

using atmospherics, / . Geophys. Res. 65, No. 7, 1933-1938. WAIT, J. R. (1962) refers to 1st. Ed., see main text of present book. WAIT, J. R., and WALTERS, L. C. (1963) Reflection of VLF radio waves from an inhomo-

geneous ionosphere, Parts I, II, III, / . Res. NBS 67D (Radio Prop.), Nos. 3, 5, and 6. WATSON, G. N. (1919) The transmission of electric waves round the earth, Proc. Roy. Soc.

95, 546.


GALEJS, J. (1961) ELF waves in the presence of exponential profiles, IRE Trans. Ant. Prop. AP-9, No. 6, 554-562.

A Note on VHF Reflection from a Tropospheric Layer]

JAMES R. WAIT

Abstract. Some remarks concerning reflection from tropospheric layers are made with special reference to a recent paper by Bean, Frank, and Lane on the subject. It is indicated that the finite vertical extent of the layer must be considered in the analysis.

In a recent interesting paper, Bean, Frank, and Lane [1963] discuss the cumulative probability distribution of field strength for VHF paths in terms of various refractive index profiles. It is claimed that the best agreement is obtained using a layer model with a linear «-profile. As indicated in their paper, a typical value of layer thickness is 100 m with a horizontal dimension of the order of tens of kilometers. The particular layer profile used by Bean et al.9 has a linear decrease of n over a height interval h. Using a method attributed to Brekhovskikh [1960], they state that the reflection coefficient has a magnitude given by

\p\ = Δ«· λ/(8ττ A sin3 a),

where An is the total change of n, λ is the wavelength, and a is the grazing angle of incidence. However, on close inspection of the matter, it turns out that this result is applicable to an infinite layer which has a gradient numerically equal to Anjh. This fact is confirmed by examining the structure of the exact solution of the problem in terms of Bessel functions of order one-third [Wait, 1958].

The correct form of the solution, for the present application, is

\p\ = An - (sin A i

A x ) \ 12 sin2 <

where χ = (2TT/A)A sin a. It may be readily verified that

T . , , An An Lim \p\ = ^ . 0 ~ -— = on, χ_+ο m 2 sin2 a - 2a2 Ηυ'

t Radio Sci. J. Res. NBS 68D, No. 7, July 1964. 477


which is the value corresponding to a discontinuity of amount An. Limitations of this type formula have been discussed extensively by the author [Wait, 1962].

To illustrate the significance of the corrected formula, |p| in dB is plotted as a function of a in milHradians in Figs. 1 and 2. The layer thickness h is 100 m and the wavelengths chosen are 4.18 m and 1.67 m, corresponding to

H z

Ld O

O

υ

V

-20

-40

-60

- 8 0

- inn

1 1

—

i 1

1 1 «

S N j / 3 !

I 1 i

1 '

^L. * L

1 ,

' l ' I h = lOOm λ = 4.18m

Δη = 10" 5

£1

I 1 I I

1

—

—

-

1 15 20 25 30

a IN MILLIRADIANS 35 40

FIG. 1. Reflection coefficients at λ = 4.18 m for finite and infinite linear profiles and for a discontinuity of refractive index.

? - 2 0

^ - 4 0

o o

-60 H

Û! - 8 0 r

-100

T

" \ *

L

1 i 1

VsN^1

1 ; 1 1

r I

J L^

' i

V^,

.. .1

1 1 ' h= 100m λ= 1.67m Δη = 10"5

1 1 ,1

1 I

\P\

J _ J .

—

—

A

15 20 25 30

a IN MILLIRADIANS

35 40

FIG. 2. Reflection coefficients at λ = 1.67 m for finite and infinite linear profiles and for a discontinuity of refractive index.

A Note on VHF Reflection from a Tropospheric Layer 479

the examples in the paper by Bean et al. [1963]. For the examples shown in Figs. 1 and 2, An is taken to be 10~5, which is also considered typical. Along with the curves for \p\, the reflection coefficient po for a sharp discontinuity, and the curves for />oo, corresponding to Bean's, Frank's, and Lane's infinite layer are also shown.

It is evident from the graphical data given here that the angular dependence for the finite linear layer is quite different from the infinite linear layer with the same slope. Of course, both of these reflection coefficients are somewhat below the corresponding value for a discontinuous layer with a change of refractive index of amount An. It is suggested that the oscillatory dependence of | p\ on the parameter χ is compatible with the spread of experimental points in Fig. 5 of the paper by Bean et al. [1963]. It would be worthwhile to make a more systematic study of the radiosonde data with a view to correlating observed fields with the vertical extent of the layer structure.

REPLY BY J. A. LANE

"The authors are indebted to Dr. Wait for drawing their attention to the limitations of a formula which has been widely used for calculating reflection coefficients of layers with linear gradients of refractive index. We support the suggestion that studies of field strength variation and layer thickness would be valuable."

REFERENCES

BEAN, B. R., FRANK, V. R., and LANE, J. A. (1963) A radio meteorological study, part II. An analysis of VHF field strength variations and refractive index profiles, / . Res. NBS 67D (Radio Prop.), No. 6, 597-604.

BREKHOVSKIKH, L. M. (1960) Waves in Layered Media, Academic Press, New York and London.

WAIT, J. R. (1958) Transmission and reflection of electromagnetic waves in the presence of stratified media, / . Res. NBS 61, No. 3, 205-232.

WAIT, J. R. (1962) refers to 1st. Ed., see main text of present volume.

Concerning the Mechanism of Reflection of Electromagnetic Waves from an Inhomogeneous Lossy Plasma]

JAMES R. WAIT

Abstract. Reflection of electromagnetic waves from a stratified lossy inhomogeneous plasma is discussed. The profile for the conductivity is idealized as an exponential curve with a superimposed Gaussian-shaped perturbation. By examining the change in the calculated reflection coefficient for various locations of the perturbation, some insight into the reflection process is gained. In particular, it is shown that reflection does not take place at a single level but, instead, a wide range of levels is important. The parameters of the problem are chosen to be representative of the D layer of the ionosphere and the wavelengths cover the range 10 km to 30 km.

1. INTRODUCTION

It is well known that the reflection coefficient of a stratified medium is determined explicitly by the profile of the complex dielectric constant. For example, if the profiles of electron density and collision frequency in the lower ionosphere are specified, it is possible to calculate the reflection coefficient for an obliquely incident electromagnetic wave. Although there is now a vast literature devoted to this general subject, some questions concerning the nature of the reflection process remain unanswered. For example, the determination of the effective reflection level in the ionosphere for VLF radio waves is not a clear-cut task. In fact, even the concept that waves are reflected at some specific level is open to question.

It is the purpose of the present paper to investigate which levels in the lower ionosphere are most influential in determining the reflection coefficient. A number of simplifications are made in order that extraneous considerations are removed. For example, in the lowest ionosphere in the daytime it is assumed that the following inequalities hold:

(a) Electron collision frequency ve > electron gyrofrequency Ωβ. (b) Ion collision frequency i>< > angular wave frequency ω. (c) Density of neutral particles No > both density of electrons Ne and

density of ions Ni. Î Radio Sci. J. Res. NBS 69D, No. 6, June 1965.

481

Under these conditions, it is then a straightforward matter [Wait, 1962] to show that the ionized medium may be characterized by a scalar complex dielectric constant e(z) which is a function of height z of the form

Φ) _ Ί _ * «o L(z)9

where

L(z) eoœ \meve mmY

In the above, €Q is the dielectric constant of free space, e is the charge of the electron and the ion, and me and rm are the masses of the electrons and the ions, respectively. The only difference between this formulation and a previous one [Wait and

Walters, 1963], is that here the ions are considered explicitly. The form of the dielectric constant e(z) is just the same. Thus, the numerical method for calculating the reflection coefficient is identical and need not be described here.

2. THE IDEALIZED PROFILE

To illustrate the salient features, it is assumed that the conductivity parameter l/L(z) of the D layer may be written in the following analytical form:

z<b = h [exp m+Ao exp [- {^rfil The resulting exponential profile obtained by setting Ao = 0 may be described as the "undisturbed profile". Typically, for daytime conditions, β would be about 0.3 km - 1 when z is measured in km above some reference level at z = 0. The second term in square brackets is then regarded as a perturbation which is centered at z = F, while its vertical extent is determined by the parameter D. In an abnormally ionized D region, such as discussed by Crain and Booker [1964], the ions can contribute significantly to the perturbation. The function 1/L(z), as defined above is plotted in Fig. 1 as a function of height z where Lo is set equal to 1/2. The smooth curve in this figure corresponds to the undisturbed exponential profile. The effect of locating the perturbation at various heights is illustrated in Fig. 1 by choosing a sequence of F values from —8 km to + 4 km while D is chosen to be 0.5 km and Ao is taken to be 0.5.

It is stressed that the perturbation as described above is highly idealized. It is not intended that it should represent the D layer under actual disturbed conditions.

Reflection of Electromagnetic Waves from an Inhomogeneous Lossy Plasma 483

VERTICAL DISTANCE" z ( 'km)

FIG. 1. The idealized profile of the inhomogeneous lossy plasma. The ordinate is proportional to the conductivity of the medium as a function of height above and below the reference level at z = 0. The perturbation is shown at a

number of levels from z = F = —8 km t o + 4 km.

3. DESCRIPTION OF NUMERICAL DATA

It is now contended that considerable insight into the reflection process is obtained by examining the changes in the reflection coefficient as the perturbation is moved along the profile as indicated in Fig. 1. Results of this kind are shown in Figs. 2a and 2b for the amplitude and phase of the reflection R for a vertically polarized incident wave. The cosine C of the angle of incidence is denoted on the curves. Also, as indicated, the wavelength λ is taken to be 15 km corresponding to a frequency of 20 kc/s. The results for the undisturbed exponential profile are shown in Figs. 2a and 2b by dashed lines. Following previous conventions [Wait and Walters, 1963] the phase is referred to the level z = 0.

It is evident from the curves in Fig. 2a that the perturbation only has an influence on \R\ if it is located below the level z = F = 0. For i^in the interval from about —1 to —6 the reflection coefficient is increased somewhat. Thus, the perturbation at these levels actually enhances the reflection process. Of course, for increasingly larger negative values of F the opposite tendency becomes evident. Then the perturbation acts as an absorber, tending to diminish the reflection coefficient.

It is suggested that the "levels of reflection" in the unperturbed profile are characterized by the interval in F where \R\ for the perturbed profile exceeds


F(km)

FIGS. 2a and 2b. The (vertically polarized) reflection coefficient as a function of F which is the vertical level of the perturbation.


\R\ for the unperturbed profile by some detectable increment. For example, when C = 0.2, the "reflection levels" extend from about F= —2 to —10 km. In this range, \R\ for the perturbed profile exceeds \R\ for the undisturbed profile by at least 5 per cent. It is interesting to compare this deduction with a statement of Crain and Booker [1964] that the reflection takes place in the height range near the level given by

€oœ\meve mm]

For the undisturbed exponential profile, this is equivalent to solving the equation

2 exp (ßF) = 2C2.

Thus, taking ß = 0.3 and C = 0.2 as in the above example, it is readily found that

F= - 11km.

Crain and Booker [1964] do not explain how they arrive at their criterion for the reflection level but, presumably, it is related to the phase integral approach of Eckersley [1932]. The latter is straightforward for lossless media at high frequency where such geometrical-optical notions are meaningful. For lower frequencies in relatively lossy media, the phase integral method loses its validity. This would appear to be the reason for the apparent discrepancy between the Crain and Booker "reflection level" at F = —11 km and the results in Fig. 2a which would indicate that reflection takes place at a range of levels from about —2 km to —10 km. The difference between the phase integral approach and the full wave solution becomes even greater for angles nearer grazing incidence. In general, the reflection level predicted by the Crain and Booker criterion is at the bottom or somewhat below the range of levels in the lower ionosphere where VLF radio waves are reflected. However, this view does not modify the conclusions regarding the importance of ions which was the principal topic in the paper by Crain and Booker [1964]. Furthermore, their "reflection level" is probably quite representative at the height region where the bulk of the energy is actually returned.

4. CONNECTION WITH MODE THEORY

An important parameter occurring in the theory of waveguide propagation of VLF radio waves is the complex quantity a [e.g., Wait and Spies, 1964]. In terms of the reflection coefficient R, it is defined by

R = — exp (aC),

where C is the cosine of the angle of incidence. Writing a = a± + /a2, the

real quantities ai and a2 are shown in Figs. 3a and 3b for C = 0.1 and 0.2, where we have used the perturbed exponential model described above. The magnitude of ai is approximately proportional to the attenuation of the dominant mode in the earth-ionosphere waveguide. Thus, for F greater than zero, the attenuation is not influenced by the perturbation. However, for a range of F from about —1 to —10 or so, there is a detectable change in the attenuation rate.

The variation of the reflection parameters ai and a2 as a function of the angle of incidence is shown in Figs. 4a and 4b for a range of F values. It is important to observe that the functional dependence on C is markedly changed when the perturbation is lowered from F = 0 to F = —10 km. The consequence is that the relative attenuation of the first and second order waveguide modes is appreciably modified by a low-lying perturbation of a smooth undisturbed profile. The general tendency is to increase the relative attenuation of the second order mode as the perturbation is lowered. The phase parameter α2 in Fig. 4b indicates that the perturbation generally increases the phase velocity of the dominant waveguide modes. For example,

4.5

4.4

a" 4.3

4.2

'-I0 -8 - 6 - 4 - 2 0 2 4

F ( k m )

9.0

8.6

8.2

7.8

T.4

'-I0 -8 - 6 - 4 - 2 0 2 4

F(km)

FIGS. 3a and 3b. The reflection parameters ai and <i2 which are defined by R = —exp (aC) where a = ai -f /<X2.

1 1

! 1

1 1 1 1 1

~s/s

! ! .1 ! 1

1 1 \ \ 1

X"=l5km A0=0.5 D =0.5 km β = 0.3 km"1

_ L _ L . i 1 I

"Ί

-

-

I

\ 1

- \

-

!

1 1 1 1

1 1 1 1

1 1 1 1 | 1

X = l5km A0=0.5 D = 0.5km /3=0.3 km"1

W - i i 1 i

1 1

-

-

-

4—f—


COSINE OF ANGLE OF INCIDENCE , C '-'

0.05 0.10 0.15 0.20 0.25 0.30 0.35

COSINE OF ANGLE OF INCIDENCE C

FIGS. 4a and 4b. The reflection parameters as a function of the cosine of the angle of incidence.

at a frequency of 20 kc/s, the phase velocity (relative to c) of mode n = 1 changes from 0.9986 to 0.9989 as the location of the perturbation moves from F = 0 to —10 km. In this case, it is assumed that the C = 0.15 and the reference level (z = F = 0) in the ionosphere is located at a height h = 70 km above a perfectly conducting earth. Other quantitative estimates of this kind may be obtained by using the waveguide parameter curves issued in a recent NBS publication [Wait and Spies, 1964]. //


5. WAVELENGTH DEPENDENCE

The results discussed above all pertain to a wavelength of 15 km. To illustrate the wavelength dependence, the amplitude and phase of the reflection coefficient are plotted as a function of C in Figs. 5a and 5b, respectively, for wavelengths extending from λ = 10 km to 30 km. The function Lo is defined by Lo = 30/λ where λ is the wavelength in km. It is interesting to observe that the curve in Fig. 5a for λ = 10 km has reflection levels extending from

0.78

0.74

0.70

0.66 K

0.62 h

0.58

0.54

0.50

I I 1 1 1 1 1 1 1 1 1 1 1 1 \—

L-

-

-

DISTURBED PRO

~ ~ ^ ~ ~ ^ - - ^ _ _ _

-UNDISTURBED PROFILE

C = O.I D = 0.5 km A0=0.5 0=0.3 km"1

FILE

λ = 30 km

25

20

15

X=l'Okm

— — —

—

-j

- j

-\

1 1 1 1 1 1 1 1 1 1 1 1 1 1 -10 - 4 - 2 0 2 4

F (km)

-10 -6 - 4 - 2 0 2 4

F(km)

FIGS. 5a and 5b. The wavelength dependence of the reflection coefficient.


F = - 2 km to well below F = - 1 0 km. The Crain-Booker [1964] reflection level would be found by solving

3 exp (ßF) = 2C2 - 0.02, which yields F = —16.5 km which is not far below the physically important height range where the reflection actually takes place.

6 . CONCLUSIONS

The graphical results given in this paper would indicate that reflection does not take place at a discrete level in an inhomogeneous lossy medium with a smooth profile of its dielectric constant. In fact, the wave is reflected at a range of levels in the medium. The upper limit is determined where the incident wave has been sufficiently attenuated by a result of conduction losses. This is in contrast to the mechanism which produces reflection in a lossless inhomogeneous medium. In that case, there is a fairly well determined reflection level which is located where the vertical propagation constant vanishes.

It is important that the reflection process be understood if experimental data on VLF fields are to be interpreted in relation to the properties of the lower ionosphere. All too often, variations of the amplitude and the phase of the YLF field are ascribed to changes in the parameters of some simplified model which bear little resemblance to the actual physical changes which have taken place.

The author thanks Mrs. L. C. Walters who programmed the digital computer for the calculations discussed in this paper. The work was supported by the Advanced Research Projects Agency under ARPA Order No. 183-62.

7. REFERENCES

CRAIN, C. M., and BOOKER, H. G. (1964) The effects of ions on low frequency propagation in an abnormally ionized atmosphere, / . Geophys. Res. 69, No. 21, 4713-16.

ECKERSLEY, T. L. (1932) Radio transmission problems treated by phase integral methods, Proc. Roy. Soc. A136, 499-527.

WAIT, J. R., and SPIES, K. P. (1964) Characteristics of the earth-ionosphere waveguide for VLF radio waves, NBS Tech. Note No. 300.

WAIT, J. R., and WALTERS, L. C. (1963) Reflection of VLF radio waves from an inhomogeneous ionosphere, / . Res. NBS 67D (Radio Prop.), Part I. Exponentially varying isotropie model, No. 3, 361-367 (May-June); Part II. Perturbed exponential model, No. 5, 519-523 (Sept.-Oct.) ; Part III. Exponential model with hyperbolic transition, No. 6, 747-752 (Nov.-Dec).

WAIT, J. R. (1962) 1st. Ed.


JOHLER, J. R., and BERRY, L. A. (1965) On the effect of heavy ions on LF propagation (NBS Technical Note).

BOOKER, H. G., FEJER, J. A., and LEE, K. F. A theorem concerning reflection from a plane stratified medium, Radio Sci. 3, No. 3, 207-212, Mar. 1968.

Influence of an Inhomogeneous Ground on the Propagation of VLF Radio Waves in the Earth-ionosphere

Waveguide^

JAMES R. WAIT

Abstract. Propagation of radio waves in the earth-ionosphere waveguide is considered for the case where the lower boundary is an inhomogeneous smooth surface. An integral equation for the problem is formulated in a direct fashion by utilizing the compensation theorem. After some simplifications, several special cases are considered explicitly. For example, in the case of a two-section path consisting of a long stretch of sea and a short section of land, a relatively simple working formula is obtained. The result shows that the modal excitation factors at VLF for an all sea path are significantly reduced when the foreground is poorly conducting. Another special case considered is when the propagation path is all sea except for a short intermediate land section. In this case, it is found that energy from low-order modes will be transferred to high-order modes with a subsequent reduction of field strength.

1. INTRODUCTION

A great deal of attention has been given to the problem of predicting groundwave fields for mixed land/sea paths. Furthermore, extensive calculations have been made which show the interdependence of the various parameters. Since a recent review of progress in this area is now available [Wait, 1964], it is not necessary to discuss this particular topic here. However, it is rather surprising that, in propagation via ionospheric reflections, little attention has been given to the influence of an inhomogeneous earth. It is the purpose of this paper to consider this problem with special reference to VLF radio propagation.

2. FORMULATION

The mutual impedance zm between two vertical antennas at A and B (separated by a great circle distance d) located on a spherical earth of radius a is considered. In order to account for the presence of the ionosphere, an equivalent reflecting layer is located at height h. For present purposes, we assume this layer is characterized by a surface impedance Zi which does not

t Radio Sci. J. Res. NBS 69D, No. 7, July 1965.

491


vary along the path. If over the earth's surface between the terminals A and B the surface impedance is Z everywhere, the mutual impedance may be expressed as a sum of modes as follows :

d\a }[ώ 1/2

W(Z, d), (1) [sin (d/a) where zo is the mutual impedance between the dipoles A and B if they were located on a perfectly conducting flat ground plane and separated by a distance d. In the above expression, W(Z, d) is an attenuation function defined by [Wait, 1962]:

W(Z, d) = ( -^-2 e-«"/* 2 exP (ikd^)A^ (2) n

where Cn is the cosine of a complex angle, and An is the excitation factor for modes of order n. The CVs are solutions of a modal equation which involves the following dimensionless parameters:

(-/»)i/2 = Cn(kal2)V*

yo = khQca/l)-1^

q=-i(Zho)(kal2)W

qi=-i(ZtlVo)(kal2)V*9

where k = 2π/wavelength, and 770 = 120π ohms. After making a number of simplifying assumptions, the modal equation mentioned above may be written in the form [Wait, 1962]

w&n) - ffw2fa)1 r^ifa - yo) + gtmCtn - jo)1 = Q_i2rrn ^ w'i(fn) — qwi(tn)\ \w'2(tn — Jo) + qiW^ifn — Jo)J

where wi(t) and w%(t) are Airy functions while the primes indicate derivatives with respect to the arguments. An alternative form of (2) is

W{q, χ) = y ^ 2 e - ' ^ y e x p (-ΐχΐη)Λη (4) n

where

x = {dld){kaßyi\ Numerical values of tn and the excitation factor Λη are now available for a

wide range of parameters [Wait and Spies, 1964]. Thus, for the purposes of this communication, the solution of the waveguide problem with constant wall impedances Z% and Z is taken to be known. What is of interest here is the extension to the same waveguide when the surface impedance of the lower boundary is a function of position along the path connecting A and B. For

Propagation of VLF Radio Waves in the Earth-ionosphere Waveguide 493

example, as indicated in Fig. 1, it is now assumed that, over some area S on the earth's surface, the surface impedance is Z' which may be different from Z. The method of approach is very similar to that used by the author [Wait, 1964] in studying mixed-path groundwave propagation. As in that case, the problem may be formulated in terms of the compensation theorem in the form derived by Monteath [1951]. For example, the mutual impedance zm between dipoles A and B over an inhomogeneous ground of variable surface impedance Z' is given by

4 = zm + ^Jj(Z' - Z)HarUMUS (5) s

where zm is the mutual impedance if the surface of the earth were homogeneous with surface impedance Z everywhere. The tangential magnetic field of dipole A over the homogeneous earth is Hat while the tangential magnetic field over the inhomogeneous earth is H^. The currents in the dipoles are both taken equal to Io for convenience. The surface of integration S extends over the region of the earth which is characterized by a surface where Z' differs from Z as indicated in Fig. 1.

The formal equivalence of (5) with the formula of mixed-path groundwave theory is a result of the impedance boundary condition on the upper wall of the waveguide. In other words, the surface impedance Z« is assumed to be the same for both the homogeneous and for the inhomogeneous earth cases. If this were not permissible, there would be an additional surface integral over the upper boundary of the waveguide.

FIG. 1. Plan view of the inhomogeneous region on the earth's surface.

To simplify the present problem, the mutual impedance zm is defined in terms of an attenuation W'(Z, Z', d) such that

Γ dia 11/2


in analogy to (1) for the homogeneous earth. The next step is to express the tangential magnetic field vectors at the point of integration P in terms of attenuation functions. Thus

2ns I 1 \ Γ sia 1^2

[l + äciimm W,40. xU) (7) and

_ikhfo ikl. M" ~ 2»/ ' / l\[ Ila I 1 ' 2

(8)

where s and / are great circle distances from A and B to P and is and ij are unit vectors in the directions of increasing s and /, respectively. In the above, ha and hb are the effective heights of the dipoles A and B while \n is a unit vector normal to and into the surface of the spherical earth.

Equation (5), when combined with (6), (7), and (8), leads to the equation ikd ff Q-ik(s+i-d)

w(z, z \ d) = w(z, d) + - — 11 G(s9 /) —7 l— s

x (Ζ' - Z)W(Z, s)W'(Z, Z\ Ϊ) cos 8 dS, (9) where δ is the angle subtended by is and ij and where

«'Λ =ky Ι Α Π 1 + έΐί1 + m)' (10) Γ rf/α I1 / 2 / 1 1 \

τ = [sin(diâ)\ \ ì + 7kd~ Ίαϊη' (11)

This is a two dimensional integral equation for the unknown function W'(Z, Z', d). To solve such an equation directly appears to be hopeless. Therefore, some simplifications are made at this stage in order to achieve tractability.

3. APPROXIMATE FORM OF THE INTEGRAL EQUATION

We note that the function exp [—ik(s + I — d)] is rapidly varying compared with other factors in the integrand. Therefore, one may expect that the principal contribution to the integrand will occur when s + / ~ d, provided that the surface impedance contrast Z' — Z does not change rapidly in a direction transverse to the path. Therefore, in the other factors in the integrand, / may be replaced by a, and s may be replaced by d — a, where a is the great circle distance from B to the point Q on the great circle between A and B. (The arc QP is perpendicular to AB.) Furthermore, over most of the range of integration, (l/ks) and (l/kl) may be neglected compared with unity and similarly cos δ may be replaced by — 1. The latter approximations


are obviously violated when the terminals are near discontinuities of the surface impedance Z'.

The reduced form of the integral equation is

W'(Z, Z', d) - W(Z, d) = - lkd ikd ÇÇ Q-ik(s+i-d)

X W(Z, d - a)W'(Z, Z', a)dS, (12) where

„ . fsin {dm V* [sin (ala)] ~W [sin [(d - α)/α]Ί -V*

where all quantities, except the exponential factor, vary with a only. The exponent s + / — d is now expressed in terms of the angle coordinates Ω and ß with reference to Fig. 2; these are defined by

Ω = αΙα = QB/α and β = QP/α.

From spherical trigonometry cos (BOP) = cos Ω cos β

and cos (AOP) = cos (θ — Ω) cos j8.

Thus, / ["cot ΩΊ

BOP = - = ß + - y - ψ + terms in ß\ ßQ, (14)

^ = Λ

- = β - Λ

FIG. 2. Coordinates for describing the integration over the spherical surface of the earth.

and r c o t (β — Ω)~\

A O P = - = θ-Ω + \ ^ }- \β2 + terms in ß\fP,.... (IS) *α=θ-Ω + [^θ-Ω)

Therefore, to a first order in β2,

s+l-d^ ^ [cotß + cot (θ - Ωψ. (16)

For the present development it is further assumed that Z' — Z does not vary in the transverse direction (i.e., with j8) over the surface S. Thus, Z' is regarded only as a function of a.

Within the region of validity of the present approximations, the element of area dS may be approximated by α2άΩάβ. The integral now has the form

W'(Z, Z', d) = W(Z, d)

ikda Γ2 Z'(a) — Z - 2 ^ o Ja , W ^ M ^ Z , Z>, a)W(Z, d - a)

X exp [-/(far/2) [cot Ω + cot (θ - Ω)]βψβάα, (17) J/?l<a>

where ßi(a) = yi(a)la, j82(a) = j2(a)/a. In this case, the limits of the surface S are yi(a) < y < y^a) and ai < a < a2 as indicated in Fig. 1. After a change of the β variable, (17) may be written in the form

(ikaV2 fa2 Z'(a) - Z W'(Z, Z', d)=W(Z,d)-[—\ d\ - ^ /(a)

\Ζπ J Jai 770 W(Z,d-a)W'(Z,Z',a)

F(lii,U2)da (18) a(d — a) cot(â)+cot(V)]

where

and

ÇU2 F(uu u2) = (//2)1/2

u exp (-inu2/2)du, (19)

K! = (^/π)1/2[οοί ί2 + COt (Θ - ß ) ] 1 ^ , (2θ)

W2 - (kafry/*[cot Ω + cot (0 - fl)]1^. (21)

As indicated before, the rapidly varying function exp [—ik(s + I — d)] in the integrand of (9), determines the portions of the earth's surface which are significant. The phenomenon may be described in terms of Fresnel zones. These are determined by the locus of the points where

k(s+l-d) = rmrß (for m = 1, 2, 3 . . .)

or, to within a good approximation.

(kaß) [cot {ala) + cot ((d - α)\ά)ψ = rmr\2.

The width of the first Fresnel zone at any point a is then obtained from

fm = 2βα = (2maA)i/2[cot (ala) + cot ((d - a)la)]-^.

The maximum width, denoted fm, occurs where a = d\2. Explicitly,

fm = (mMI2)WXW,

where

tan [dl(2a)] [dj{2d)f X ~ d/Cta) ~~ 1 + 6 + e

Under most conditions Zmay be replaced by unity even when dis comparable with the earth's radius a. Within this approximation, the Fresnel zones are ellipses and the semi-minor axes are fm\2 while the semi-major axes are (d/2) + (mX/S). Tn the classical sense, the "first Fresnel zone" corresponds to m = 2. It is immediately evident that F(m, U2) being replaced by unity in (18), is equivalent to saying that the width of the surface S extends to several Fresnel zones on both sides of the propagation path.

It is convenient to rewrite (18) in the following form:

fa2 Z'(a) - Z Ui

(22)

(23)

likd\V* [ W'(ZZ',d)= W(Z,d)- [—) j

ηο F(ui, u2)

x W(Z,d-a)WXZ,Z'9a)

[a(d - a)]l/2

where we have made use of the trigonometric identity

sin (d/a) V/2 l - α \ 1 1 / 2 H3sin(V)j Hî)+cot(V)]

= 1.

(24)

(25)

When F(ui, U2) is set equal to one, (24) bears a formal equivalence to the one dimensional integral equation developed for groundwave propagation over mixed paths [e.g., Wait, 1964]. It is interesting to note that (24), in the form given, is not restricted to distance d such that d\a < < 1. This comes about because of the normalization factor \Xd\a)\sm (d/a)]1^ which is not included in the definition of the attenuation function W(Z, d). In the case of ground-wave propagation, this distinction is not of any consequence since d\a is small in any case. However, in YLF propagation in the earth-ionospheric waveguide, the ratio d\a may be comparable with unity and the normalization factor mentioned above may exceed unity by a significant amount.

Equation (24) is in a reasonably tractable form for direct numerical calculation of the attenuation function W'(Z, Z', d) when the limits of the

surface S and the surface impedance function Z ' are specified. For present purposes, some simple limiting cases will be considered rather than attempting a frontal assault on (24).

4. Two SECTION PATH

An important special case of the general mixed path problem is when Z' is sectionally homogeneous, as indicated in Fig. 3. For example, if the path between A and B is characterized by a constant surface impedance Z, from 0 to d — d\ and Z\ from d — dito d, the integral equation (24) simplifies to

x f: W(Z,d-a)W'(Z,Zi,a) da, (26) [a(rf-a)]l/2

when d\ > 0 and when the function F(m9 Uz) has been replaced by unity. Of course, if d\ < 0 such that A and B are both over the surface of the earth of surface impedance Z, it is seen from (24) that

W'(Z, Zi, d) = W{Z, d).

At least this is true to within the stationary phase approximation which, in effect, reduces the area integration to a line integration. As a result of this reasoning it is equally justified to replace W(Z, Z\, a) where it occurs in the integrand of (26) with W(Zi, a) which is the attenuation function for propagation from the point B to distance a over a homogeneous earth of surface impedance Zi.

EQUIVALENT REFLECTING LAYER

FIG. 3. Sectional view of earth-ionosphere waveguide for a two section path, such that Z\ φ Ζ.

Equation (26) with the simplification indicated in the preceding paragraph may be written in terms of the dimensionless coordinates mentioned earlier. Thus, for the two section path,

where

q = - i (Z/ ,o) (^a/2)V3, ? 1 = -,·(Ζι/,ο) (to/2)1/3, χ = (<//α) (*a/2)V», Xi = (*/σ) (Α:α/2)ΐ/3, and χ = (α/β) (*β/2)ΐ/3.

Because of (4), we may write

(X-X)1'2 = 7oUJ Zne i28) « = 1,2,3..

and

1χΎ^=7ο\ν Zm m' ( } m = l ,2,3. .

where Λη is the excitation factor corresponding to the earth of surface impedance Z while Am is the excitation factor corresponding to the earth of surface impedance Z\. The tn are roots of (3) while tm are roots of an equation identical to (3) if q is replaced by q±.

On inserting (28) and (29) into (27), the integration with respect to χ may be readily carried out to yield

W'(q,qi,x)=W(q,X) 4 Χ ^ ' Ό - eWm-trJXi 1

+ - i t^l^xf'Kqi ~9>y y AnAmt-ixtn — -±-- . (30) SO Δ—IJL—I \Jn — lm)

n m

The physical significance of this result is best seen by examining the special case where the distance d — d\ or χ — χι is sufficiently large that only the n = 1 term is needed. Thus

2 M 1 ' 2 - 4 W'fa qux)~ H -7 Λιβ-'**ι + -2 tiWHjnpiHai ~ iMie-'*«i

Γ fe«*i-*"i)^i—1) , λ e ^ i ^ i - 1 - e ^ V ^ - 1 ] X U i — 7 : fT~ + ^2 — — - + A3 - 7 ; =rc- + . . . . (31)

The first term on the right of (31) is the attenuation function for a path which is homogeneous throughout its length (with a surface impedance Z). The remaining terms, proportional to q\ — q or Z\ — Z are corrections which result from the inhomogeneity extending over the path of length d\ (proportional to χι). The terms proportional to A2, A3 . . ., etc., represent conversion of energy in the waveguide from mode 1 to mode 2, 3 . . ., etc.

A somewhat simpler approach to the two section problem is appropriate when the section of the path of length d\ is very small compared with the total length of d (i.e., χι < < χ). Then, from (27), it is seen that

W'(q, qi, x) ~ W(q, X) + g = ^ J*1 W(q, χ-χ) W{qu χ) , (32)

which is a slight simplification. When the latter result is applied to a long sea path (of length d — di) and a short land path (of length d±), further simplifications are possible. For example, W(q, χ — χ), in the integrand, is replaced by W(q, χ) and, thus,

W'(q,qhx)=W(q,X)[l+Q] (33)

where

Ω=* (/V)l/2 αχ· (34) (x)1'2

On inserting the mode series expansion given by (29) into (34), the integration with respect to χ is readily carried out. However, because of the assumed smallness of χν the resulting expansion would be very poorly convergent. An alternative approach is to recognize that W(x, qi) may be replaced by the groundwave attenuation function for χ ranging from 0 to χι.

The form appropriate for short distances is [Wait, 1964]

W(qu χ)= Σ Ame<>*«l*qm1(£)m/*9 (35)

#w=0 , l , 2 . .

where

Ao=l9Ai=- / » * , A2 = - 2, A3 = ΐ(π)*( 1 + ^-X

Using (35), the integration indicated in (34) is carried out to yield the following series form for the correction factor:

2 / / Z \ Γ (^χ)ΐ/2 2 Ω=-Μ1~^Γ [ l-~2-~3Pl

fal/ty/2/ I \ 4p2/ M "I

where

and

ρψ = (Wi/2)V2(Zl/^0)e-^/4 (37)

il = - Kkaßy/HZxho). It is seen from this series expansion that the earth curvature only influences the correction factor Ω through the higher order terms in powers of p\>2. Also if Z and Z\ correspond to sea and land respectively, the factor 1 — (Z/Zi)


may be replaced by unity. Furthermore, at VLF, displacement currents in the ground are negligible, which means that σ9ι + Ϊ€9ιω may be replaced by σ9ι in the definition of p±. Thus

p{l* cz (1Ιλ)(α!ΐσβ1)ν*(πΙ60)ν*

is a real quantity. Then, to a first order

Ω ~ Ω' + iQ"

where

Ω' ~ - P l a n d ß " = - ρ/π*/*)/^.

Thus, the fractional reduction in the amplitude of the field is p\ while the phase lag is increased by 2(/?ι/π)1/2 radians. For example, if the length d\ of the land path is 100 km, for a land conductivity σ9ι of 1 millimho/m and a frequency of 15 kc/s (i.e., λ = 20 km), it is easily found that

Ω' = -2 .62 x IO"2 and fi" = - 0.129 rad = - 7.4°.

For this example, the short land section has a negligible effect on the propagation over the total distance d.

5. PROPAGATION ACROSS A STRIP

An interesting situation occurs when the path between A and B is homogeneous with surface impedance Z everywhere except for a relatively short stretch of length d% where the surface impedance is a constant Z2 as indicated in Fig. 4. Assuming that the inhomogeneity is effectively a strip of infinite transverse dimension, F(m, 1/2) in (24) may be replaced by unity. The resulting integral equation for the attenuation function may then be written

/ v \ l / 2 f * l + X2 W(q, x ~ xW'(q, q2,x ) a ,1βΛ -α,χ (38) [(x - χ)χ]1/2

L EQUIVALENT REFLECTING LAYER

L* d2

FIG. 4. Sectional view of the earth-ionosphere waveguide for an intermediate section where Z2 Φ Z.

where q2 = - W ) 1 / 3 (Ζφ0), χι = &αβψ* dx\a and *2 - (]fce/2)i/s d2\a. The function W, as it occurs in the integrand in the preceding equation, may be identified as the attenuation function appropriate for propagation from the point B to a variable point on the strip (i.e., χι + χ2 > χ > χ{). In accordance with the previous discussion, this particular function W may be replaced by the appropriate form for a two section path which, in effect, ignores reflection at the boundary χ = χ2 (i.e., at distance d\ + d2 from B). If the series representation given by (30) is used, the subsequent integration leads to a triply infinite series for the resultant attenuation function for the path A to B. Convergence of this type of expansion is satisfactory provided that the distance parameters χ, χι, and χ2 are all somewhat greater than one.

A somewhat simpler approach to the strip problem is to regard the midsection as a perturbation to the homogeneous path. In this approximation, W\q, q2, χ) in the integrand of (38) is replaced by W(q, χ), which is the attenuation function for propagation over a homogeneous earth of surface impedance Z. Furthermore, if the strip is relatively narrow (i.e., χ2 < χλ and χ), the integrand may be replaced by its value at the midpoint of the strip. Thus,

W ~ W(q, x) + (^)1/2(<72 - q) [ ^ Z ^ p s WÜ> X ~ *<>) W(g, X0)> (39)

where Xo = Xi + (X2/2) = (kal2y;*(dola)9 d0 = dx + (</2/2).

The perturbation term, which is proportional to q2 — q, may now be regarded as a first order or single scattering from the strip.

Using the modal representation of the form given by (28), it readily follows from (39) that

W ~ W(q,x) + MV, (40) where

4ΤΓ/Υ\1/2

yl (ä)1/2 (q2 ~ q)x2 ye~ i ( r x o ) <^»ye-^o^m. (41)

^2 - q)x2[Ai + A2 Q-^2-h)xo

If x is sufficiently large, only the n = 1 term is needed and the above representation for AW may be written in the more meaningful form,

àw a w ~~~ y0

+ Az e-*<f3-£i>xo + At e-Wt-tJxo + . . . ] (42)

or, what is the same thing,

+ A3 e-*(53-Äi)fcdo + . . ], (43)

where

and cm = {-tmyi\2ikayi\

Thus, as indicated, the strip will modify the strength of the first mode by a factor proportional to A\ and, at the same time, it will produce higher order modes proportional to A2, Λ3, . . ., etc.

For negligible displacements in the earth, the multiplicative factor in (43) may be written

where aff2 is the conductivity of the earth over the strip of width dz. For σ9 ^ σ*72> ^2 == 100 km, h = 70 km, / = ω/2π = 1 5 kc/s, σ92 = 1 milli-mhos/m, it follows that

— ~ -0.040 e^/4 [A± + Λ2 β-*<52-5ι>*Λο

+ A3 e-U^-s^äo + . . . ] . (45)

For this example, the excitation factors Λ±, A2, . . . are not appreciably different from unity and they are nearly real. Thus the strip produces approximately a 0.03 fractional diminution of the amplitude of first mode and a change of phase of the order 0.03 rad. The higher modes are excited with a relative strength of about 0.04. The resultant effect of the higher modes, of course, depends on the magnitude of the electrical distance kdo and the relative phase velocities of the modes. In general, the higher modes are attenuated as a result of the increasing value of the imaginary part of Sm as m increases. As indicated above, numerical values of the excitation factors Am and the propagation factors Sm are available [Wait and Spies, 1964] for a variety of conditions appropriate in the VLF range.


It would appear that the inhomogeneity of the ground is an important factor in the propagation of VLF radio waves in the earth-ionosphere waveguide. In certain practical applications, such as to navigation systems and worldwide communications, the influence of inhomogeneous land sections on the path may alter significantly the normal behavior of the transmission. In particular, abrupt changes in conductivity from sea to land may KK


convert appreciable amounts of energy to the higher modes. Not only will this change the resultant attenuation, but the effective phase velocity will be modified as a result of modal interference.

I thank K. P. Spies and R. L. Gallawa for their helpful comments.

7. REFERENCES

MONTEATH, G. D. (1951) Application of the compensation theorem to certain radiation and propagation problems, Proc. Inst. Elee. Engrs. (London) pt. IV, 98, 23-30.

WAIT, J. R. (1964) Advances in Radio Research, ed. J. A. Saxton, 1, 157-217, Academic Press, London. (Contains many references to earlier work.)

WAIT, J. R., and SPIES, K. P. (1964) Characteristics of the earth ionosphere waveguide for VLF radio waves, NBS Tech. Note 300. (Available from Superintendent of Documents, U.S. Government Printing Office, Wash., D.C. 20402, price 50 cents.)

WAIT, J. R. (1962) 1st Ed.


KING, R. J., WAIT, J. R., and MALEY, S. W. (1965) Experimental and theoretical studies o propagation of groundwaves across mixed paths, Department of Electrical Engineering University of Colorado (presented at URSI Symposium on Electromagnetic Theory, Delft, Sept. 1965).

WAIT, J. R. On mode conversion of VLF radio waves at a land/sea boundary Trans. I.E.E.E. AP-17, No. 2, Mar. 1969 [shows, among other things, that eqn. (30) can be simplified to the equivalent form

W\g, gv χ) = 4 &ί3πμ (^)1/2tei-£)

> Λη A w e - ^ n ττ~* ]. έ-4 itn-tm)

n m

Propagation in a Model Terrestrial Waveguide of Nonuniform Height: Theory and Experiment]

E. BAHAR AND JAMES R. WAIT

Abstract. Propagation of electromagnetic waves in multimode waveguides of variable height is investigated. The model consists of two uniform rectangular waveguides connected by a linearly tapered waveguide section. Using a generalized reciprocity theorem for waveguide junctions, a previous quasi-optic solution of this problem is extended to account for reflected waves. The results have application to the theory of VLF radio propagation when the effective height of reflection of the ionosphere boundary varies along the path. The analytical investigation has been complemented by laboratory measurements taken from a two-dimensional microwave model, and good agreement with calculated results was achieved.

1. INTRODUCTION

The diurnal variation of the electrical properties of the ionosphere results in a change of the effective height of the ionosphere lower boundary along the path of propagation. These spatial variations of the effect height of reflection modify the field patterns across the waveguide. If the change is sufficiently gradual the waveguide theory developed for a uniform earth-ionosphere waveguide may be generalized in a fairly straightforward manner [e.g., see Wait, 1964]. Furthermore, the observed data [Crombie, Allen, and Newman, 1958; Wait, 1959, 1961] are consistent with this picture. However, it is now apparent that near sunset and sunrise lines, mode conversion effects may be important. It is the purpose of this paper to consider this problem.

A two-dimensional dual model waveguide described previously [Bahar and Wait, 1964] is the basic tool in the present investigation. In this scaled model, rather ideal conditions are assumed. At the ionosphere boundary (considered to be sharply bounded), the tangential magnetic field of the VLF radio wave is assumed to vanish, corresponding to a reflection coefficient Ri= — 1 . The earth's boundary is assumed to be perfectly conducting, corresponding to a reflection coefficient Rg = 1. Consideration is restricted to the lower order modes (which account for most of the energy of the VLF

t Radio Sci. J. Res. NBS 69D, No. 11, Nov. 1965. The research reported in this paper was sponsored by the Advanced Research Projects Agency on Contract C.S.T. 7348 (ARPA Order No. 183-62).

505


radio waves at large distances from the source). It has also been assumed in this work that

hiao 4 \σ\ and (kao/2)1^ ReC > 2,

where ao is the radius of the earth, h is the effective height of the sharply bounded ionosphere, k is equal to 27r/free-space wavelength, and C is the cosine of the angle of incidence at the earth's boundary. With these restrictions a flat earth approximation can be introduced. Furthermore, the effects of the earth's magnetic field have been neglected.

It has been shown [Wait, 1964; Bahar and Wait, 1964] that, with the assumptions stated above, the modal equation governing the propagation of TM modes in the actual earth-ionosphere waveguide is the same as for the TE modes in the dual model waveguide of half-height corresponding to the height of the ionosphere in wavelengths. The cosine of the angle of incidence of the pth mode on the boundaries of the waveguide, derived from the appropriate modal equation [Wait, 1960], is given by

π(2ρ — 1) Cp=-2kh~~> P=1>2>1 0 )

where k = 2π/λ is the wave number mentioned above. The original curved earth-ionosphere waveguide is represented by the

upper (or lower) half of the rectangular dual model waveguide (see Fig. 1). The upper (or lower) conducting boundary of the model waveguide represents the ionosphere boundary and the plane of symmetry (x, z) represents the earth boundary. Also TM modes propagating in the earth-ionosphere waveguide are represented in the dual model by TE modes. Of course, it is necessary that only waves with electric fields that are symmetric about the *-axis of the model waveguide are initially launched into the waveguide; thus the tangential magnetic field at the plane of symmetry (x, z) vanishes, which conforms with the vanishing tangential electric field at the earth's surface, it should also be noted then that the/?th mode in the earth-ionosphere waveguide corresponds to the (2/7 — l)th mode in the model waveguide since the even modes with asymmetric electric fields must be excluded. A more detailed discussion of the modelling technique is given elsewhere [Maley and Bahar, 1963,1964; Bahar, 1964].

The diurnal change in the effective height of the ionosphere derived from phase velocity measurements as reported by Pierce [1955] and Crombie, Allen, and Newman [1958] have been discussed by Wait [1959, 1961]. From the viewpoint of mode theory of VLF propagation, Wait obtains a value of ΔΛ (the change in the effective height) between 16 and 18 km, in good agreement with values obtained by Bain et al [1952], who analyzed the interference pattern of the groundwave and the first hop sky wave at 16 kc/s. The model

Model Terrestrial Waveguide of Nonuniform Height : Theory and Experiment 507

waveguide with a half-height ha = 12.7 cm (operated at 9 Gc/s) would thus represent the earth-ionosphere waveguide 88 km high (at about 13 kc/s) under more or less normal conditions prevailing at night. Similarly, the model waveguide with half-height ha = 10.16 cm would represent the earth-ionosphere waveguide 70 km high under the conditions prevailing during daytime. This paper deals with a special case in which the transition between the day and night propagation paths is assumed to take place over a length of several wavelengths rather than in the abrupt manner discussed in an earlier paper by the same authors [Bahar and Wait, 1964]. The transition region is assumed to be wedge-shaped, and it is bounded by uniform waveguide sections of unequal cross-sections representing the day and night propagation paths.

2h(x)

ELECTRIC CONDUCTING WALLS

ELECTRIC FIELD LINES

Z

v H t = 0 ON THE PLANE OF SYMMETRY (DUAL OF THE PERFECTLY CONDUCTING EARTHJ

ELECTRIC CONDUCTING WALLS

FIG. 1. Cross-section of the two-dimensional dual model of the idealized earth-ionosphere waveguide used in the experimental investigation.

2. FORMULATION OF THE PROBLEM

The half-height h(x) of the waveguide, as a function of the distance x along the axis, is expressed analytically by

h(x) = ha + [u(x) — u(x — L)] —-—- x + u(x L) {ha - ha) (2)

where u{x) is the unit step function defined as

(0 for x < 0 u(x) = |

[lfor*^0, L is the length of the transition path, and ha and ha are the heights of the uniform waveguide sections at the terminals of the transition region. The gradient of the height profile in the transition region is restricted in the following analysis by the condition

<£ ^ tan <£ = h-a^-a < 0.25. (3)

The total variation in height is assumed to be about a wavelength and, thus, the length of the transition region is restricted by

L > 4(A* - ha) *> 4λ. (4)

This is not a serious restriction judging from the nature of the problem under consideration.

The linear transition described here has been investigated by Wait [1962], who considered a TM wave incident on a wedge-shaped transition between two parallel-plate waveguides. These solutions are extended in this paper. Here the transition region between the uniform waveguides is analyzed as a waveguide junction.

To derive the scattering of the waveguide modes for the case of propagation in the day-to-night path, a quasi-optical method is used in which reflections are totally neglected. This approximation is very suitable whenever a low-order mode propagates through a multimode waveguide in the direction of increasing cross-section, as was illustrated rather rigorously in the investigation of mode conversion for the case of propagation of grazing modes in a multimode waveguide with an abrupt height discontinuity [Bahar and Wait, 1964]. The above solution is then used to derive the scattering when the direction of propagation is reversed (night-to-day path) by invoking the reciprocity theorem. In this manner, reflections for the night-to-day propagation path need not be neglected.

In order to apply the reciprocity condition at each discontinuity in the gradient of the height profile (x = 0, x = L in Fig. 2), it is necessary to derive a generalized interpretation of the reciprocity theorem for waveguide junctions.

At the terminal surfaces in the wedge region (ports A, B, C, and D in Fig. 2), the tangential fields are described by a finite number of unattenuated modes and an infinite number of evanescent modes. The electric "basis" fields (transverse field pattern) for the nth mode in ports A and B, for instance, are

Model Terrestrial Waveguide of Nonuniform Height: Theory and Experiment 509

-PORT C

FIG. 2. Wedge-shaped transition region between the day and night propagation paths.

denoted by Φ^ and Φ* respectively. In the problem under consideration

Φ^ - cos k*ya, \y\ < ha and Φ* == cos νηφ, \φ\ < φ (5)

where

kn~2ha (6a)

and

vn

Ππ Υφ' (6b)

It will be found very convenient to use matrix notation throughout this work and thus, before proceeding with the analysis, the definitions of the symbols for the matrices and their elements are stated.

Let ΦΑ and ΦΒ denote the "basis" field row vectors whose elements are Φ^ and Φ* respectively, and let Φ denote the total "basis" fields row vector, representing all the "basis" fields in both the ports A and B, of the junction under consideration. Hence Φ is defined as

Φ = [φΛφβ]. (7) The quantities a* and bA are defined as the nth mode complex wave

amplitude (referred to port A) of the wave travelling towards and away from the waveguide junction respectively. Thus \a*\ is the magnitude of the forward travelling nth mode and arg (aA) determines the phase of this wave relative to the other waveguide modes. The symbols aA and bA are wave-amplitude column vectors whose elements are aA and bA respectively. Similarly, with respect to terminal B, aB and bB are wave-amplitude column vectors whose elements are a% and bB

n respectively. The total wave-amplitude column vectors are defined as


a = and b =

ïbAl

\bB\

(8)

Then the electric field at port A expressed in terms of the "basis" fields and wave amplitudes is

EA(y, t) = Re {ΦΑ[αΑ + bA] exp (/'ωί)}.

Matrices YA and YB are diagonal characteristics admittance matrices whose elements YA and Υξ are the ;?th mode characteristic admittances for ports A and B respectively. The total characteristic wave admittance is defined as

Y = YA 0

.0 YB

(9)

The symbols ZA, ZB, and Z are characteristic impedance matrices. They are equal to the inverse of the matrices YA, YB, and Y respectively.

Matrices WA and WB are diagonal matrices whose elements WA and W% are the power normalization factors defined by

WA = [0Afdy = ha and W» = Γ [Φ^ράψ = Ρι>φ. (10) ha J ^

The total power normalization matrix is defined as

[ha 0 W

WA

LO

0

WR\ o ρί»Ά (Π)

where / is the unit diagonal matrix. The symbol SA is a diagonal matrix whose elements are Si; SA can be

identified as the sine of the angle of incidence of the wth mode on the narrow walls of the model waveguide in port A. It is given by

SÛ = [1 - (ttlkn12, " = 1 , 3 , 5 , (12)

where kA is given by (6a). These elements are related to the mode characteristic admittance through the equation

YA VQA 1 n ~~ / 0 n >

(13)

where Y is the free-space wave admittance. They are also related to the mode propagation constants βΑ through the equation

ßi = kS*, (14)


where k is the free-space wave number. Symbol SAA is a square reflection scattering matrix, whose element SA^ is

the complex amplitude of the «th reflected mode when the mth mode of unit amplitude is incident on the junction from port A. Similarly, SBB is the reflection scattering matrix related to the waves in port B.

The symbol SBA is a square transmission scattering matrix, whose element ΞξΑ is the complex amplitude of the nth mode transmitted through the junction to port B when the mth mode of unit amplitude is incident on the junction from port A. Similarly SAB is the transmission scattering matrix related to transmission through the junction from port B to port A.

The total scattering matrix is then defined as

S = SAA SAB

(15) ISBA SBBi

In terms of the matrices defined above, the scattering matrix equation of the waveguide junction becomes

b = Sa. (16)

Similarly, the above matrix quantities can be defined corresponding to ports C and D (Fig. 2).

LIST OF SYMBOLS

a, aA = forward travelling wave amplitude vectors ufo = radius of the earth

b, bA = backward travelling wave amplitude vector h = height of the ionosphere, or half-height of model waveguide

ha = half-height of "daytime" portion of waveguide ha = half-height of "nighttime" portion of waveguide

i = V ( - 1) k = free-space wave number

m, n, p, q, s = integer indices u = unit step function

x, y, z = Cartesian coordinates E = electric field Ê = electric field amplitude matrix H = magnetic field tì = magnetic field amplitude matrix JR = reflection coefficient S = scattering matrix

SAA = reflection scattering matrix SBA = transmission scattering matrix

SA = sine of the angle of incidence for the mth mode

W = power-normalization matrix Y = free-space wave admittance, characteristic admittance matrix Y = equivalent admittance matrix Z = characteristic impedance matrix ß = propagation constant e = dielectric constant λ = free-space wavelength μ = permeability

φ? φΛ = basis-field row vector p, φ, z' = cylindrical coordinates.

3. DAY-TO-NIGHT TRANSITION PATH

The incident wave travelling in the direction of the axis (from the narrow to the wide uniform waveguide as indicated in Fig. 2) is assumed to be the TEm, o mode for which the electric field in the narrow waveguide is given by

Ez = ai exp ( - ikxSffli>*(y) = < exp {ikxS*} cos k*y (17)

where k£ and S£ are defined by (6) and (12) respectively. For the case considered, the width of the model waveguide increases

monotonically (or remains constant) as the wave advances along the waveguide, and the cross section of the multi mode waveguide never changes abruptly. Thus, the assumption that reflections may be neglected can be applied; this renders SAA = 0, Scc = 0 (actually this assumption was also verified experimentally).

The waveguide region shown in Fig. 2 is regarded to possess junctions for the sake of analysis. For example :

Junction I is bounded by the x = 0 plane and the p = pb cylindrical surface, Junction II is bounded by the cylindrical surfaces p = pb and p == pC9 and Junction III is bounded by the cylindrical surface p = pc and the plane

x = L. Junction II, being a uniform linear wedge region, will transmit the incident

waves without mode conversion. (a) Scattering through junction I. Let x, y, z, be the coordinate system connected with port A (x = 0 plane)

and let p, φ, ζ' be the coordinate system connected with port B (p = pb cylindrical surface). The axes z and z' are parallel as indicated in Fig. 2.

The distance Δχ (along the x-axis) between the terminal surfaces x = 0 and p = pb of junction I, as a function of the aximuthal angle φ, is

Ax = pb (cos φ — cos ψ) ~ -y (φ2 — φ2) (18)

where ψ is given by (3) and cos ψ is replaced by the first two terms of its

Maclaurin series expansion. The radius pb is related to the parameters of the height profile (2) by the equation

pb = -—: ™ -y, (19) sin ψ ψ

with a similar relationship for pc. On neglecting reflections, the electric field at p = pb, by substituting (18)

into (17), is given by

Ez{p = Pb) = ai exp { - ikS*&x} Φ£ ~ < exp {- ikPbS^ß} .exp {ikptSfafiß} cos fc^. (20)

The electric field in a uniform radial waveguide is expressed in terms of the orthogonal basis fields ΦΒ by the equation

where Φ^ is given by (5), and / / ^ is the Hankel function of the second kind and order vn. The form of (21) follows immediately from an earlier analysis of the wedge region [Wait, 1962].

It now immediately follows that the continuity condition of the electric field at p = pb is given by

< exp {-ikPbS*W2} exp {ikPbS^I2yS>i = 2 Φ Χ = Σ Φ ^ χ . (22)

Now, subject to the restriction of (3) on the gradient of the height profile,

7 ^ π ) ; mTT sin ψ ηΐπφ k-y = 2ÄT = TsinJ ~ "2^ = ^ (23)

To solve for the scattering coefficients S%£, premulti ply by Φ* and integrate with respect to φ over the interval (0, ψ); thus

S5*=*exp{-ikPbSi*t,*l2}x exp {ikpbS^cp2/!} cos νηψ cos νιηψάψ o

= ^ e x p { - / Ä r p & 5 ^ / 2 } / ^ . (24)

As indicated before [Wait, 1962], the integral Ιξ£ in the above equation can be expressed in terms of a Fresnel integral F(x) defined by

F ( x ) = [ e x p j ^ j d x . (25)

To be explicit,

* - Î t è ) > F ^ [ ' { m ' + ' ^ ) ♦'{(?rc-^)}]

+ F<

where

2 ' (27)

(b) Transmission through the uniform radial waveguide junction II. In this junction no scattering takes place, and it can readily be shown

[Wait, 1962] that the transmission coefficients in the forward and backward directions between the terminal surfaces pb and pc are given, respectively, by

SSi = m»(kpc)IH%(kpb)Smn (28)

SSZ = m»(kpb)IW»(kpc)8mn. (29)

The relationship between the coefficients S%% and S*% checks with the general reciprocity theorem (appendix A) applied to uniform radial waveguides. The //i£j and Η}%> are the Hankel functions of the first and second kind (of order vm) respectively.

(c) Scattering through junction III. One should appreciate that the wave incident on junction III comprises

several modes even when only one mode is incident on junction I. However, in the analysis it is necessary to consider only one of these modes incident on junction III. On using superposition the result for several incident modes is readily obtained. The incident wave approaching junction III from the radial waveguide is thus taken to be

E _ ac ESÊA Φο m

where Φ£ = Φ^ is given by (5). The basis fields in the rectangular waveguide are

«>S = cos*2y (31)

where

The electric field in the rectangular waveguide expressed in terms of these basis fields is

Ez=lb% exp [-/*(* - L) Si] (33) n

where

S2 = [1 - (k°Jkni*. (34)

Neglecting reflections, and following the analysis applied to junction I, it readily follows that the continuity condition for the electric field at x — L is given by

ag exp {ikpcSS/Pß} exp {-ikPcS^ß} == 2 KK = I *&£<& (35) n n

in which Sg = S°. To solve for S%£, premultiply by Φ° and integrate with respect to y over

the interval [0, ha]. This gives

SZ = exp {ikpcSgPß} 2Tdexp {-ikpeS^ß} cos vmf cos kDnyay

Off

exp {ikpcSffiß} - exp {—ikpcSgcp2/!} cos vmy cos νηφάφ

= exp{ftpcSS0a/2}?(/2S)*, (36)

where, as in (23), it is assumed that

Ky *> νηψ and -7- ™-j (37) rid ψ

and (Inm)* is t n e complex conjugate of /££. The latter is defined by

JnÎ = eXP {/α£<Ρ2} COS * W COS vm<pd<p, (38)

which in turn can be expressed in terms of the Fresnel integrals as in (26). Thus

a£ = kpcSg (39)

Combining the results of the scattering and transmission in the separate sections of the wedge region and denoting the scattering coefficients of the


composite junction (with terminal planes at x = 0 and x = L)T, it can be readily shown that the transmission scattering matrix (from port A to port D) is given by

TDA = SDCSCBSBA (40)

where SBA SCB and SDC are given by (24), (28), and (36). The reflection scattering matrix for port A is assumed to be the zero matrix

TAA = 0. (41)

The electric field at x = L is, therefore, given by the matrix multiplication

Ez(x = L) = Φ DTDAaA. (42)

4. NIGHT-TODAY TAPERED TRANSITION PATH

When the cross section of the model waveguide decreases as the incident wave propagates in a nonuniform waveguide (propagation in the direction of decreasing p or x), the reflections cannot be considered negligible unless the gradient of the effective height profile is very small. As in the case when the height profile had an abrupt discontinuity [Bahar and Wait, 1964], it was observed experimentally that there were substantial reflections in the case of a wedge transition region (for the night-to-day transition path), even when the gradient (3) of the height profile was ψ ~ tan ψ = 0.25. In view of this, the method developed in the previous section to derive the scattering matrices is not effective in this case. However, since a very good approximation to the solution of scattering in the day-to-night propagation path is already available from the preceding analysis, the solution to the problem of scattering in the night-to-day path can be obtained readily from it by the use of the reciprocity theorem. The reciprocity theorem, for waveguide junctions with uniform waveguide ports of constant cross section and a linear axis of propagation, is given by the matrix equation

WYS= WYS (43)

where the matrices WY and S are defined in section 2 and the curled symbol above the matrices represents the transpose operation. Equation (43) states that the matrix product WYS is symmetric. The corresponding relationship between the transmission scattering matrices SAB and SBA is

WA YASAB = SB~k YB WBt ( 4 4)

In terms of the elements of the matrices, the reciprocity theorem is given by

WiYÛ>Si°=W*Y*S%. (45)

The derivation of the above theorem is given by Kerns [1949]. A generalization of this theorem for the case when the cross section of the terminal ports


of the waveguide is not constant is given in appendix A. The scattering matrix SAB can, therefore, be obtained directly from the previous solution for SBA

without involving any further approximations, since the reciprocity theorem is exact. Hence

SAB = [WA YA]-1SBA γΒψΒ^ ( 4 6 )

The fields transmitted through the wedge region to port A can be derived in a straightforward manner by means of the scattering matrix TAD in the following matrix equation :

Ez(x = 0) = ΦΑΤΑ Da D, (47)

where the transmission scattering matrix TAD is related to the matrix TDA, given by (40), through the reciprocity theorem as in (46),

TAD = [WA ΥΑΛ-IJDA γΏ\γΏ _ _ ZAJDA γΏ (4g\ a

where the power normalization matrices WA and WD and the impedance and admittance matrices ZA, YA, YD are defined in section 2.

In order to derive the fields within the wedge region or to obtain a first-order approximation of the reflection scattering matrix TDD, it is necessary to apply the generalized reciprocity theorem (appendix A) to each of the subsections of the wedge region (since ports B and C are not rectangular). In particular it is necessary to determine the scattering matrices SAB, SCD, SBB, and SDD. This shall be done in the remainder of this section.

Using the generalized reciprocity theorem, it can be readily shown that the transmission scattering coefficient through junction I (Fig. 2) from the terminal surface p = pb to the terminal surface x = 0 is given by

TUB γΤΒ VBA TI TI r^T) Λ *~»m*i VAB _ vv n * n OB A _ _ ° m n / 4 0 Λ

mn \λ/Α~γΤΑ mn / __u \ V >

^m Ç-ψ) \miXkPb)\2 where ΥζΑ and Y%B are the total wave admittances defined by (A. 10) and (A. 11) for the uniform rectangular waveguide A and the uniform radial waveguide B respectively. The power normalization constants defined by (10) are

Wi = ha ™W%= Ροφ (50)

since

pb ^ αΐφ.

Assuming the incident field at the terminal p = pb of junction I is given by Φ£Λ£, the transmitted field at the plane terminal x = 0 is

Ez(x = 0) = Σ Φ ^ = Σ Φ Α < . (51) n n

The total field (including reflections) at the surface p = pb is approximately

£(p = P>) = Σ *ÄSi£ exp {ikS^x} <- (52)

This field can be expressed in terms of the incident and reflected modes at the terminal surface p = pb by the equation

E(p = Pb) = 2 Φ? {8pm + S** }al (53) V

where S ^ is the reflected /?th mode due to an incident rath mode at the terminal surface p = pb. Hence, the continuity condition on the electric field yields

Σ < W m + ϋξ£Κ « Σ Φ Α exp{ifcSÎA*}fl£. (54)

To determine the reflection coefficients S™, premultiply by Φ^ and integrate with respect to φ:

^ W e xP ί'α»02} I e xP {-*<4<Ρ2} c o s "* c o s *Wd(P > Sim n

= 2 ^ x p { / a ^ 2 } / J Ü ^ ^ (55)

(VBA\*VAB

where (S££)* is the complex conjugate of S££ which is given by (24). Hence, in matrix notation, the reflection scattering matrix is given by

SBB ^ (SBA)*SAB _ L ( 5 6 )

The scattering matrix SCD for the waves that are transmitted through junction III (in the direction of decreasing p), and the reflection scattering matrix SDD at the terminal x = L, are derived in the same manner as were SAB and SBB, by the use of the reciprocity theorem and the continuity condition of the electric field. It can be verified that

il/D γΤΌ (ττΙϊηΛ ^τηη ~ TI/C VTC ^ηιη ~~ ° η y \nvm\K?c)\ ^nm \J I)

** m * m

and

SDD « (SDC)*SCD - I (58)

where (57) and (58) are analogous to (49) and (56) respectively. The reflection scattering matrix TDD for the composite junction (the entire

wedge region) can now be obtained either by cascading the scattering matrices of the three elementary junctions or by direct inspection:

TDD = SDD + sDcsCBSBBSBCDCD, (59)

where the first term represents the reflections due to the discontinuity in the gradient of the height profile at x = L, and the second term represents the reflections due to the discontinuity in the gradient at x = 0.

The field at any point within the wedge region may be derived in a similar manner, since all the scattering coefficients have now been derived.

5. PROPAGATION THROUGH A WEDGE REGION WITH A VERY SMALL GRADIENT IN THE HEIGHT PROFILE

Experimental data obtained from the model waveguide show that the reflection coefficients for either direction of propagation can be entirely neglected when the gradient of the height profile in the wedge region is restricted by the condition

φκίζηφ = hd~ha <ξ 0.03. (60)

Hence, for the case {ha — ha) = 0.76A, L must be about 25 λ before the reflection coefficients Τξ^ can be neglected together with the reflection coefficients T£*.

Provided that the condition on the gradient (60) is satisfied, it is also seen that

kPf^^<0m3m (61) 2 Λ

Hence, a further simplification in the computation of the scattering coefficients can be made. The exponent in the integral of (24) can be substituted approximately by the first two terms in the Maclaurin series expansion such that, for the propagating modes (S£ < 1),

exp {ia^} = exp { * > § * £ } « 1 + * ^ » 8 ? · (62)

The scattering coefficients can now be evaluated in closed form [Wait, 1962] for either direction of propagation, since reflections are now neglected in both directions. With the mth mode approaching the junction from the narrow waveguide (port A), the scattering coefficients can be shown to be given by LL

SBni = exp {-ikptSZPß}. i

Ί + ikPbs$p n 2 (\-&*)]''m = "

- myi- (—1) \m Φη

~ exp ^ ^ r _ 1 = J

7Γ2 (m 2 — n2)2

\3 ~^n2) \ l i

(63)

exp[^tp^(i-^,,i;,» = " ihaSjQia - ha) 32 wit 2" M ^ n

wLA (m2 — w2)2

In the expression for SB£ the following approximation has been introduced :

ikhaSM l\_ 2 \ [ikhgSjt (\ 1 + >^^r ^ e x p

since it has been assumed that

^3 <n2m2) j '

kptSjP _ khgSAJj < < χ

and

φ & tan φ =

From this expression it is clear that the incident mode essentially undergoes a phase shift on propagating through junction I. In the earlier analysis by Wait [1962], this phase factor was neglected.

From the above results it is also seen that, for n Φ 1, the scattering coefficients decrease rapidly as \\nz and that very little energy is converted into the higher order modes. It is also seen that the magnitude of the scattering coefficients is inversely proportional to the length of the taper region (the scattered power is inversely proportional to the square of the length of the wedge region [Solymar, 1959]).

Examination of (49) and (56) shows that, provided the condition (61) is satisfied, S%% can be neglected, since not only are the reflection coefficients S^fn negligible, but even the transmission coefficients S%£ (n Φ m) are small in magnitude.

Finally, it should be pointed out that the results of the analysis in this section should be restricted to the cases when the incident wave can be assumed to be essentially comprised of lower order modes.

6. EXPERIMENTAL RESULTS

Calculations have been carried out for the case of the principal mode


propagating across the night-to-day path with ha = 3.05 λ and ha = 3.81 λ (Fig. 2) for the following cases:

(a) L = 20X (b) L = 10λ.

The transverse electric field variations (amplitude and phase) obtained from the calculated scattering coefficients are plotted in Figs. 3 and 4, together with the experimental data from the model waveguide. The scattering coefficients 7^f are tabulated in Tables 1 and 2.

For the case L = 20 λ, the approximate formulas (63) for the evaluation of the scattering coefficients between the rectangular and radial sections have proved to be quite accurate. [Equation (48) has been used to obtain the scattering coefficient of the entire junction. This involves the application of the reciprocity theorem so that reflections are not neglected for the night-today path.]

It should be noted that even in this case where the gradient of the height profile is very small, the amplitude of the third mode is approximately 12 per cent of the principal mode in the narrow waveguide section. The amplitude of the higher-order modes m ^ 5 drops substantially (the amplitude of the fifth mode is about 2 per cent of the principal mode). This indicates that for the case of nonperfectly conducting boundaries a surface impedance concept would be applicable since most of the incident wave is scattered into near grazing modes.

From the computed data it is obvious that for the case L = 10λ the approximate formulas for the scattering coefficients (63) are not appropriate and the more accurate formulas (24) involving the Fresnel integrals should be used. Special care should be taken in evaluating the transmission coefficients in the tapered region (28), (29) since the values of the order and the argument of the Hankel functions cross over, with the order being less than the argument for the near grazing modes and the argument being less than the order for the higher order modes. In all the above calculations only the first six (even) modes have been accounted for (n = 1, 3, 5, 7, 9, 11), this being justified by the fact that most of the incident energy is scattered into the lower order modes.

It is rather revealing to investigate in some more detail the coefficients that are involved in the calculation of the amplitude of the third mode transmitted into the day path with the principal mode of unit amplitude incident from the night path. This is obtained by the following summation:

N rrAD Y OABCBCCCD J 3 1 Λ ° 2 P ° P P ° P 1 '

V

Since it has been noted that only the near grazing modes are of particular significance, in order to simplify the following discussion only the first and

third mode will be considered; hence

TAD ^ CABCBCOCD I CAB OBC CCD 7 3 1 ^ ò 3 3 ò 3 3 ò 3 1 ^ ° 3 1 ° l l ° l l '

The first term is clearly the contribution to 7^f due to mode conversion at junction III (Fig. 2), while the second term is due to mode conversion at junction I.

EXPERIMENTAL

40°

< I a.

< cr

•EXPERIMENTAL

h_= ZJ05K

-10·«-

FIG. 3. Amplitude and phase variations at the transverse plane x = 0 for the case when a TEio mode travelling in the direction of the negative *-axis is

launched into port D with L = 20λ (Fig. 2).

EXPERIMENTAL

a. < >

v> < X a. Id

>

40

30'h-

20*

m

-ιο·ι

CALCULATED

I ~**χτκχχ*'

EXPERIMENTAL

ha = 305 λ'

FIG. 4. Amplitude and phase variations at the transverse plane x = 0 for the case when a TEio mode travelling in the direction of the negative *-axis is

launched into port D with L = 10 λ (Fig. 2).


TABLE 1. Scattering coefficients T£[J for the night-to-day propagation path ha = 3.05λ ha = 3.18λ L = 20λ

n

1 3 5 7 9

11

Re (7%D)

0.1051E + 01 0.1887E - 01 0.7920E - 04 0.8953E - 02

-0.2212E - 02 -0.8585E - 03

Im (7ft) 0.3583E - 00

-0 .1351E- 00 0.1940E - 01

-0.2083E - 02 -0 .1139E- 02 -0.7392E - 03

Where E ± XY = 10±*^

TABLE 2. Scattering coefficients T£» for the night-to-day propagation path ha = 3.05 λ ha = 3.81 AL = 10λ

n

1 3 5 7 9

11

R e ( ^ A ) 0.1090E + 01

-0 .1194E- 00 -0.3798E - 01

0.1022E - 02 -0.2000E - 02

0.1086E - 02

Im(T^)

0.1765E - 00 -0 .1125E- 00

0.6171E- 01 -0.1364E - 01

0.9251 E - 02 0.5656E - 03

Where E ± XY = 1 0 ± ^

I.G

" 0 -25 -50 -75 -100 -125 λ

DISTANCE IN WAVELENGTHS FROM EDGE OF DAY PATH X=0

FIG. 5. Amplitude variations along the negative x-axis for the case when a TEio mode is launched into port D with L = 10λ (Fig. 2).

Now it is interesting to note that, as was predicted by the preceding analysis, the amplitude of the scattering coefficients S£P and S${* for L = 20 λ is half that for the caseL = 10λ. Nevertheless, the amplitude of the conversion coefficient Tj£ is 12 per cent of the amplitude of the transmission coefficient Τ^ξ for the case L = 20 λ compared with only 13 per cent for the case L = 10λ. It should be noted that this is not due to a large reconversion coefficient from the second mode back into the first mode at junction I for the caseL = 10Λ since |#ff| £ 1 in both cases where L = 10λ and L = 20λ.

__ ' ' — ~ r» -1

-

r !

« , | 1 1 1 i | ,

^EXPERIMENTAL

/ v f ^ /i\

)*J ^ — CALCULATED

! I 1 ! 1 f ! 1 I

. 1

Wy

! !

i [

//

I !

1 ί

JA *>>

1 !

I ' '

^\],

! i ,

1 ' J. — j

y / n

J

1 !


The behavior of the coefficient 7£f is best understood by noting the phase relationship between the two major contributions to Τ$? (i.e., conversion at junctions I and II). While these contributions are only about 10° out of phase for the case L = 20 λ, they are about 90° out of phase for L = 10λ.

The above discussion sheds light on design considerations of a taper section (between two multimode waveguides of different cross-sections), with minimum mode conversion. While it has been pointed out in the discussion of the wedge region with the very small gradient that the conversion coefficients decrease as 1/L, this is strictly true only if each junction (Fig. 2) is taken separately. On treating the composite junction as one unit it is clear that the total conversion factor does not decrease monotonically with increasing L. Thus in order to suppress mode conversion it is necessary to choose the length of the linear taper L such that the two major contributions to T^ are exactly out of phase and thereby tend to cancel out (destructive interference). This aspect of the problem is not pursued any further in this paper, which is principally concerned with the analysis of the propagation problem.

The amplitude of the electric field along the axis of the waveguide is plotted in Fig. 5 for the night-to-day propagation path with L = 10λ. The undulation of the electric field for x > L is clearly seen to be principally due to the interaction of the principal mode with the third and fifth modes.


It is evident that the quasi-optical technique is applicable to problems of propagation in multi-mode waveguides when the direction of propagation is from the narrow to the wide waveguide. However, for propagation in the opposite direction it is necessary to modify the quasi-optical approach. In this paper the required extension has been carried out by applying a generalized reciprocity theorem for waveguide junctions with nonuniform cross-sections. Substantiation for this approach is obtained by using a modelling technique which at the same time has provided insight into the nature of the problem.

Finally, it might be mentioned that transmission through a transition region with a gradually varying height profile (not necessarily wedge-shaped) between two multimode waveguides with finite surface impedance boundaries can also be solved by a generalization of the methods described here. This case will be discussed in a subsequent paper by the first author.

The authors thank C. T. Johnk, S. W. Maley, D. D. Crombie, and A. G. Jean for their encouragement and assistance.

8. APPENDIX A. THE GENERALIZED RECIPROCITY THEOREM FOR WAVEGUIDE JUNCTIONS

The characteristic admittance for a waveguide of nonuniform cross-section


is generally not the same for a forward and a backward propagating wave. It is also a function of the distance along the axis of the waveguide. Only in straight cylindrical waveguides with constant cross-sections is the characteristic admittance the same for both directions of propagation, and independent of the distance along the axis of the waveguide.

For a radial waveguide, for instance, the characteristic admittance for the principal wave travelling in the direction of increasing radius p (junction II, Fig. 2) is [Wait, 1962],

n~ Ef~ WlKkp)9 t j

where Y is the wave admittance in free space and where E+ and //+ are the tangential electric and magnetic fields, respectively. Here H{

vf'(u) is the derivative of the Hankel function of the second kind and order vn with respect to the argument u. For perfectly conducting walls the argument of the Hankel functions is

ηπ

where φ is the azimuthal boundary of the radial waveguide as indicated in Fig. 2.

The characteristic wave admittance for the wave travelling in the direction of decreasing radius /> is, on the other hand,

Y«-£;- ,Ymi\kP) (A2)

where the Hankel function of the first kind replaces the Hankel function of the second kind in (Al).

It should be noted that 7+ is equal to the complex conjugate of Y~, and, that for large kP, Y+ ~ Y~ ~ Y.

Now, for the region under consideration, it may be shown that [Kerns, 1949]

ί (£' x H") - (E" x Η') -uàS = Ç> (A3)

where S is the closed surface bounding the region of integration, n is the unit vector normal to S, and Ë\ H' and E", H" are any two electromagnetic fields of the same frequency that can exist within the source-free region bounded by S. Equation (A3) is valid in the present context since the electromagnetic characteristics of the medium are isotropie. Furthermore, for the present

application, S consists of the inner conducting surface »So of the waveguide junction plus the terminal surfaces SA and SB of the two-port. At the terminal surfaces of the junction, the electric and magnetic fields can be expressed in terms of the basis fields Φη (defined in section 2), and thus (A3) may be written as

f , Σ Σ [φ™(<4 + b'JWia; - Y-b"n) J SA' SB m n

- <&„(<£ + θ Φ » ( 1 ϊ * ; - YûK) ]àS = 0 (A4) where am, b'm and ά^, b'^ are elements of the total wave amplitude vectors (8) related to the electromagnetic fields E\ H' and is", H", respectively. For waveguide ports with boundaries corresponding to orthogonal surfaces of separable coordinate systems, the basis fields are orthogonal; hence, in this case (A4) reduces to

h(am + b'JWm(Yid^ - Y-b'm) - {am + b%Wm(Yi dm - Υ-b'j] = 0 (A5)

where Wm is the power normalization coefficient defined in (10). In matrix notation this can be written as

(α' + b')W(Y+a" - Y~b") = (a" + b")W(Y*a' - Ybf) (A6) where ä is the transpose of a.

Since the transpose of a scalar is equal to itself, (A6) reduces to ä" WYTSa' = ä"SWYTSa' (A7)

where YT, the total characteristic admittance, is defined as YT=Y++ Y-

and b is replaced by Sa, where S is the total scattering matrix defined by (15). Now since a' and a!' can be chosen arbitrarily, the reciprocity theorem

for waveguide junctions is, from (A7),

WYTS = (WYTS) (A8) since

(WY*) = WYT. For completeness, the relationship between the scattering matrix S and

the equivalent admittance matrix Y of the junction can be shown to be given by

Y+ — YS Y+ — Ϋ Y= T , I , S = - i . (A9)

1+ S Y-+ Y


The equivalent admittance matrix Y relates the amplitude matrix of the electric field Ê=a + b to the amplitude matrix of the magnetic field H= Y+a — Y~b through the equation Ê = YÊ. For the case of waveguide ports for which Y+ = Y~, (A8) and (A9) reduce to the more familiar forms.

The total characteristic admittance for a radial waveguide is

4 7 γτ = Y+ _L γ- = — . (AIO)

For waveguides with constant rectangular cross-sections, such as port A in section 6 for example, the total characteristic admittance is

Y*A = 2Yi = 2YS* (All)

where S„ is defined by (12).

9. REFERENCES BAIN, W. C , BRACEWELL, R. N., STRAKER, T. W., and WESTCOTT, C. H. (1952) The iono

spheric propagation of radio waves of frequency 16 kc/s over distances of about 540 km, Proc. IEE 99, Pt. IV, Monographs 12-54, 250-9.

CROMBIE, D. D., ALLEN, A. H., and NEWMAN, M. (1958) Phase variations of 16 kc/s transmission from Rugby as received in New Zealand, Proc. IEE 105, Pt. B, 301-4.

BAHAR, E. (1964) Model studies of the influence of ionospheric perturbations on VLF propagation, Technical Report, ARPA, Contract No. CST-7348, Elect. Engr. Dept., University of Colorado, Boulder, Colorado.

BAHAR, E., and WAIT, J. R. (1964) Microwave model techniques to study VLF radio propagation in the earth-ionosphere waveguide, Quasi-Optics, ed. Jerome Fox, pp. 447-64, Interscience Publishers, New York, N.Y.

KERNS, D. M. (1949) Basis of the application of network equations to waveguide problems, / . Res. NBS 42, 515-40, RP1190.

MALEY, S. W., and BAHAR, E. (1963) Model studies of the influence of ionospheric perturbations on VLF propagation, Technical Summary Report, ARPA, Contract No.

Ï CST-7348, Elect. Engr. Dept., University of Colorado, Boulder, Colorado. MALEY, S. W., and BAHAR, E. (1964) Effects of wall perturbations in multimode wave

guides, Radio Sci. J. Res. NBS 68D, No. 1, 35-42. PIERCE, J. A. (1955) The diurnal carrier phase variations of a 16-kilocycle transatlantic

signal, Proc. IRE 43, 584-8. SOLYMAR, L. (1959) Mode conversion in pyramidal-tapered waveguides, Electronic Radio

Engr. 36, No. 12, 461-3. WAIT, J. R. (1959) Diurnal change of ionospheric heights deduced from phase velocity

measurements at VLF, Proc. IRE Al, No. 5, 998. WAIT, J. R. (1960) Terrestrial propagation of VLF radio waves, / . Res. NBS 64D (Radio

Prop.), No. 2, 153-204. WAIT, J. R. (1961) A comparison between theoretical and experimental data on phase

velocity of VLF radio waves, Proc. IRE 49, 1089-90. WAIT, J. R. (1962) An analysis of VLF mode propagation for a variable ionospheric

height, / . Res. NBS 66D (Radio Prop.), No. 4, 453-61. WAIT, J. R. (1964) Two-dimensional treatment of mode theory of the propagation of VLF

radio waves, Radio Sci. J. Res. NBS 68D, No. 1, 81-94.


KERNS, D. M., and BEATTY, R. W. (1965) Basic Theory of Waveguide Junctions and Introductory Microwave Network Analysis, Pergamon Press Limited, Oxford, England.

Transverse Propagation of Waveguide Modes in a Cylindrically Stratified Magnetoplasma\

JAMES R. WAIT

Abstract. An exact solution is given for the problem of a magnetic line source in the presence of an isotropie conducting cylinder which is surrounded by a concentric stratified plasma. The d-c magnetic field is aligned with the cylindrical axis. The harmonic series representation is transformed to a more highly convergent form which facilitates computation for large cylinder radii. In this case, the field is expressed as a sum of modes which propagate around the cylinder in opposing directions. The presence of the d-c magnetic field destroys the usual azimuthal symmetry in problems of this type. It is indicated that the situation is somewhat analogous to the propagation of radio waves in a spherical earth-ionosphere system.

1. INTRODUCTION

The solutions of boundary value problems involving anisotropie media are somewhat more difficult than those for isotropie media. In spite of their complexity, the solutions have considerable value in providing an understanding of propagation of waves in such media as the ionosphere and other magnetoplasmas. Fortunately, there is a class of two-dimensional problems involving anisotropie media which yield to exact treatment [Wait, 1961; Brandstatter, 1963]. An example which is considered in this paper is an ionized region which has cylindrical symmetry and an axial d-c magnetic field. To achieve further simplicity, all variations in the axial directions are taken to be zero. Thus, propagation of the waves is constrained to directions which are transverse to the d-c magnetic field.

2. FORMULATION

The general problem considered is shown in Fig. 1. Cylindrical coordinates (p, φ, z) are used with the z axis taken normal to the paper. The region P < ao is taken as an isotropie homogeneous medium of conductivity σ, permittivity 2, and permeability fi. The region (0) is defined by ao < p < a± which, for convenience, is regarded as free space with permittivity eo and

f This paper was prepared originally for the Proceedings of the Radio Physics Symposium, University of California, Los Angeles, 7-9 August, 1963. Sometime later the symposium organizers cancelled the plans for publication of the symposium proceedings. (Radio Sci. 1 (New series), No. 6, June 1966.)

529

permeability μο. The anisotropie region (1) is then defined by a\ aPi is regarded as a homogeneous anisotropie region.

The anisotropie plasma regions are described in terms of a dielectric tensor (e) which has the following form:

/ « ' -iQ 0 \ ( € ) = ( iQ «' o , (i)

\ ° ° «"/ when the time factor is exp (/W). As is well known, the quantities c', e", and Q are functions of the density of electrons and the ions and the frequency of collisions between them. They also depend on the strength of the d-c magnetic field.

FIG. 1. The geometry of the problem.

The case usually considered is that in which the electromagnetic forces influence only the electrons. Furthermore, the motions of the ions are commonly neglected. For this situation, the properties of the plasma can be approximately described, in a macroscopic sense, in terms of the following quantities:

ωο, the (electron) plasma frequency, ωτ, the (electron) gyrofrequency, v, the effective collision frequency for electrons with the neutral particles.

Transverse Propagation of Waveguide Modes 531

The elements of the dielectric tensor are then given explicitly by [Bremmer, 1949]

e ' , _ Kv + "")tug/tu e0 « | + (v + to)«' W

_β _ —ωτω%Ιω eo - ω | + („ + /ω)2' (3)

where eo is the permittivity of free space.

3. BASIC ELECTROMAGNETIC EQUATIONS Maxwell's equations in a source-free region with a (tensor) dielectric

constant (e) and a (scalar) magnetic permeability μ are

i(€)c«iE = curlH, (5)

—ΐμώΗ = curl E. (6)

The first of these may be written in the form

fYüE = (€-i)curlH,

where (c-1) is the inverse of the dielectric tensor. It is not difficult to show that

l M -iK 0 \ €o(e-i)= [iK M O , (7)

\ 0 0 €0l€" J where

and

M= ( 0 2 - g 2 > (8a)

K=~IZ)^~QZ' ( 8 b )

This formula is quite general. The only restriction is that the z axis is to be taken in the direction of the impressed magnetic field. If, in addition, the fields are assumed not to vary in the z direction, Maxwell's equations may be written

• r mr dHz _J_ rdH* ™ 1€«ωΕρ = Μ-ρϊφ+ιΚΤΡ> ( 9 )

ΐ€0ωΕφ = iK ^ - M *ψ (10)

Ϊ€0ωΕζ = ( e o / O

-ΐμ0ωΗρ^Ν-~, (12)

dEz ΐμ0ωΗψ=-Ν-^, (13) dp'

IP dP Ρύ(ι (14)

where Ν = μοΙμ. (15)

The transverse components Ep, Εφ, Hp9 and Ηψ may be eliminated to yield

[V2 + k^MNY^Hz = 0 (16) and

[V2 + k*N-Ke"leo)]Ez = 0, (17)

where k2 = ω2μο^0, and where

1 d d a2

*■ = ; s-, ' a-, + ;**» (18)

is the two-dimensional Laplace operator in cylindrical coordinates. Equations (16) and (17) are valid only if the magnetic permeability and the elements of the dielectric tensor are constant for a given region.

The fact that Hz and Ez individually satisfy a wave equation means that any solution of the problem may be regarded as the linear combination of two partial solutions. In the first of these, Ez = 0, and in the second, Hz = 0. Thus, without any subsequent loss of generality, attention can be restricted to these two cases. It should be emphasized that this decomposition into independent partial fields is valid only when the derivatives with respect to z are zero.

It should be immediately evident to the reader that the solution for the fields with Hz = 0 is relatively trivial, since the d-c magnetic field has no influence. Consequently, in what follows, attention will be restricted to the partial fields in which Ez = 0.

4. FORM OF THE GENERAL SOLUTION We now return to the specific configuration shown in Fig. 1. A magnetic

line source is located at the point A where p = po and <f> = 0. The line source has a magnetic current Im which does not vary with z. The primary magnetic field at B, with coordinates (p, <f>9 z), has only a z component and it is given by

Hï^ — KW), (19)

where H™ is the Hankel function of the second kind of zero order, and

P=[pl+ P2-2ppo cos φ]ν*. (20)

Employing an addition theorem [Watson, 1945] for Hff\kp)9 this can be written

+ 00

m = — co

for p < po, where H ® and Jm are Hankel functions of the second kind, and Bessel functions of order m, respectively. When p > po, kpo is to be interchanged with kp. The primary components of the electric fields may be obtained from differentiation; for example,

+ 00 ikl *S^

m= — co

To obtain the complete field, Hz, in the concentric space ao < p < tfi, indicated in Fig. 1, a general solution must be added to Hv

z. Evidently, it must be a linear combination of two independent solutions of

(V2 + k*)Hz = 0. (23)

Some considerations show that

Hz = V [AmH%\kp) + BmJmikp)} Q-^Φ ! = — CO

, [H%(kpo)Jm(kp)] c-^Φ; p < po, ' [H<2(kp)Jm{kp0)]e-">*i p>po ( 2 4 )

where Am and Bm are undetermined coefficients. An equivalent form of (24), which is more convenient later on, is obtained by using the identity

2Jm(kp) = H™(kp) + &2(kp), (25) where H^(kp) is the Hankel function of the first kind of order m. Thus

8

+ co β ο ω / , y [amH%(kp) + bmH^{kp)\ e-*»«5

_, [H%Xkpo)H%(.kp)]e-^, m= — oo

[H%Kkp)H^Kkpo)]t-*^ ( 2 6 )

where am and bm are also coefficients yet to be determined.

5. APPLICAI ION OF NONUNIFORM TRANSMISSION-LINE THEORY Using a somewhat classical procedure, one could set up expressions for the

Hz field in each of the anisotropie plasma regions. Each of these would have two sets of undetermined coefficients. By matching boundary conditions, such as requiring the continuity of Hz and Εφ, one would be led to a system of linear equations to solve for the unknown coefficients. While this is a straightforward procedure, it is somewhat more convenient to use an iterative method which is analogous to Schelkunoff's [1943] nonuniform transmission-line theory. To facilitate this approach, it is necessary to define certain impedance and transmission parameters of a typical homogeneous section of the nonuniform transmission line [e.g., the/?th region].

From Maxwell's equations it is noted that, for the pth region,

dHz A/f dHz ΐεοωΕφ = iKp —-- — Mp —?, (27)

where the field Hz must satisfy

(V2 + ßl)Hz = 0 (28)

where ß* = Ι&ίΜρΝρ)'1. The quantities Kp, Mp, and Np are defined by (8a), (8b), and (15) with the subscript p to indicate that they are to apply to the /?th region bounded by ap ), respectively. For example, when \βρρ\>> m these solutions are proportional to exp(+#W) and exp (—ißpp), respectively. To distinguish these two cases, a superscript (/) is used for the "ingoing" solutions and the superscript (0) is used for the "outgoing" solutions.

The wave impedance of order m for ingoing waves is defined by

Ef. m

whereas for the outgoing waves, £(0)

K™>P = + jar"- (3°)

Using (27), it readily follows that

'"-^b^-fel wm\ (31)

**·>--"»[-&-(*) WM>\· (3)

and

where ηο = GW€o)1/2.

Reflection coefficients for the interface p = ap are now defined by

^mtP-l + Z™> V Η(Μ m J p=ap

J -(O) y zy(i) η

, m J / and

fro, u - l — 7 7 7 ^ ΓΤΤ^ — FT^~" > V-*4J

where Zm, p is the wave impedance of the total field looking outward at P = ap; Zm,p may be expressed in terms of the reflection coefficients Rm,p and rm, p by making further use of some of the concepts from nonuniform transmission-line theory. Thus

Zm,p= + ^ \ , (35) or

z = K(o) 1 + rm, Vxt\ e (aP> aP+l)Xm, e (fp+U <h>) m-P m'p 1+Λ»,,χ»'Αίαρ,βρ+ι)χ«:»(αρ+ι,βρ)* (36)

where (o) ,- - Λ _ ÊtJto+Ù

^φ, m\uP)

^*«·*>=Ί§Α· (38)

^ U a P , a P + i ) = ^ ^ , (39)

*Ua~ua,)=-$^. (40)

The x's are transmission factors which describe the fractional change of the wave as it propagates from one interface to another within a layer. For example, the numerator in (36) is a measure of the electric field at the pth interface which takes into account transmission through the layer, reflection at the (p + l)th layer, and transmission back to the pth layer. The denominator has a similar interpretation. Because of this method of constructing transmission factors, multiple reflections are automatically taken care of.

The specific forms of the transmission factors are

(o) (n n \ - MvßpH%'(ßvav+1) - mKpH™(ßpap+1)lap+1 Xm. .W» ap+D - MpßpHW(ßpap) - mKpH^(ßpap)/ap ' <41 '

(<) (n n x MpßpH%'(ßpap) - mKpH%(ßpap)lap Xm, A<b+u <h) - MpßpH%'(ßpap+1) - mKpH%(ßpap+i)\ap^ {*Z)

MM

tìM^-flH^,). (44)

The iterative process begins with the reflection coefficients Rm, P - I and rm, p- i , which are given by

ίζ(0) _ tf(C) A m , P - l — ~ ω , „ ( 0 ) [f*0)

and

"m*p-1_ w , - ! +w. ; · (46)

Then, using (33) and (34) with/7 = P — 1, we obtain explicit expressions for •#m, p-2 and rw, p-2, which, in turn, enable Rmt ps and rm, p-3 to be determined. The process is continued until the innermost interface at p = a\ is reached, whence

Am.O ~ „ ( i ) _Γ ~ — ( 4 / )

and

r" ' 0- i /A5i!o+"' i /z. ."i ' ( 4 8 )

where K - = im WW (49)

and //«»'(tei)

^îo=-» ïo^^y . (50)

The iterative method outlined above may also be used to obtain the reflection coefficients at the boundary p = ao. In this case, it is desirable to regard the medium p < ao as a nonuniform transmission line. Looking inward, the impedance is readily found to be

^ rJm&ao)

^m = — 1η ~ z , (51) Jm(kao)

where η = [(//Ϊω)/(σ + ϋωψ*

and ih = [(ίβω) (σ + /60j)]l/2

in terms of the constants Θ, e, and fi of the isotropie and homogeneous medium occupying the space p < ao. The generalization to a concentric stratified core is straightforward but, for simplicity, this region has been regarded as homogeneous. The waves in this region are characterized by Jm(kp) exp (— im <f>), which is finite at p — 0.

Reflection coefficients Rmto and rw,o for the interface p = ao may be defined in terms of appropriate impedances. Thus

& » , „ = Α ^ ~ ^ ' (52)

and

where

and

_ 1/Ä& - 1/ZW r m - 0 - i / ^ % + i/zm' (53)

jg» — fa, ^ * * » (54)

Km = im WZ*Ù ( 5 5)

6. EXPLICIT FORM OF THE FINAL SOLUTION The final expression for the resultant field Hz in the space öo ( Μ

X 1 Α - · ο Α , · ° Α » ( ^ ) Äff(tei)J C ' ' (57)

which is valid for p < po. The corresponding expression, valid for p > po, is obtained by interchanging p and po everywhere.

The φ component of the electric field in the region a\ > p > ao is found from

Εφ= + f ^ / n e - ^ (58) m— — co

where . „ dHz, ΐ€0ωΕφ> m = — . (59)

It may be readily verified that the final solutions as given for Hz, m and Εφ, m satisfy the boundary conditions

Εφ, m = Zm, i Hz, m at p = d\ (60) and

Εφ, m = — ZmHz, m at p = ao. (61) Equation (57) is an exact solution for the problem as posed. In its present

form it could be used for calculations if the circumference of the outermost cylindrical region were not large with respect to the wavelength. Otherwise, the series solution in the form given is very poorly convergent. However, with the use of high-speed computers, numerical results may be obtained even for relatively large cylinders.

An interesting property is immediately evident from (56) by writing it in the equivalent form

oo oo

Hz = Σ (<W2)(Hz, m + Hg, -m)cosmj> — i Σ (Hz, m — Hz,- m)sinηιφ, (62)

where e0 = 1 and em = 2(m Φ 0). As is now immediately evident, Hz is an even function of φ when Hz, m = Hz, _ m. This condition holds only if the d-c magnetic field is zero. Otherwise, the odd part of the solution is finite.f Consequently, in the general case the field Hz is nonsymmetrical about the plane φ = 0.

7. APPLICATION OF THE WATSON TRANSFORMATION For cylindrical surfaces, which are quite large in terms of a wavelength,

it is really better to transform the series over integral values of m to a more highly convergent form. This is accomplished by a direct application of the Watson transformation [Bremmer, 1949; Wait, 1959; Felsen and Marcuvitz, 1966].

The essential step is to express Hz as a contour integral in the form

Ht=2J cdvn

Ηζ,ν*-1νφτ^Γ- àv, (63) le sin νττ

where v is a complex variable, and C is a contour which encloses the poles of the integrand (i.e., the zeros of sin νπ), as shown in Fig. 2. The equivalence of (56) and (63) is shown by noting that the integral is of the form

4 ^ d.v (64) sin VIT '

where A(y) has no singularities within the contour C, and therefore / = 2πΐ x sum of residues within C

+ 00 ^ A(v)

"-I . = 2πί / { (d/dv) (sin νπ) =2 ίΣ in7rA(n). (65)

t A related problem has been discussed by Crombie [1958] and Wait [1962] for planar models. The nonreciprocity results from the magnetic field dependence of the wave impedances, as indicated by (31) and (32).


V PLANE

+1 +2 +3 +4 +5

FIG. 2. The complex v plane and the contours C and C". (The contour C is to be closed by infinite semicircles in the upper and lower half spaces.)

It is not difficult to establish that the contributions to the contour C, which cross the real axis, approach zero as the ends of the contour approach ±00. In the limit, the contour C becomes two straight lines, one just above and one just below the real axis. Following the Watson prescription, the two line integrals are deformed to enclose the poles in the upper and the lower half plane, respectively, as indicated in Fig. 2.

The deformed contour C" is equivalent to the contour C if the contributions from the infinite semicircles may be neglected. The latter appears to be the case in view of the asymptotic relation

Lim jR-> oo

^ίν(ττ-φ) ~ M exp [-.RHsin θ\ + (π - Θ) sin 0)], (66)

sin V7T ' where M is bounded and v = RQÎÔ. It is also assumed that the behavior of Hz, v does not introduce exponentially divergent factors.

Assuming that justification has been provided to transform the contour C to the contour C", it follows that

A(v) | csin vu

The poles of A(y) are, of course, the (complex) zeros of (HVt z)~l which occur =L 2ni x residues enclosed by C". (67)

in the second and fourth quadrant of the v plane. Thus OO

1

m = l,2,. £W*J 00

exp [ivm(jr — Φ)] sin vmTT

exp [iv-m(ir — </>)]

m=l,2,.. . . 7Γ ( l / / / z ,„ ) sin v-mTT

(68)

The first summation includes all poles in the fourth quadrant, whereas the second summation includes all poles in the second quadrant.

As it turns out, in a number of specific problems which have been treated [Wait, 1962], the poles in the v plane arrange themselves into families or groups. This is indicated in Fig. 2 where the principal group in the fourth quadrant is designated v±91% vz,.... The corresponding poles in the principal group in the second quadrant are deisgnated v-\9 v-2, vs, The numbering on these is ordered such that — Im v± < — Im v2 < — Im v$ . . ., and Im ν-ι < Im v-2 < Im v-v. . . . Another typical set of poles is also shown in Fig. 2. These are designated vq, vq+i, vq+29. . . and v-q, v-q-i9 v-q-2,. . . , and they are located in the fourth and second quadrants, respectively. Depending on the complexity of the plasma media, additional sets of poles may occur.

It is desirable to give a physical intepretation to these residue terms. For this purpose, it is convenient to imagine φ to be greater than zero and less than 2?r. Then, if the media have finite losses, Im vm < 0, and

sin vmTT / f p = 0 , l ,2 , . . . .

Clearly, this corresponds to waves travelling in the positive φ direction with complex propagation constants ivm. The value p indicates the number of revolutions which the waves undergo. In a similar fashion, Im*>-m > 0, and

exp iv-nfa - φ) = _ 2 / e .^m ( 2 π_φ ) y e<2p_m

sin v-mTT / ( p=0, 1 ,2 ,3 , . . . .

which corresponds to waves travelling in the negative φ direction through a distance 2π — φ in addition to the p complete revolutions. In this case, the propagation constants are iv-m.

As indicated above, the positive- and negative-order residue waves correspond to waves propagating in the positive and negative φ directions, respectively. Because of the finite value of the d-c magnetic field, vm and v-m are not symmetrical, in general, about the origin. Thus, these azimuthal waves, corresponding to positive-order and negative-order poles, may have quite different properties.

where

8. NATURE OF THE PRINCIPAL ROOTS The central remaining task is the determination of the locations vm and

v-m of the complex poles in the v plane. As seen from (57), these are to be obtained from

L1 ~ R" ° M- ° Hf\ka0) m\kai)\ - °- (71)

When the Hankel functions occurring in this expression are replaced by Debye approximations, there is a great simplification. The appropriate Debye representation [Watson, 1945] is

/ 2 \ 1 / 2

H^x) = ( « ) 5 ' 1 ) ( X ) ' ( 7 2 )

/ „ \ 2-1 -1/4 Γ Cx Γ „2-11/2 Ί 1_wJ "n'U1-*! d4 (73) and similar expressions for Hf\x). This result is valid when \vjx\ < 1, but not near 1. Then it is not difficult to show that (71) may be written in the form

pc - Δ1 [C - Δ1 Γ P Γ 2zS*]1/2

ί^Τΐ]ί^ΤΔ]6ΧΡί-2ΊοΓ+ 'ail dz = e-2 ' ™ where C - (1 - S2)1/2, S - vl(kao)9 h = a± - a0, C = [1 - (S')2]1/2, S' = v/kai, Δ = ΖνΙηο, Δ = ZVti fao, and ηο = 120 π.

Equation (74) has an interesting physical intepretation. The complex quantity C may be regarded as the cosine of a complex angle φ, whereas C is the cosine of a complex angle φ'. In fact, φ and φ' are the angles of incidence of the (complex) rays at the interfaces p = do and p = cti9 respectively. Thus (C — A)/(C + Δ) is a Fresnel-type reflection coefficient at the boundary p = ao, while ( C — A)/(C + Δ) is the corresponding reflection coefficient at p = ai. Furthermore, the exponential factor may be interpreted as the (complex) phase for the ray paths between the two interfaces.

It should be mentioned that the impedance parameters Δ and Δ are also functions of C via the complex parameter v. However, in solving for the principal groups of poles (i.e., v±9 i% · . · and v-i. v-2, . . .), the parameters Δ and Δ may be regarded as constants. A detailed discussion of the reflection coefficient ( C — A)/(C" + Δ) for real values of C" is available [Walters and Wait, 1963] for various profiles of the electron density and collision frequency.

In considering propagation in systems of curved layers, it is not always permissible to employ the Debye approximations [Watson, 1945] for the Hankel functions. An improved approximation, adapted from Sommerfeld [1949], is

where

with

H?Kx) = ( ~ ) 1 / 2 S?K*)T?Kx), (75)

w-(fr«,[->(%-ß) HSm (76)

It may be readily verified that ΤΙυ(χ) approaches unity asymptotically as β becomes large. For numerical work [Wait, 1962], it is convenient to replace the Hankel functions of order one-third by the Airy functions H>2(0 and w±(t) of argument t. The connecting relations are

w2(t) = exp (2ΑΓ/3) (-ntl3)U*H® [f(-i)1/2] (77) and

wi(0 = exp (-2/π/3) (-vtWWSl [f(-01/2]· (78) To simplify matters a bit, it is assumed that h/ao « 1. This results in the approximate forms

H?Xkaò) s - ~ (^-o)1/3 w2(0 (79a) and

/ / 2 \1/3

Hf(kao) ^ + ^ i^J W1Q), (79b)

where / = (l/kd)1^ — kao). Similarly,

HOKkaù s - ^ (^-J 1 ' 3 w2(t - yi) (80a)

i / 2 \ 1 / 3

Hïtkai) S ^ \j^j wiif - yi), (80b)

/ 2 \1 /3 / 2 \1/3

^=(w *<*-*>=te) **· (80c) Using these Airy function approximations, (71) may be written in the

compact form [»!'1(.t-y1) + m(.t-y1)l K(Q-gw2(Qj = e_2î7iBi; ( 8 ] )

where /9 = (Α:αο/2)1/3Δ

and # = (£α0/2)1/3Δ.

and

where


When the real parts of the arguments of the Airy functions are sufficiently negative, these functions may be replaced by the first term of their asymptotic expansions. Then, if we recognize that (—01/2 = (teo/2)1/3C and (yi — 01 / 2 = (&ao/2)1/3C, equation (81) reduces to a form which is equivalent to (74). Thus, in a nutshell, the Debye-approximated pole equation is valid, provided Re [(kaoßy^C] > 1.

9. FINAL REMARKS

In the discussion of the pole determinations we have referred mainly to the principal group of the residue terms. These are the waves which propagate in the free-space region bounded by the concentric plasma and the conducting core. In this sense the development is very similar to that in the mode theory of VLF radio waves in the space between a spherical earth and a concentric ionospheric plasma [Wait, 1962].

The properties of the roots vq, vq+i9 vq+2, . . . and v-q, v-q-i,. . . and other secondary groups have not been considered in this discussion. However, again in analogy to VLF mode theory, they can be expected to be waves of relatively high attenuation if the plasma regions have finite loss. In limiting case of a planar model (i.e., #o -> oo), these secondary groups of rootsf can be identified with the lateral waves [Brekhovskikh, 1960] which are well known in seismology.

The excitation of the modes or residue waves has not been explicitly considered here. This question is related to the magnitude of the derivative with respect to v in (68). An explicit determination of the excitation factors has been carried out in the analogous VLF radio problems [Wait, 1962]. As indicated there, one may find that certain modes of low attenuation are poorly excited by a source which is outside the effective waveguide. In the earth-ionosphere waveguide, these are known as whispering-gallery modes, since they propagate via the lower edge of the ionosphere without being influenced appreciably by the ground. There is some evidence that additional modes of this kind may propagate in concentric shells in the outer ionosphere. This aspect of the subject warrants further study.

10. REFERENCES

BRANDSTATTER, J. (1963) Introduction to Rays, Waves, and Radiation in Plasma, p. 315, McGraw-Hill Book Co., Inc., New York, N.Y.

BREKHOVSKIKH, L. M. (1960) Waves in Layered Media, Academic Press, New York, N.Y. BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier Pubi. Co., Amsterdam and New

York. WAIT. 3.R. (1962) Refers to 1st. Ed.

t For a planar model, one sequence of poles coalesces into a branch point, thereby providing a continuous spectrum of waves guided parallel to the interfaces. Lateral waves of this type have been discussed thoroughly by Tamir and Felsen [1965].


CROMBIE, D. D. (1958) Differences between east-west and west-east propagation of VLF signals over long distances, J. Atmospheric Terrest. Phys. 12, 110-20.

FELSEN, L. B., and MARCUVITZ, N. (1966) Alternative representations of source-excited vector and scalar fields, Radio Sci. 1 (New Series), No. 6, 619-40.

SCHELKUNOFF, S. A. (1943) Electromagnetic Waves, Van Nostrand Co., Ltd., New York, N.Y.

SOMMERFELD, A. N. (1949) Partial Differential Equations, Academic Press, New York, N.Y. TAMIR, T., and FELSEN, L. B. (1965) On lateral waves in slab configurations and their

relation to other wave types, IEEE Trans. Ant. Prop. AP-13, No. 3, 410-21. WAIT, J. R. (1959) Electromagnetic Radiation From Cylindrical Structures, Pergamon

Press, Oxford and New York. WAIT, J. R. (1961) Some boundary value problems involving plasma media, / . Res. NBS

65B, No. 2, 137-50. WALTERS, L. C , and WAIT, J. R. (1963) Numerical calculations for reflection of electro

magnetic waves from a lossy magnetoplasma, NBS Tech. Note 205. WATSON, G. N. (1945) Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge Univ.

Press, Cambridge, England.

Cavity Resonator Modes in a Cylindrically Stratified Magnetoplasma]

DAVID B. LARGE AND JAMES R. WAIT

Abstract. The exact harmonic series solution for the fields in a cylindrically stratified plasma is utilized to examine the cavity resonant frequencies. The problem is restricted to two-dimensional geometry such that the d-c magnetic field is parallel to the axis of the system. Some numerical results are given for the special case of an annular air region bounded by a perfectly conducting core and an homogeneous external plasma. It is shown that the d-c magnetic field shifts the resonant frequencies from their isotropie values and causes certain "line-splitting" effects.

1. INTRODUCTION In a previous paper [Wait, 1966], an exact expression was given for the

fields of a line source in the presence of an isotropie conducting cylinder which is surrounded by a concentrically stratified plasma. The d-c magnetic field was taken to be parallel to the cylindrical axis of the system. In that paper the nature of the waveguide modes which propagate circumferentially around the cylinder was discussed in some detail. It was indicated that these modes were analogous to VLF radio waves propagating in the earth-ionosphere waveguide for directions along the magnetic equator. A related problem, which is discussed in the present paper, is to calculate the natural resonant frequencies of the system. While we restrict attention to a cylindrical model, the results are analogous to the so-called "Schumann resonances" which are actually observed in the sphercial earth-ionosphere cavity [Baiser and Wagner, 1960; Madden and Thompson, 1965; Galejs, 1965].

2. GENERAL CAVITY MODE EQUATION The geometry of the problem is identical to that in the previous paper

[Wait, 1966] which, hereafter, we shall refer to as paper I. The "cavity resonances" may now be obtained by examining the condition for which the denominator of (57) in paper I vanishes. Explicitly, this condition is written

' HSKkmì Hmfknìi t Radio Sci. 1 (New Series), No. 6, June 1966. The research reported in this paper was

supported by the Advanced Research Projects Agency, Washington, D.C., under ARPA Order No. 183-62.

545

where the symbols have the following meaning: ao is the radius of the isotropie core; a\ is the inner radius of the concentrically stratified magnetoplasma; Em, o is the cylindrical reflection coefficient, for the interface at p = ao, as defined by (52) in I; Rm, o is the cylindrical reflection coefficient, for the interface at p = o\, ar defined by (47) in I; H$ and H™ are Hankel functions of the first and second kind, respectively, with arguments as indicated; k = (eo/xo)1/2"> is the wave number where ω is the (complex) angular frequency.

The "cavity resonances" occur at the complex frequency ω/2π where (1) above is satisfied for integer values of m. Thus, it is appropriate to call m the cavity resonance mode number. This is to be contrasted with the waveguide modes which would be formally obtained from (1) by solving for the non-integral value of m (i.e., denoted v in I) for a real frequency ωβπ.

3. SPECIAL CASE OF A HOMOGENEOUS EXTERNAL PLASMA It is evident that there are many solutions of (1) when the plasma is con

centrically stratified, since each individual layer will have its own set of resonances. Furthermore, in general, these various sets of modes will be coupled to each other. In order to simplify the problem, we shall consider a highly idealized special case. Specifically, the core (i.e., p < ao) is taken to be perfectly conducting, while the external plasma (i.e., p > cti) is homogeneous and nonmagnetic (i.e., μι = μο). Using the appropriate forms of Èmy o and Rmt o from I, we find readily that (1) above may be written

+ kH£Xkai)Jn(kao) - kH^r(kaoK(kai) = 0. (2)

Here, Mi and K\ are related to the tensor dielectric constant (ei) of the plasma as follows:

where

eoC*!1) =

( € l ) =

Mi -Mi 0 iKi Mi 0

LO 0 e0/e"J

< -iQ o 1 Iß e' 0 0 0 €

As indicated in I, the plasma wave number βι is given by

βι = fc(Mi)-i/2,

(3)

(4)

(5)

which is chosen to have its imaginary part negative and real part positive for a plasma with finite collisions. Other definitions from I are

Cavity Resonator Modes in a Cylindrically Stratified Magnetoplasma 547

L _ 1 *Q + Ϊω)ωΙ/ω Q _ —ωτω^ω ç^ _ ίω* *0 ω | + (ν + ίωψ €0 ω% + (ν + /ω)2' e0 ω(ι> + ι'ω)'

Λ^ € ' e ° η, Q€0

(e')* - Ô2 (O2 - Ô2

where c0 is the permittivity of free space, ω0 is the electron (radian) plasma frequency, ωτ is the electron (radian) gyrofrequency, and v is the electron-neutral collision frequency. If the effect of ions is to be included, we should use more elaborate definitions for the quantities e', e", and Q; however, for present purposes, we shall ignore the influence of the ions.

4. INFINITE CONDUCTIVITY LIMIT It is interesting to note that if the plasma becomes effectively a perfect

conductor, βιαι -> oo and, therefore,

Η<£>(βια{) "* '■ w

At the same time,

M-ißi = kM\V -^ 0.

Thus, the modal equation (2) reduces to

H%'{kai)Jm(kao) - H^XkaoKQcax) = 0, (7)

which could have been derived directly by very elementary means. In fact, its solutions are well known [e.g., Dwight, 1948]. For example, if we choose a0 = 6 x 106m and a± = 6.5 x 106m, it follows that the two lowest resonance frequencies are

m = ± 1, / i = / _ i = 7.65 c/s,

and

m =±2, f2=f-2= 15.3 c/s.

The value used here for a\ certainly exceeds the actual height of the lower ionosphere. This excessive height was used in all calculations to insure the numerical significance of the sum of the last two terms of (2). A lower value of a\ would result in slightly higher values for the fm. As indicated, for this special case, the cavity resonator modes of order — m and +m are the same. This follows from the well-known relations

J-m(Z) = (-l)^MZ) and

H™JZ) = (-1)*/7«>(Z).


In fact, the resonance frequencies fm and/_m are always the same, provided the d-c magnetic field is zero. This is evident from (2) on setting Ki = 0.

5. SOME NUMERICAL RESULTS FOR THE ANISOTROPIC CASE The determination of the resonances for the anisotropic case (i.e., K\ Φ 0)

requires a numerical solution of (2). The iterative method employed is facilitated by using the known real solutions of (7) as a starting point. To present a few concrete results, we adopt plasma parameters which are roughly equivalent to those of the night-time lower ionosphere. Thus, we choose the collision frequency v = 7 X 104 c/s and the electron density Ne = 2 x 109

m- 3 . As before, #o = 6 X 106 m and a± = 6.5 X 106 m. The results for the real part of the resonant frequencies / i , / 2 , / - i , and/-2 are listed in Table 1 along with the appropriate strength of the d-c magnetic induction Bo in webers/m2. (The parameters are chosen primarily to facilitate calculation rather than for physical significance. Actually, the separation a\ — a^ = 500 km is unrealistically large.)

The results in Table 1 indicate that an impressed magnetic field with a strength comparable to that of the earth's (i.e., 0.5 x 10-4) has a negligible influence on the cavity resonances. This result was predicted by Madden and Thompson [1965]. It is interesting to note, however, that for a larger value of Bo the resonant frequencies are lowered somewhat. Also, asymmetries between/i,/-i and/2,/-2 are evident.f These asymmetries, which occur only in the anisotropic case, give rise to what is known as "line-splitting."

TABLE 1

Bo X 104

0.0 0.5 5.0

50.0 500.0

R e / i

7.61 7.61 7.57 7.04 4.12

R e / 2

15.24 15.23 15.14 14.08 8.21

R e / - i

7.61 7.61 7.63 7.62 7.48

Re/-a

15.24 15.24 15.27 15.25 14.95

The results shown in Table 1 refer to the real parts of the complex resonant frequencies. The fact that the imaginary parts are nonzero means that the resonant oscillations are actually damped with time when the cavity is shock excited [e.g., Wait, 1964]. In fact, for an impulse-type excitation, the energy density in the air space in the cavity decays with time t approximately in the manner exp (—4^|Im/m|i). A commonly used criterion for the ability of the cavity to maintain the oscillations is the quality or Q factor which, for mode

t It is interesting to note that an electrostatic solution (i.e., m = 0,/Ό = 0) is also implied by (2), but this case is of little interest to us here.

Cavity Resonator Modes in a Cylindrically Stratified Magnetoplasma 549

m, may be defined by the relation Qm = Re/m/|2Im/m|. The Q factors of the system for the same range of parameters are given below in Table 2. These numerical results indicate that the Q of the cavity region increases significantly with the magnitude of the impressed d-c magnetic field, at least for the range of values indicated. This effect is opposite to that of the radial component of the magnetic field in the spherical earth-ionosphere system; i.e., the radial component of the earth's field is known to increase the damping of the cavity resonances [Wait, 1965]. The fact that the Qm increase from m = 1 to m = 2 is in accord with earth-ionosphere measurements. However, the actual Q values shown in Table 1 are unrealistically large because of the excessive value of a\ — tfo used in the calculations.

TABLE 2

Bo X 104

0.0 0.5 5.0 50.0

Ôi

152 152 252 1760

02 212 214 399 3520

Q i

152 152 224 1910

ß-2 212 212 402 3820


We make no attempt here to give a detailed comparison of the present results with the extensive experimental data available for the actual earth-ionosphere cavity. The cylindrical model adopted here is intended only to demonstrate, in a rather qualitative fashion, some of the features present in the actual spherical system. It is unfortunate that quantitative solutions for spherical cavities with anisotropie walls are usually restricted to azimuthal symmetry [e.g., Wait, 1965]. In these cases, no line-splitting phenomena occur.

Assistance with the computer programming was obtained from Jean Troyer and Mathew Lojko of the Radio Standards Laboratory, National Bureau of Standards. The comments of Dr. Robert Gallawa are also much appreciated.

7. REFERENCES

BALSER, M., and WAGNER, C. A. (1960) Observations of earth-ionosphere resonances, Nature 188, No. 4751, 638-41.

DWIGHT, H. B. (1948) Tables of roots for natural frequencies in coaxial-type cavities, / . Math. Phys. 27, 84-89.

CALEIS, J. (1965) Schumann resonances, Radio Sci. J. Res. NBS 69D, No. 8, 1043-56. MADDEN, T., and THOMPSON, W. (1965) Low frequency electromagnetic oscillations of the

earth-ionosphere cavity, Rev. Geophys. 3, No. 2, 211-54. WAIT, J. R. (1964) On the theory of the Schumann resonances in the earth-ionosphere

cavity, Can. J. Phys. 42, No. 4, 575-82.


WAIT, J. R. (1965) Cavity resonances for a spherical earth and a concentric anisotropie shell, / . Atmospheric Terr est. Phys. 27, No. 1, 81-89.

WAIT, J. R. (1966) Transverse propagation of waveguide modes in a cylindrically stratified magnetoplasma, Radio Sci. 1, (New Series), No. 6, 641-54. (See previous paper in this collection.)

Electromagnetic Propagation in an Idealized Earth Crust Waveguide]

JAMES R. WAIT

Abstract. An electromagnetic waveguide in the earth's crust is idealized as a homogeneous lossy dielectric slab which is bounded by two homogeneous semi-infinite conducting media. The source is a vertical electric dipole which is either above or below the upper surface of the horizontal slab. The exact integral formula for the field is evaluated by function-theoretic means. It is shown that the dominant contributions are the propagating waveguide modes in the slab, although the lateral waves should not be ignored. The excitation of the waveguide modes and the relevant height-gain and depth-gain functions are also discussed. The attenuation characteristics of the modes are considered in some detail. It is shown that the principal loss mechanisms are the ohmic dissipation within the waveguide and the energy transmitted through the upper and lower walls into the bounding media. A numerical example is chosen where all three loss mechanisms are of the same order of magnitude.

1. INTRODUCTION

There is evidence that extended layers of low-conductivity material exist in the earth's crust. { The possibility that these layers may guide electromagnetic waves with low loss is intriguing. The analogy with VLF electromagnetic waves in the earth-ionosphere waveguide immediately suggests itself. A major shortcoming, however, arises in that the excitation mechanism for subsurface waveguides is inherently poor unless, of course, the source itself is located within the semi-insulating stratum. Also, there is a fundamental difficulty in that nearly all geological materials in situ are electrically conductive by virtue of connate waters in the pore structure.

It is the purpose of this paper to present some theoretical solutions which should provide insight into the mechanisms of propagation in subsurface waveguides. To reduce the complexity, the model is highly idealized. Nevertheless, it is believed that the inherent features of this type of subsurface propagation are adequately displayed.

f Radio Sci. 1 (New Series), No. 8, Aug. 1966. The work in this paper was supported, in part, by the Advanced Research Projects Agency, Washington, D.C., under ARPA Order No. 183-62.

X Special issue on Electromagnetic waves in the earth (May 1963), IEEE Trans. Ant. Prop. AP-11, No. 3, 207-370.

551 NN

2. FORMULATION

We start with a simple model which is illustrated in Fig. 1. As indicated, the subsurface waveguide is bounded by two semi-infinite homogeneous media. The source is taken to be a vertical electric dipole which, for the moment, is in the upper medium. The conductivity, permittivity, and permeability of the media are σ*, e*, and μχ, respectively. The subscript / takes

z = ^h '

>

<*1 » ε Χ » M l

σ2 , €2 , μ 2

σ 3 , e3> ßs

t

z

FIG. 1. The homogeneous lossy dielectric (subsurface) waveguide bounded by homogeneous conducting regions.

the values 1, 2, and 3 when referring to the upper, middle, and lower regions, respectively.

With reference to a cylindrical coordinate system (p, <f>, z), the waveguide (i.e., the middle region) is bounded by the planes z = 0, and z = hi. The source dipole, which has a current moment Ids, is located at z = — h. The fields, in any of the regions, may be derived from a Hertz vector which has only a z component Πΐζ (i = 1, 2, or 3). In what follows, the time factor is taken to be exp (iœt).

The formal solution for the problem, as stated, is known [Wait, 1962].| For the upper region,

t Wait, J. R. (1962) Electromagnetic Waves in Stratified Media, chapter II, Pergamon Press, Oxford. (Gives numerous references to earlier papers.) (See main text of present

volume.)

Electromagnetic Propagation in an Idealized Earth Crust Waveguide 553

7dj Γ6χρ(-&ιΚ) expi-ilaR') 1

where

Ä = [p2 + (* + Κ)ψ\ R = [p2 + (2 _ A)2]i/2)

^ ι = [(ίμιω) (σι + /^ιω)]1/2, and Fis an integral which is a spectrum of plane waves. If the upper surface of the waveguide were effectively a perfect magnetic conductor, F would be identically zero. However, in general, it is given exactly by the integral

J - o where

_ Kz + K2 tanh (u2h2) 2 2 ^2 + ^3tanh(w2/*2)·

where

Ki = ~ V - , Ui = (Λ2 - ^ 2 ) 1 / 25 ^ 2 = - (i/Ηω) (at + ι *,ω),

and Η^Χλρ) is a Hankel function of the second kind of order zero and argument Xp. The integration contour for F is along the real axis of the λ plane. For this contour, the square roots for u% are chosen so that the real parts are taken positive. On the other hand, the real part of ki is positive and the imaginary part of ki is negative.

The integrand in (2) has a rather complicated analytical form, and any closed-form evaluation is out of the question. However, we may achieve some simplification if we deal with the case where the conductivity σ2 of the middle region is small compared with the conductivities σ± and as. Thus, the situation is similar to the earth-ionosphere waveguide and, therefore, we anticipate that the important contributions to the received field come from the waveguide modes propagating in region 2. However, some important differences exist. In the first place, the source may be within the upper highly conducting region. Also, we wish to investigate whether or not the (usually neglected) lateral waves will contribute in any significant way. Thus, it seems that a careful evaluation of the integral F is warranted.

3. THE RESIDUE SERIES

To recast the solution in a form for displaying the waveguide character of the problem, we write

2Kl = / ( * ) , „ p. * , ,...7Tv (3) Ki + Z2 y v ' \ — i?«J?jexp(—2u2h2Y


where Ru and Ri are reflection coefficients defined by

Ru '' K2 + Äi' A = κ2 — K3 K2 + Kï and

2Κχ(Κ2 + Kz tanh u2h2)(ì + Q~2u^) Λ ) ~ (K2 + tfiX/fc + K2)

From the identity given by (3), it is evident that the integrand of F has poles at λ = λ where Xj are solutions of

1 — RuRi exp (-2u2h2) = 0. (4) By making use of this modal condition, it is a simple matter to show that

f(h) = 4KiK2

Kl- i Kfh ■ λ} (5)

The locations of the pole singularities Xj in the complex λ plane are indicated in Fig. 2 by small crosses. They are in two sets located (skew-symmetrically) in the second and fourth quadrants. The integrand also has branch points at λ = ±ki, ±/c2, and ±&3. Following the usual prescription, the original contour is deformed in such a manner as to enclose the pole and branch point singularities. The process is illustrated in Fig. 2. For reasons

Deformed contour

FIG. 2. The complex λ plane and the integration contours.

which are evident later on, the branch lines are drawn from ki, k2, and £3 down to infinity in the lower half plane. (This is appropriate when the distance p is somewhat greater than both h and z.) The contour is closed by a semicircle in the lower half plane. By an application of Jordan's lemma, it may be shown that the contribution to the integral from the latter vanishes as the radius of the semicircle approaches infinity. Thus, we are left with the contribution from the portion which encloses the poles λ;· and the contributions from the branch lines. It may be demonstrated that for the present problem the branch line integration associated with k2 is identically zero.f Thus, it does not play any further role.

The pole contributions to F are designated Fv and, by the theorem of residues, it is given by

Fp = _Ì7T yßXi)H?K*JP)Wui, ,·)-1 exp [ -m, jjh - z)]

[d\(l " Rn Ri exp (—2u2h2)) (6)

x=xj where m, j = (A* — k'f)1^2. In order to exploit the analogy with the earth-ionosphere waveguide, we write

λ, = k2Sj = k2(i - qyi\

where Sj and Q are the sine and cosine of a complex angle associated with the waveguide modes of order j .

By carrying out the differentiation indicated in (6) and making use of (4) it follows that

Fp s ~ | y C^E? W f e S y p ) exp [-uu ,(A - z)], (7) j

where 8, = __ J , (8)

I + l IkzhzRuRxlc-ci

« < ( > - $ * ) " -ik\ ik2

ηι = —~7—.— and η2 = Ι€\ω σ2 + Ι€2ω

The vertical electric field in region 1 which corresponds to the sum of the modes is obtained from

-=(* f +S) IdS FP] . (10) 4π(σχ + Ι€\ω)

t Actually, this follows immediately from the fact that Z2 is an even function of m. In some problems concerned with anisotropie media, this may not be the case.

Thus, under the condition that μι = μ2 = μζ = μο, we readily find that

E{2 = E<tk2\ktfW^ (11) where

Ζπ p

and

Wi = ~ξ? e'*.' 2 δ> c 3 Ä 2 ^o^Sjp) exp [-uh ,(A - z)]. (13) j

The corresponding expression for the field Εξ2 is obtainable from the general solution or, more directly, it is gotten by noting that

[k\E2z = k\El]z^

and by remembering that in the middle region U2z must satisfy (V2 + kl)U2z = 0. Thus, we readily find that

ΕΙ = Εο^2ΐ^Ψ2, 04) where

W2=~T2 ***' 2 Si ^ Δ 2 Ht2°)(k2Sjp) fl(z) CXP [~"1*}h]' ° 5 ) j

where βα,σ,χ + jty Q-ik2ciz

fiV) = rqr^ü) > 0 6) and where

RU) = _J? ; (17)

Another simple extension of the solution is that when the source itself is located in the waveguide. The corresponding solutions are then

Ef2 = Eo(k2lk1fWi (18)

E*x = EoWi (19)

where the primed quantities W[ and W'2 have the same form as W\ and W2 except that exp [—ui, jh] is now replaced by fj{h') where z = h' is the location of the source. The function W'2 is identical in the form to the corresponding result derived for the earth-ionosphere waveguide.

4. THE LATERAL WAVES

We shall now discuss the (nonvanishing) branch line contributions. Specifically, we shall consider the portion of the integration which runs

from ks — ico to ks on the left side of the branch line and then from ks to ks — ico on the right side. It is evident that the contribution to the integral F, which we denote F^ is given by

Çkt-l* Γ ΚΛ ΚΛ Ί λ (20)

where Ζ{2+) and Zl

2~] are the values of Z2 evaluated on the right and left portions of the branch line contributions. We observe immediately that, provided \k3p\ > 1, the Hankel function may be well approximated by the first term of its asymptotic expansion. That is,

/ 2/ \V2

which indicates immediately that the major contribution to the integral i7*3* is when λ is near ks- This suggests that we introduce a new variable of integration s defined such that

λ = ks — is2.

Thus, on the ( + ) side of the contour we note that

us = (λ - ks)1!2 (λ + ks)1'2 = Q-^'^ilks - is2)1'2 z e - ^ ' M W 2 ,

whereas on the (—) side,

us = -Q-W*s(2ks - is2Y'2 Ξ -e-«*/4A(2fc8)1/?. The Hankel function in the integrand is thus

/ 2/ VI2

H^\Xp)^ \-j—\ e-«*s/> e-"1 , (21)

which emphasizes the rapid decay as s becomes greater than zero. Elsewhere in the integrand for F@\ we note that

U2 = (\2 - jt|)i/2 ~ (jfcg - jtJ)i/2

and

Ul = (X2 - kl)''2 ~ (kl - itf)V2 = /(/Cf _ £2)1/2.

Also, in order to achieve further simplification, we note that

tanh W2A2 ^ 1 — 2 e " 2 ^

provided \kshs\ > 1. With the approximations indicated above, the integral F<3> is thus pro

portional to Poo π 1 / 2

j^e^d,-^, (22)

provided the slowly varying factors are taken outside the integral. Thus, we find that

F<8> Ξ - 1 8 ki exp {-ikzp) exp [-2(fcg - kj)1^ - (k\ - kf^i\h - z)] k\ (k* - kt)ksp* Γ + (kl- k*p* kf\ *

(23) (kl - kffw kl

The vertical electric field corresponding to this wave is

F<3>~ Ids ir*Fm ~ -2iIàs^œkt expf-ifcsp) 12 = 4π(σι + ΐ€ΐω) 3 ττ k\ /c3/>2

X exp [-2(iki - fc!)1/2^2] exp [-(*§ - k^\h - z)\ (24)

This corresponds to a heavily damped wave, since the arguments of the first two exponential factors have real parts somewhat greater than one.

Similar arguments apply to the branch line associated with k±. The corresponding branch line contribution in this case leads to the result that Ε{ιζ χ p"2exp(—ikip), which is also very heavily damped, particularly in view of the relative largeness of |fci| compared with both |&2| and \ks\.

5. DISCUSSION OF THE MODE SUM

As indicated by (13) and (15), the actual field must be calculated as a sum of modes. Provided \k%p\ > 1, we must evaluate

Wu 2 £ (2ffff1/2 e'**' e"*-/4 V SjSp e"« W^1'a ,> (25) j

where Pj is an appropriate "height gain" or "depth gain". For example,

P,(1> = exp [-wi, j(h - z)]C*(Cf - Δ;)-ι (26) and

Pf = Mz) exp [-uhjh]Cf(Cf - Aj)-\ (27)

depending on whether the observer is in medium (1) or medium (2), respectively.

The summation in (25) is a sum of waveguide modes which have a complex propagation constant ik^Sj in the radial direction. The modal equation (4) may be written in the form

( ^ ) e - = exp[-2./(,·-*)], (28) where Δ = (&i/£2)(l - S2*:?/*!)1'2, S = (1 - C2)1/2, and a is defined such that

fa = —exp (aC), and C = «2/(^2).

Solutions of (28) yield the permissible value Q . Thus, we note that the attenuation of the mode of order y is —Im k2Sj nepers per unit distance. For convenience, we shall assign the mode numbers j such that the imaginary parts of — k2S\, —k2S2, —k2S%, —k2S^ . . . , are monotonically increasing.

For convenience, we now assume, for the modes of low attenuation, that |aC| <ξ 1 and |A/C| < 1. The modal equation (28) is then expressible in the form

C(a - i2k2h2) - 2A/C = -2ni[j - (1/2)]. (29)

While this appears to be simple, we must remember that both a and Δ are functions of C. Fortunately, however, for the dominant modes, these are slowly varying and, for a given mode they may be regarded as a constant. For example, since for modes of low attenuation S is near unity, we may approximate

Δ ; ^ Δ - (fci/feoo - k i i k i y v , which is equal to the (normalized) surface impedance for a plane wave at grazing incidence on to the upper surface (i.e., at z = 0).

The solutions of (29) are

2π (y - 1Λ ± Γ(2ΤΓ)2 (/ - ^ 2 + SM(2k2h2 + ια) Cj 2(2k2h2 + ια)

11/2

(30)

When the + sign is taken before the radical, the solutions for Q reduce to

2, {J -1) C = C<° > = . , \ , / (31)

under the condition Δ = 0. Thus, regarding the second term within the radical of (30) as a perturbation, we readily find that

k2S} s Âr2S;l0) —^L. , (32)

h> (l + A ) ' where

Sjo> = [ l _ (C<0))2]1/2.

To illustrate the application of the above results, we assume that the lower medium has a conductivity which is not too great. First of all we have the exact result

hfn κ^-_κ3 n±A ~^ _, = _

C+kz

Clearly, this may be approximated by

2C

kz]

or Ri £ - 1 + aC,

where a = - 2(*8/it2) (1 - ifcîS2/*!)-1/2.

The simplification indicated above is evidently valid for modes where \aCj\ <ζ 1, which means that the boundary is acting as a "magnetic wall," since the reflection coefficient Ri is near — 1. Furthermore, for the modes of low attenuation, | Q | is small and, thus,

a - - 2/Δ,,

where Δ* = (fe/fcs) (1 - kllk§w

is a normalized surface impedance for plane waves at grazing incidence onto the lower interface (i.e., at z = A2).

Setting Ϊ

z *1/*| = *r - 7

where ^Tr and L are real, it is evident that

2 /α ~ ( ^ 1 / 2 - Z ^ ) ( c o s ( ^ - ^ / s i n ^ - ^ ) )

[1 + L2(AV - l)2]i/2

(35)

where , Φ = - arc tan L(^r — 1).

This result may be used in (32) for calculating kzSj. Further simplification is now achieved by choosing the loss tangent δ in

the waveguide region to be very small. Thus,

k2 Ξ N O - /δ/2) where δ < 1.

At the same time, the upper medium is taken to be well conducting. That is,

ki £ |fci|e<7r/4, provided *ιω/σι < 1.

As a result, \k%\ ^ A^W^o, where N% = (^2/^ο)1/2 is the refractive index of

the waveguide medium, δ = σ2/(<Γ2ω), and \ki\ £ (σχμοω)1!2. Under these conditions, we then find that

„ / / N (cJWKrLW cos - L-i/2 sin) ΙΔΙ k2Ss Z N2(œlco)sj + A 2 [ 1 + W r _ _ 1 ) 2 ] i / 2 + 2172^

-4 NMcoWP + h2[l+LKKr_mm + 2Ï7i/T2 - (36) where

|Δ| ~ |ÀTi/Aa| S (e jw/oiF = (eow/oi)1/2^, «r S «8/«a S (esl€o)Nf, L S €2ω/σ3 = (€οω/σ3)ΛΓ2

2, J, = (1 - C*)1/2,

-M) ii-jb C / = " W Â T = Afe*, ' C° = (€0/i0)_1/2'

sin = sin (2~~ 0)> and cos = cos \-z — ^ ) .

If in the somewhat unlikely case that e2 = e3, then ATr = 1 and ψ = 0. Then (36) may be written in the somewhat simpler form,

- / [ * * Μ « + » | Μ ] . (37) When N% = 1 and δ = 0, corresponding to a vacuum for region 2, the situation becomes fully analogous with the earth-ionosphere waveguide.

In the latter case, L = ων\ω\ where ωο and v are the plasma frequency and collision frequency of the ideal sharply bounded ionosphere. In this situation, Ä2 is the ionospheric reflecting height while σι and ei are the equivalent ground constants.

In order to illustrate the order of magnitude of some of the relevant quantities, we select an idealized earth crust waveguide with the following properties :

fi2 = 500 m, σ2 = IO-6 mho/m, e2 = 7V|e0 = 9e0,

σι = IO-1 mho/m, €ΐω/σχ <ζ 1, σ3 = 5 X IO-4 mho/m, €3 == 9eo,

and the operating frequency is ω/2π = IO6 c/s. For the conditions stated, we find that

Ci -H)/*

which is relatively small for the low-order modes (i.e., j = 1, 2, and 3). Thus, Sj £ 1 — (Cj/2) £ 1 for most cases of interest.

The attenuation is conveniently broken into three parts. Thus,

—Im kzSj = aw + <*>u + <*z,

where aw is the attenuation due to the ohmic losses within the waveguide, whereas au and ai are the attenuation due to the losses associated with the upper and lower walls, respectively. For the present example, it is found that

60Τ7 aw = ^ΙΜδ/2 = ΤΓ °^SJ = 0.063 x Sj nepers/km,

iV2

a« (eoœ\V2N2

at z - - 21/2/, 2 Ξ 1/ - j I X 0.0282 nepers/km.

It is evident for the example chosen that all three components of loss are comparable to one another. However, for the lowest-order modes (e.g., j = Ì and 2), the dissipation in the upper medium is the dominant loss mechanism. However, as the mode number increases, the loss in the lower medium becomes more important. For example, aj > au when the mode order j > 3. On the other hand, aw due to the dissipation within the waveguide does not depend significantly on mode number, at least, provided the modes are not near cutoff (i.e., c? not near unity).

It is interesting to note from the above that ai is approximately proportional to A~3 whereas au is'only proportional to A^1. Thus/for thinner waveguides (i.e., A2 < 500 m), ai will become dominant, whereas the converse is true for thicker waveguides (i.e., A2 > 500 m). Frequency, of course, also plays a role. It is evident (provided δ remains < 1) that aw is, to a first order, independent of frequency. On the other hand, au is proportional to ω1/2, while αχ is roughly proportional to ω~2.

Of special interest is the factor L1/2 + L-1/2 in the expression for a;. This has a broad minimum where L = 1, which is the case of our example. Thus, if the conductivity σ3 is less than or greater than 5 x 10-4, the attenuation will be increased somewhat (provided, of course, all other parameters are kept fixed).

For the example chosen, we may also go back and compute the excitation factors for the various modes. However, for the modes of lowest attenuation, it appears that the factor δ; is within a few percent of unity for the example chosen, provided./ < 8. Also, the quantity SjC*(Cf — Δ|)-ι in (13) and (15) may be replaced by unity for all intents and purposes. On the other hand, the ratio falkiY appearing in (14) and (18) plays a key role. For the example considered above, its magnitude is 5 x 10~3, which suggests that every effort

should be made to locate both the source dipole and the observer within the waveguide. Otherwise, some other mechanism of excitation should be used such as a horizontal electric dipole.


In this paper, a rather idealized model of an earth crust waveguide has been considered. Nevertheless, it is believed that the three contributions to the attenuation rate will be the limiting factors in radio propagation under more realistic conditions. Which loss mechanism is dominant will depend in a rather complicated manner on the numerous parameters of the problem. However, it appears the concept that the lower boundary of the waveguide may be equivalent to a magnetic wall is worth exploiting.

7. LIST OF PRINCIPAL SYMBOLS

<*u €u M electrical constants of three regions (/ = 1, 2, and 3) (p, φ, z) cylindrical coordinates

/?2 thickness of lossy dielectric slab Ids current moment of source dipole

h height of source dipole above slab Πΐ2 z-component of Hertz vector for region 1

k\ = — ΐμιω{σί + ietœ) is the square of the horizontal wave numbers for region /

ui = (λ2 — kffil2 is the vertical propagation constant for region / Ui

K* ■—;— is the vertically looking wave impedance for region /

F is a basic integrafdefined by (2) Fp is the contribution to F from the poles

FW is the contribution to F from the branch line integration associated with the branch point at λ = k%

Z% which is defined after (2), is the surface impedance for the upper layer

Ru and Ri are reflection coefficients H{

02)(Xp) Hankel function of second kind of order zero

j mode number Sj normalized vertical wave number Cj normalized vertical wave number δ;· excitation factor defined by (8) Aj normalized surface impedance for upper interface

Fu vertical electric field in upper region Efz contribution to E\z from poles (i.e., the waveguide modes)


£((1,3) contribution to E\z from the branch points at k\ and kz (i.e., the lateral waves)

Wi, 2 attenuation functions for the sum of modes a defined by Ri = — exp (—aC) and approximately given by

(35) δ loss tangent for lossy dielectric slab

JVf dielectric constant of slab relative to free space Ki and L real parameters defined by k\\k\ = Kr — iL~x and approxi

mately given by Kr = e3/e2 and L = €2ω/σ3 and ai are components of the attenuation rates resulting from the

losses within the waveguide, in the upper wall, and in the lower wall, respectively.

Illumination of an Inhomogeneous Spherical

Earth by an LF Plane Electromagnetic Wave] JAMES R. WAIT

Abstract. Starting with a previous formalism, a method is developed for computing the tangential magnetic field on the surface of an inhomogeneous spherical earth for excitation by a plane wave. The model consists of a two-section earth which is piece-wise homogeneous. As has been shown before, the line integral approximation for the pattern function may be evaluated in terms of various representations for the pattern function and the groundwave attenuation functions for homogeneous earth. In the present paper, the results are cast in a form suitable for numerical work. This involves some nontrivial transformations of the relevant contour integral representations. Some specific examples for 50 and 100 kHz show that a marked recovery effect takes place as the receiving antenna is moved from land to sea.

1. INTRODUCTION

The reception of a downcoming ionospheric skywave at low frequencies is influenced to a considerable extent by the properties of the ground around the receiving antenna. Because of the low grazing angles involved, geometrical optics is not valid when calculating antenna patterns in such applications. In an earlier study [Wait and Conda, 1958] of this problem, the earth was assumed to be homogeneous. In this note, we wish to indicate the extensions required to account for certain types of inhomogeneities which occur, for example, when the receiving antenna is located in the vicinity of a coastline.

The analytical formulation is applicable to any spherical (or cylindrical) surface for the range of parameters indicated. However, the specific examples shown correspond to frequencies of 15, 50, and 100 kHz for two-section paths on a spherical earth with atmospheric refraction neglected.

2. GENERAL FORMULAS

A cross-sectional view of a spherical (or a cylindrical) surface is indicated in Fig. 1. A vertically polarized wave is incident from the left. The observer is at a point P which is at an angular distance Θ from the horizon point T for the incident ray. One should note that Θ is negative if P is to the left of OT (i.e., the observer is on the illuminated part of the surface).

For the portion of the path to the left of Z>, the surface impedance is t Radio Sci. 2 (New Series), No. 1, Jan. 1967.

565


(2)

constant and is denoted Z. To the right of D, the surface impedance is constant and is denoted Z± which, in general, is different from Z. The field at P is proportional to the pattern factor V'(ß, Z, Zi) which is a function of Θ, Z, and Z\. It may be expressed in terms of the following normalized parameters:

q = -i{kaßyi\ZM qi = -UkaffliyZiho) X = (ka/iy-m xi = (kaliymx = {kaßfl\dila\

where k = (2π/λ), λ is the wavelength, ηο = 120ττ, and a is the radius of curvature of the surface. The integral formula for the pattern is [Wait, 1961, 1964]

where j r Q-iXt

and

^ , , , , ' ί Τ ί ^ , . (3) Here, the contour Γ for the integrals in the complex t plane may be taken from oo e~*27r/3 along a straight line to 0 and then out along the real axis to +00. In addition,

wi(0 = V(r)[Bi(t) - iAi(t)l (4) and

WiW = V t o l W ) - iAi\t)l (5) where y4/, Bi, Ai', and J5// are Airy functions in the standard notation [Miller, 1946].

It may be noted that if Zi = Z (or q\ = q), the pattern function V becomes equal to V(X, q), which is closely related to the pattern factor for a homogeneous surface. Thus, the integral from 0 to x\ is the correction due to the section of the surface with surface impedance Zi. The integrand involves the pattern factor V(X — X, q) and the groundwave attenuation factor W(X, qi) for surfaces of impedance Z and Zi, respectively. Numerical values for these two functions are available for various special conditions. Thus, in principle, the integration from 0 to x\ could be carried out directly by numerical integration. A more attractive procedure is to replace the functions V and Win the integrand by series expansions which are valid over the whole range of integration of X from 0 to x±.

3. ALTERNATE EXPANSIONS FOR V(X, q) Expanding the exponential in (2) leads directly to the power series repre

sentation œ V(X,q) = Ian(g)X«9 (6)

Illumination of Earth by an LF Plane Electromagnetic Wave 567

where Q—inn/2 Ç fn

αη(4) = "»i /pi ~ΤΛ 77~\ άί· (7) 2^im\)v wL(t) — qwi(t) v y

The integral may be decomposed as follows : e-^/2 r r ^ + e-^/3 Γ ιηΛ i (8) an{q) =

where we have made use of the rotation formulas

Wl(r ο-^ΐη = e-^/3w2(r) = e-*«/**W[Bi(r) + iAi(r)]9 (9) and

w'x(r e-*2*/3) = e^/3Vi;;(r) = e*»/W*[Bi'(r) + M/'(r)]. (10)

In the important special case q = 0, it is found that

ö»(0) = — ^ - [Μη + e : ^ « « : ] , (11)

where _ _L f00 tnAt 1 ΓΓ00 fwdf

(12)

and M* is the complex conjugate of Mn. The integral in square brackets in the preceding expression has been evaluated numerically by Riley and Billings [1959].

It is seen that (6) may also be obtained by writing (2) in the following equivalent form :

v(Yn\- 1 Γ Γ « Ρ (-**') , exp [ - (V(3 ) -Q^ /2 ] 1 n*,9) 2 7 T i / 2 j o [w;(0-îwi(0^wa(0-iexp(-/27r/3)iV2(0j ' K }

which, itself, is suitable for numerical integration provided | X\ is not large (say, <3).

A residue-series representation for V may be obtained by noting that the integral (2) has poles at t = ts, where

wl(0 - qwi(t) = 0. (14) Thus,

00

Σ Ζ-iXt

fc=rtïW 05) s = l

which, of course, is only usable when Zis somewhat greater than zero. It is worth noting that the roots ts are related to the roots rs of the follow

ing equation, which involves Hankel functions of orders 1/3 and 2/3: 8 e,,/, gj[(l/3)(-2Ts)3/2J_ „ Ò e / f<f)[( l /3)(-2r s )3/2]- ( 2τ«> ' <lb>

oo


where δ = — /(^o/Z)(fca)_1/3. This follows from the identity

wi(0 = exp [-27Γ//3] (φψ\-ίγΐ^Ι\{2β){-ΐ)^1 (17)

which also shows that ts = 21/3rs. The roots rs have been investigated in great detail by van der Pol and Bremmer. A good summary of their work is given in Bremmer's book [1949].

While the series expansion (6) for V(X, q) is suitable for values of X in the vicinity of zero, the residue series (15) is useful when Xis a positive quantity somewhat greater than zero. For large negative values of X, a stationary phase evaluation of (2) yields

v(x a) „ exp(i^/3) ηχ>ν = i _ KqjXy (18>

As discussed elsewhere [Wait and Conda, 1958], this simple formula corresponds to the predictions of geometrical optics, since the observer is well within the illuminated region. Logan [1959] has developed an asymptotic expansion for V(X, 0) which is of the form

Γ 1 1 469 V(X9 0) ~ exp (/ΛΤ3/3) 1 - / — + - + / I 6 ' 64 XQ

5005 _ 1122121 1 + 64 X™ ~l 1024 χπ + · · ·_! ' ( 1 9 )

which is adequate for computation provided — X is somewhat greater than about 1.5.

4. ALTERNATE EXPANSIONS FOR W(X, qi)

The groundwave attenuation function W{X, q{) as defined by (3) is the other part of the integrand in (1). It is important to note that the argument ranges from 0 to x±, and thus a suitable representation for W(X, qi) must be valid over this range. The well-known residue series representation [Bremmer, 1949],

s - 1 , 2 , 3 , . . .

is obviously useful over only part of the integration range, since its convergence becomes bad for small values of X. A more satisfactory approach is to employ the following expansion [Wait, 1961, 1964]:

W(X,qi)= Σ 4 e » ^ ; ( J ) ^ (21) m - 0 , 1 , 2 , . . .

where Ao = 1, A\ = —iy/π, Az = —2,


(i+4)' ^=-Ml+^+aî)'etc· (22) iVl/2 As = — ~zr-

5. SERIES EXPANSIONS FOR F^Z, q, qi)

Using (21) in conjunction with (6), it readily follows that (1) may be written

V(X9 q9 ?i) = V(X9 q) + ^ = | f f o»( i )^foi) e ^ ^ 9 » P n> , (23)

where Cxi

Pnf m=l = \Xl {X- Xnxfj1 dX. (24) 2 Jo

The integral Pn, m-i may be reduced by using the following formula: 2

r » » » " ™ - · -f- - ^ η - 1 . 2 + 1 · ^ ^

Successive applications enable Pw, m_-i to be expressed in terms of Po, m where 2 2

Λ , ( J ) 2 ' d J - ^ — . (26) o m + 1

The above double-series representation for V converges quite rapidly when both \X\ and x\ are somewhat less than unity.

6. SIMPLIFIED APPROXIMATION WHEN d\ is SMALL

A slightly different approach is to rewrite (1) in the following form:

where o _ g i -g Γ1 r (*-*U)m£gi) . s. m .

The quantity ß is the fractional change of the pattern function as a result of the inhomogeneity extending from 0 to xi. This suggests that we utilize the expansion

V(X-X,q) _ ^V\X,q) (X? V"(X,q) _ V{X,q) AV{X,q)^ 2 V(X,q) " · K£*>

to obtain a useful representation for the integrand in (28). An interesting and important limiting case occurs when important limiting case occurs when

V'(X,q) \X\- <i.

where ika

Pi= - - S -

(31)

V(X,q) This inequality would be satisfied for \q\ -4 1 and small values of \θι\. Then,

(W)l/2jo (X)11* ( j

Using the expansion given by (21), it readily follows that

2/[1_-^Ζ/Ζ0] Γ _ / ( ^ _ 2 - »ri/a ^ 1 [ 2 3 ^

IW * The formula for ß given by (30) and the resulting expansion in (31) are appropriate when the section of length d\ is small, and at the same time Z\ is sufficiently large that |/?i| is comparable with or greater than unity. For example, if the path of surface impedance Z is sea water while the portion with surface impedance Zi is land, then (31) should be applicable, and at the same time 1 — (Z/Zi) may be replaced by unity. In terms of the conductivity σι and the dielectric constant ei of the land,/?i, the numerical distance factor, may be written

wdi / €0ω \ / Ϊ€0ω \ pi=Yc U T / W i1 -^+ΰ^ί' (32)

At low frequencies where ειω <ξ σι, it is seen that pi is a real quantity and, therefore,

Thus, to a first order, ß ~ Ω' + /ß" , (34)

where fl' = -/>i and β " = -(2/ττΐ/2)^ι/2. (35)

Consequently, the fractional reduction of the amplitude is/?i, while the phase lag is increased by 2(/?ι/τ7)1/2 rad. For example, if the length d\ of the land path is 100 km, for a land conductivity ag of 1 mmho/m and at a frequency of 15 kHz (i.e., λ = 20 km), it is easily found that

ß ' ~ -2 .62 x 10-2 a n d Q» ~ -0.129 rad = -7 .4° . (36)


For this particular example, it would appear that a 100-km stretch of land has a negligible effect on the launching efficiency. For a longer land section the effect could well be significant. However, in such cases higher terms in (31), which involve earth curvature, need to be considered, as indicated in the next section.

7. NUMERICAL EXAMPLE FOR LOW FREQUENCIES WITH d\ ARBITRARY

An important application of the present theory is to LF sky wave propagation. In this case, one is interested in the spatial variation of the field in the vicinity of a coastline for a downcoming plane wave. As an example, we choose the following parameters: σ = 5 X 10~3 mho/m, e/ o = 15, σ± = 4 mho/m, and ei/eo = 80. This corresponds to locating the receiving antenna on the sea a distance d± from the coastline. The amplitude of the pattern function V for this case is shown plotted in Figs. 2 and 3 for frequencies o 50 and 100 kHz, respectively. The parameter a = Θ — Θχ is the angular distance of the coastline from the tangent point T shown in Fig. 1. In other words, a = 0° corresponds to the situation where the coastline is located right at the "light-shadow" boundary in the classical sense of geometrical

FIG. 1. Cross-section of the two-section spherical earth for a plane wave incident as indicated.

1.0

0.5

>L 0.2

2 0.1 l·-

g 0.051

0.02

I i i i i I i i i—n

a = 0 - 0 , σ = 0.005 mho/m σ( = 4.0 mho/m

€/e0=!5 <T(/€0=80

f = 50kKz

I I i i 1 I i I I I i i i i 1 I I I I I I i I i 1 i i i I 1 i i 0.01 -15 -10 - 5 0 5 10 15 20

Θ (DEGREES)

FIG. 2. Amplitude of the pattern factor for a two-section spherical earth for a frequency of 50 kHz.

1.0

0.5

0.2

2 O-'tT

Ü

0.02 l·

0.01

_

-

-

-

—

-

k

1 1

ÌU-14JJJJ 1JJ4J 1 ' ' ' ' 1 ' ' '

α = - 5 ο / \ ν \

°' \ \ a = 0-0, V v σ=0.005 m ho/m ^ \ σ; =4.0 mho/m ^λλ

f = ΙΟΟ kHz ^ Λ

5 ° ^ V

L I T -·· SHADOW

1 1 1 1 1 1 1 1 1 1 1 I 1 1 I 1 1 1 1 1 1 1

1 | 1 1

1 0 1 ^

M M

1 M

-Ì

—

-J

~

-

~

1 -15 - 1 0 - 5 0 5 10 15

Θ (DEGREES)

FIG. 3. Amplitude of the pattern factor for a two-section spherical earth for a frequency of 100 kHz.


optics. Then, negative values of a correspond to locating the coastline in the lit zone, whereas positive values correspond to locating the coastline in the shadow zone. The curves in Figs. 2 and 3 then illustrate how the amplitude of the tangential magnetic field on the surface of the earth varies as the receiving antenna is moved from the lit to the shadow zone for various locations of the coastline. In both sets of curves there is an interesting "recovery" effect which takes place as the receiving antenna is moved from the land to the sea. This phenomenon also occurs in groundwave propagation across mixed paths [Wait, 1964] and it has received convincing experimental confirmation [King, Maley, and Wait, 1966].

An extensive set of calculations of the type shown in Figs. 2 and 3 is available in a report [Spies and Wait, 1967], which includes numerical tabulations and graphical plots for the complex pattern factor V\X, q, qi) as defined by (1). The frequencies considered are 20, 50, 100, and 200 kHz for the ground parameters indicated in the above example. This report also contains a detailed discussion of the numerical procedures used.


The angular distance parameter 0, shown as the abscissa in Figs. 2 and 3, is readily converted to actual distance d by multiplying by the actual earth's radius. Thus, for the airless-earth model used here, a = αθ = 6368 x Θ in km for Θ in radians. Consequently, 1° in the scale corresponds to 111 km. In previous calculations for homogeneous-earth models [Wait and Conda, 1958 ; Hyovalti, 1965], the earth's radius ae was taken as (4/3)# to account for atmospheric refraction. The equivalent angular distance 9e used in the previous calculations is then d\ae. It is evident that the two models are equivalent if the parameters X, x±, q, and q± are kept constant.

In the preceding discussion, the point P, indicated in Fig. 1, is imagined to be a small magnetic loop antenna with axis perpendicular to the paper. The results have been normalized such that V = V = 1 if the earth is flat and perfectly conducting. From reciprocity, the curves of V in Figs. 2 and 3 also apply if P is a transmitting antenna, in which case V is proportional to the far-zone radiated field.

I am grateful for the valuable assistance and collaboration of K. P. Spies, who carried out the numerical calculations leading to the curves in Figs. 2 and 3. The work reported in this paper was supported, in part, by the U.S. Naval Research Laboratories under Contract Order No. 00173-3-006354.

9. ADDENDUM

A convenient double-residue series for the pattern function V'(X, q, qi)

(39)

may be obtained by inserting the single-residue series formulas (15) and (20) into (1). Carrying out the integration leads readily to the result

V'{X,q,qx)=V(X,q)

s s'

The double summation may be simplified by first noting that, for any value of/,

.. . _ _ * J < ! ) _ = V — ! î-_ ß j n *Ί(0 - ÎHVXÎO Z , 0; - ?î) c - H) ' κ }

s' Then, setting t = ts and using the fact that w'^tg) = ^wi(^), we find that

1 ^ P 1 J q - qi ~ Z^ (X; - ql) (ts - t$ '

s'

Using (39), it is immediately evident that (37) simplifies to

v (x, q, qi)=,(?1 - q»» 2, z - Ä ^ ^ W - Ä s s*

which converges well when both (X — x{) and xi are sufficiently positive.

10. REFERENCES

BREMMER, H. (1949) Terrestrial Radio Waves, Elsevier Publishing Co., New York and Amsterdam.

HYOVALTI, D. C. (1965) Computations of the antenna cut-back factor for LF radio waves, NBS Tech. Note No. 330.

KING, R. J., MALEY, S. W., and WAIT, J. R. (1966) Groundwave propagation along three-section mixed paths, Proc. Inst. Elee. Engrs. (London) 113, No. 5, 747-51.

LOGAN, N . A. (1959) General research in diffraction theory, Tech. Rept. LMSD 288087, Lockheed Aircraft Co., Sunnyvale, Calif.

MILLER, J. C. P. (1946) The Airy Integral, Cambridge Univ. Press, Cambridge, England. SPIES, K. P., and WAIT, J. R. (1967) Calculations of antenna patterns for an inhomogeneous

spherical earth (to be published). RILEY, J. A., and BILLINGS, C. (1959) Gaussian quadrature of some integrals involving

Airy functions, Math. Tables Other Aids Computation, XIII, No. 65-68, 97-101. WAIT, J. R. (1961) On the theory of mixed-path groundwave propagation on a spherical

earth, / . Res. NBS 65D (Radio Prop.), No. 4, 401-10. WAIT, J. R. (1964) Electromagnetic surface waves, Advances in Radio Research, vol. 1,

pp. 157-217, ed. J. A. Saxton, Academic Press, London. WAIT, J. R., and CONDA, A. M. (1958) Pattern of an antenna on a curved lossy surface,

IRE Trans. Ant. Prop. AP-6, No. 4, 348-59.


BACH ANDERSEN, J. (1964) Fresnel zones for ground-based antennas, IEEE Trans. Ant. Prop. AP-12, No. 4, 417-22.

BACH ANDERSEN, J. (1965) Influence of surroundings on vertically polarized log-periodic antennas, Teleteknik 9, No. 2, 33-40.

GUSTAFSON, W. E., CHASE, W. M., and BALLI, N. H. (1966) Ground system effect on H F antenna propagation, Rept. No. 1346, U.S. Naval Electronics Lab., San Diego, Calif.

Radiation from Dipoles in an Idealized Jungle Environment]

JAMES R. WAIT

Abstract. A method is presented for computing the field of a vertical electric dipole over a homogeneous conducting half space coated with an anisotropie conducting slab. The asymptotic form of the solution indicates that the field varies essentially as an inverse square of the horizontal range. The results have application to HF radio propagation over smooth terrain with dense vegetation.

There has been a recent interest in the manner in which radio waves propagate through or over jungle-covered terrain. A discussion of current activity in this field is available in a recent publication [Katz, 1966]. Frequencies of interest are mainly in the HF band and distances are of the order of tens of kilometers. In some of the theoretical approaches, the jungle or other thickly vegetated ground is idealized as an isotropie conducting slab lying on a homogeneous conducting half space. In this note, we wish to suggest that the jungle cover might be more appropriately characterized by an anisotropie layer such that the average conductivity in the horizontal direction differs from that in the vertical direction. At the same time, we wish to point out that theory, previously developed for propagation in stratified isotropie conducting media, is applicable to the general problem. There is no need to rederive the standard results.

We consider an electric dipole of moment las located at height h over a two-layer half space which is illustrated in Fig. 1.

With respect to a cylindrical coordinate system (p, φ, ζ), the upper and lower surfaces of the anisotropie layer are z = 0 and z = —D, respectively. The source is located at z = h, where h is positive. The region z > 0 is free space with dielectric constant eo. The region z < — D is a homogeneous isotropie medium with dielectric constant e9 and conductivity σ9. The intermediate layer is characterized by a tensor dielectric constant (e) and a tensor conductivity (σ) of the respective forms

€h 0 0

(*) = I 0 en 0 I and (σ) =

I ah 0 0 \ 0 hn 0

0 0 ev I \ 0 0 σν I t Radio Sci. 2 (New Series), No. 7, July 1967.

575

where the subscript h refers to the horizontal direction, whereas the v refers to the vertical direction. In what follows, it is convenient to use the designation complex conductivity og to refer to the combination σ9 + iegœ and (σ) in place of (σ) + /(β)ω. Thus, ση and σν refer to the complex conductivities which describe the idealized jungle layer. The magnetic permeability is denoted μο which is assumed to be the same for all three regions.

The field at P may be obtained from a Hertz vector which has only a z component Ποζ. For a time factor exp (ίωί) it is given by

Ποζ = - Α ^ - [Q-«o\z-h\ + R, (A)e«o<*+*>] - /ο(λρ) dA (1) 4777 βοω J 0 WO

where

Χ ^ ^ Ί ^ τ Ψ ' K» = ^ I/o = (λ2 - fc2)l/2, fc = cu/c, Ao + Zi /€οω

_ „ K2 + ^ i tanh (vD) v u ATi + #2 tanh (i?D) σ σ

V = (A2/c + y2)1/2, κ = σηΙσν, y2 = /σΛ/χ0ω,

w = (λ2 + y*)1/2, and y^ = /σ^0ω.

The method used for deriving (1) is straightforward and has been described elsewhere [Wait, 1966] for a general stratified anisotropie half space.

Exact expressions for the field components in the region z > 0 are obtained by performing the operations Eop = d2IIoz/dpdz, Eoz = (k2 + d2ldz2)Uoz and Ηθφ = + i€oœdUoz/dp. Because of symmetry, it is evident that Εοφ = Hop = Hoz = 0 for this geometry.

The integral formula in (1) admits a physical interpretation. The first term in the integrand involving exp [—uo\z — h\] may be identified as the primary field of the source in the absence of any reflecting medium below. Thus, the second term accounts for the presence of the jungle-covered ground and, in fact, R\\(X) may be identified as a reflection coefficient for vertically polarized waves incident at an angle arc sin (λ/Α:) from the surface z — 0. Then, just as in isotropie stratified media [Wait, 1962], Ko9 Ki, and Κ% are characteristic impedances for waves whose propagation constants are wo, v, and w, respectively. Furthermore, Z\ may be identified as the input impedance of a transmission line of length D whose characteristic impedance is K\ and whose propagation constant is v, which is terminated in an impedance Kz. In the absence of the jungle layer, we see that D = 0 whence R\\(X) = (Ko — fa)! (Ko + fa). Inserting this into (1) leads to Sommerfeld's [1926] classical integral formula for the fields of a dipole over a homogeneous conducting

Radiation from Dipoles in an Idealized Jungle Environment 577

half space. The appropriate formulas for a two-layer isotropie half space are easily recovered from (1) by setting κ = 1.

We shall now write down an asymptotic development for (1) which is valid when the horizontal range p is sufficiently large. Following the treatment used for the stratified isotropie half space [e.g., Wait, 1962, ch. 2], we find that

Ta? n0:^(f» + W (2)

where

and

_ t-ikRQ c - Δ e~***i Δ _ Zi Ψα~ R0

+C + A Ri ' ~ηο (3)

λ = ^

1 x 3 1 x 3 x 5 ç-ikRi

(4) The various parameters entering into (2), (3), and (4) are defined by the relations Ro = [P

2 + (z - A)2F2, Ri = [p2 + (z + A)2]1/2, C = (h + z)/Ru S = plRx = (1 - C2y/2, p = - ikRi(A2l2), and ηο = Οο/*ο)1/2 = 120 π ohms. The geometry of the problem is evident from Fig. 2 where the angle Θ is arc cos C or arc sin S.

Clearly, φα is the geometrical optics part of the field at P. It consists of the direct or primary wave and a secondary wave which is computed as if there were a specular reflection from the surface z = 0 with a reflection coefficient (C — A)/(C + Δ). In this case, Δ is the normalized surface impedance for a plane wave incident at angle Θ onto the surface z = 0. Mathematically speaking, this comes about because the predominant value of λ in the integrand of (1) is kS, which is the saddle point.

It is important to note that, in addition to φα, we have a contribution J/*& which is given by (3) and it consists of terms which vary as \jR\> \jR\, . . . . At first glance, one might think that 0& could be thrown away for large ranges since φα varies as 1/Ri. However, please note that φα vanishes if C-> 0 or if Θ -> 90°, corresponding to grazing incidence along the top of the jungle. Thus, we need to retain both φα and φο in the general case.

Actually, there are some other contributions in the asymptotic development for Ποζ [Wait, 1962]. For example, under some conditions there is a surface wave which is excited when the phase angle of the surface impedance Zi is greater than approximately 45°. Also, there will be waveguide modes excited which propagate essentially in the intermediate jungle layer. However, both the surface wave and the waveguide modes are exponentially attenuated in contrast to the algebraic decay of the terms of φα and φο. This exponential

attenuation is of the order of (Re y) nepers per meter. Thus, for example,

.

t D

1

i z -

F -—· e o

%

%

σ V

P</>,z)

z

FIG. 1. Idealized model of an anisotropie jungle layer over an isotropie ground.

Z=h

/ Πρ,ζ)

z=-n >e-

. ^ É - - ^

FIG. 2. The geometry for the asymptotic development. Note that C = cos Θ.

if the mean jungle conductivity is 0.1 millimho/m, the attenuation for 1 MHz is of the order of 1 dB per meter. The incredibility of waveguide mode propagation in typical jungle-covered terrain was first pointed out by Burrows [1965].

In most cases of practical interest, \p\ > 1, so only the leading term in the expansion for φο is needed. An interesting case is that of h -> 0 such that the source dipole is just above the jungle top. Then,

Q-ikR φα + Φϋ~ — £ -

■ 2C_ C + Δ >Κί (5)

where Ji = O2 + z2)1/2. When both \C\ and |C/A| < 1, this result may be further simplified to

φα + Φυ ■9$)™ (6)

Radiation from Dipoles in an Idealized Jungle Environment 579

where Go(z) = 1 + ikzk is a height-gain function. In this case, the vertical electric field in the air at height z may be written

E0z = Eo(-llp)Go(z) (7)

where £b = -—τ~^~ e~ikp is the free-space field of the source dipole. Here, 4πρ

i(kPl2)A* and

ηοση where

gg(y2 + Kfc2)i/2

0 = Ä + Ai tanh (vD) I » + I ^ '■ L * Î „ tanh [(y« + ^ » D ] ΑΓι -f A:2 tanh (vD)

(9)

We see that β -> 1 when the jungle is of infinite depth (i.e., D = oo). A more explicit form of (7) is

2/y2 n - 2

^ = £ o ^r^F\ G o ( 2 ) · (10)

When the observer is located in the jungle (i.e., 0 > z > — D) but the source is still just above the jungle, we use a modified form of (7). Then,

Ez^ Eo(-llp)G(z) (11) where

ί€0ω [cvz + RUQ-2VDQ-VZ' G(z) = H —

σν L 1 +R\\e-to*> . (12)

+ R\\ and Â\\ = (Ki — Κ2)Ι(Κι + K2). The form of (11) may be verified by noting that G(z) is a superposition of upgoing and downgoing wave types within the jungle layer with an appropriate reflection coefficient at the ground surface. Also, we may note that, for z = 0, σνΕζ = Ì€QCUEQZ, which is merely a statement that the normal current density is continuous.

An explicit form of (12) is written

Giz) QVIZ _|_ r^Q~2vlDQ-vlz'

e v - ί(σνΙω) |_ 1 + r i l e v i D

where Vl = (y2 + „£2)1/2

(13)


and

P. r1 ..«»Μ + Ι / Γ , οηΜ + νγιη 11 ~ L ^ ( y 2 + ^ 2 ) 1 / 2 J ' L ^ ( y 2 + ^ 2 ) 1 / 2 J "

Usually, σ > σΛ and, therefore, f\\ = 1. In the above, we have located the source electric dipole at, but just above,

the jungle top. If the source be raised to a height h, then we generalize (7) and (11) to

EozZ Eo(-llp)Go(z)G0(h) (15) and

Ez^ Eo(-llp)G(z)Go(h)9 (16)

respectively. On the other hand, if the source dipole is located in the jungle such that h is negative, we merely replace Go(h) by the function G(h) (CTÄ/^)1/2

where G(h) is given by (13). The inclusion of the factor (σ/ι/σι,)1/2 is evident from a theorem given by Clemmow [1963], and a comment made by J. E. Spence [private communication].

Because the electrical characteristics of the jungle do not differ markedly from free space, the gain functions Go(z) and G(z) do not differ appreciably from unity. Thus, we expect that under most conditions the predominant feature is the (l/p) multiplicative factor in the field expressions. This leads to an inverse (distance)2 field dependence. Furthermore, this conclusion is not modified by a macroscopic anisotropy of the jungle medium which results when K Φ 1. Nevertheless, the quantitative aspects of the problem are certainly modified by the anisotropy.

I am indebted to C. R. Burrows and T. J. Doeppner who stimulated my interest in this subject.

REFERENCES

BURROWS, C. R. (1965) Ultra short wavelength propagation in the jungle, IEEE Trans. AP-14, No. 3, 368-88.

CLEMMOW, P. C. (1963) On the theory of radiation from a source in a magnetic-ionic medium, Electromagnetic Theory and Antennas, p. 474, ed. E. C. Jordan, Pergamon Press, Oxford.

KATZ, I. (Editor) (1966) Progress in radio propagation in nonionized media, Radio Sci. 1 (New Series), No. 11, 1341-50.

SOMMERFELD, A. N. (1926) The propagation of waves in wireless telegraphy, Ann. Phys. 81, 1135-53.

WAIT, J. R. (1962) 1st. Edition. WAIT, J. R. (1966) Fields of a horizontal dipole over a stratified anisotropie half space,

IEEE Trans. AP-14, No. 6, 790-2.


KING, R. J., and SCHLAK, G. A. (1967) The ground wave attenuation function for propagation over a highly inductive earth, Radio Sci. 2 (New Series), No. 7, 687-93.

WAIT, J. R. (1966) Electromagnetic fields of a dipole over an anisotropie half-space, Can. J. Phys. 44, No. 10, 2387-2401.

Comments on a Paper "A Numerical Investigation of Classical Approximations Used in VLF Propagation" by R. A. Papperty E. E. Gossard, and I. J. Rothmuller^

JAMES R. WAIT

WITH some interest, I have read that Pappert, Gossard, and Rothmuller have developed a comprehensive computer program to obtain numerical results for the propagation constants of an earth-ionosphere waveguide with an arbitrary electron-density profile. The overall agreement between their approach based on Budden's [1961] work and our method [e.g., Wait and Spies, 1960; Wait, 1960] is quite reassuring. If nothing else, it should lay to rest the controversies which have arisen in connection with the importance of accounting properly for earth curvature in calculating the characteristics of the least attenuated modes.

In their paper, Pappert, Gossard, and Rothmuller [1967] show good agreement with some of our numerical results for the first-order mode in daytime but they get only partial agreement for the second-order mode in daytime. Actually, the differences are not to be unexpected. As we explained in our technical note [Wait and Spies, 1964]:

"In summary, the calculations given in this technical note are strictly correct for an earth-ionosphere waveguide which is bounded by a homogeneous smooth earth and a reflecting layer at height h with a normalized surface impedance Δ. The applicability of the results to an actual diffusely bounded ionosphere requires a number of approximations which are justified mainly on physical grounds. The principal assumption is that Δ is assumed at the outset and then the mode equation is solved to yield the propagation characteristics. A fundamental question may arise in that the angle of incidence is complex for modes in a lossy waveguide. However, it is fortunate that the cosine Cof the angle of incidence at the ionosphere is approximately equal to (2h/a + C2)1/2 where C is related to the parameter t, in (13), by

(te/2)1/3C = (-01 / 2 . For all the important modes, |C2| <ξ 2h\a and, thus, C" is approximately equal to (Ih/a)1^ as assumed above. A small refinement is to use the above form for C to obtain a modified value of Δ (or a) which is then employed in

t Radio Sci. 2 (New Series), No. 11, Nov. 1967.

581


conjunction with the contour plots in section 12 to yield corrected values of the propagation parameters. The resulting corrections appear to be negligible for any situation considered in this technical note."

While it is true the attenuation rates differ by as much as 25 per cent for the daytime second-order mode, the point one should realize is that in such cases the total field at a distance of say 1000 km is only in error by a few per cent. Furthermore, for night-time conditions where the second-order mode is of practical importance, the approximation that C'may be identified with (2A/a)1/2

in estimating Δ is well justified since C is very small. This simplification is not an essential feature of the theory and does not restrict the validity of the contour plots given in section 12 of the referenced technical note.

Finally, I would like to point out that the differences in the formalisms between the two approaches are really not basic. The main point is that their modified Hankel functions h±(z) and h^z), apart from a constant, are the same as our Airy functions wvit) and wi(t), respectively (provided z = — t). Also, as discussed in some detail elsewhere [Wait, 1964], there is really nothing basically different between the modal equation in cylindrical and spherical geometry, provided h\a is sufficiently small, which is certainly the case. The main differences arise at very long ranges in connection with antipodal focusing.

REFERENCES

BUDDEN, K. G. (1961) The Waveguide Mode Theory of Wave Propagation, Logos Press, London.

PAPPERT, R. A., GOSSARD, E. E., and ROTHMULLER, I. J. (1967) A numerical investigation of classical approximations used in VLF propagation, Radio Sci. 2 (New Series), No. 4, 387-400.

WAIT, J. R. (1960) Terrestrial propagation of very-low frequency radio waves, / . Res. NBS 64D (Radio Prop.), No. 2, 153-204.

WAIT, J. R. (1964) Two-dimensional treatment of mode theory in the propagation of VLF radio waves, Radio Sci. J. Res. NBS 68D, No. 1, 81-93.

WAIT, J. R., and SPIES, K. P. (1960) Influence of earth curvature and the terrestrial magnetic field on VLF propagation, / . Geophys. Res. 65, No. 8, 2325-31.

WAIT, J. R., and SPIES, K. P. (1964) Characteristics of the earth-ionosphere waveguide for VLF radio waves, NBS Tech. Note No. 300.

On the Calculation of Mode Conversion at a Graded Height Change in the Earth-ionosphere Waveguide

at VLF\ JAMES R. WAIT AND KENNETH P. SPIES

Abstract. A multistep model is used to simulate the transition between the day and night portions of the earth-ionosphere waveguide. A convenient simplification is the justified neglect of all reflected modes. On the other hand, forward mode conversion (and reconversion) is accounted for. The analytical method involves the multiplication of square matrices which correspond individually to the conversion at the various steps. The calculations are carried through for a sharply bounded ionosphere and a perfectly conducting ground. Earth curvature is allowed for but the influence of the earth's magnetic field is neglected. A calculated example for 20 kHz indicates that the transition length Δ</ between day and night influences the amount of mode conversion.

1. INTRODUCTION

There is conclusive evidence [Crombie, 1964 and 1967] that the change of height of the ionosphere at sunrise will produce mode conversion for VLF radio waves. The analytical aspects of the subject have been discussed in a number of recent papers [Wait, 1964, 1968; Bahar, 1966]. As it turns out, rather simple models such as abrupt height changes will yield reasonably good agreement with the experimental results. A notable example is the work of Rugg [1967].

In this paper, we wish to make a number of extensions that, hopefully, provide new insight into the mode conversion problem. The same model is employed as in a previous paper [Wait, 1968]. Specifically, the transition from a constant daytime height to a constant night-time height is assumed to extend over a great-circle distance Δί/as measured on the earth's surface. To facilitate the analysis, the transition is approximated by a finite number of uniform sections. The number of steps in height change is designated N and they are located at a great-circle distance dj (j = 1, 2, . . ., N) from the ground-based source dipole at A. The receiving antenna is a ground-based dipole antenna at B. (The path connecting A and B is assumed to be perpendicular to the sunrise transition.)

t Radio Sci. 3 (New Series), No. 8, Aug. 1968. The research reported here was supported by the Advanced Research Projects Agency, Department of Defense, Washington, D.C., under ARPA Order No. 183.

583 PP

2. FORMULATION

Following common usage, we employ the "natural" parameters

and ^ = (2/L)W4| f0r7 - Ι ^ , 3, . . ., ΛΓ, ΑΓ+ 1 where hj is the height of the y'th section, k = 27r/wavelength, and a is the earth's radius. We note that d = dN+i while hi = /z(day) and hN+i = A<night>. The situation is illustrated in Fig. 1, where we show only the "natural" parameters. Here, xo = (ka/iy^do/a is an arbitrary reference distance from A which must be specified.

FIG. 1. The multistep model of the transition between day and night portion of an idealized earth-ionosphere waveguide.

For a mode of order m incident from A, the field received at B has the following explicit form for N = 3 :

£ » = ^ ° ^ e V ) y. C(l) e.-H(xi-x1)t'

^ S£« e-«<*8-*»>C V 5 $ e-*<*4-*s>'' (1)

where JSO is a reference field; Λ^} is an excitation factor for the source dipole at A in waveguide region (1). The coefficients t£\ t{*\ etc., are solutions of the waveguide mode equation appropriate for the region under consideration. The coefficient S(£m describes the amount of scattering into mode n for an incident mode m at junction (1). Similarly, the coefficient S™n characterizes the scattering into mode/7 for an incident mode n at junction (2). Finally, the coefficient S{^v accounts for the scattering into mode q for an incident mode of order p.

3. THE SOLUTION The scattering process described above amounts to a multiplication of

square matrices, each of infinite dimension. Under any practical scheme

On the Calculation of Mode Conversion 585

these may be truncated, since, from a physical standpoint, modes of sufficiently high order do not contribute if the number of steps remains finite. It is also central to our argument that reflections at each step are neglected consistently. For most applications at VLF to the earth-ionosphere waveguide, this is well justified. The derivation of an explicit expression for S^ is outlined in the appendix.

For the observer at B to the right of the transition indicated in Fig. 1, the field may be written in the form

Em = I * e - ' < > AlaUmi y Sfm e-«*4-*o>C (2)

where Sffm is the effective scattering coefficient, referred to X — XQ, for the whole transition region. Clearly (1) and (2), by definition, are identical. Thus, on equating the right-hand sides of these two equations, we get

Seff = e - i(^-a:0)i ( 1> V 5(1) mQ-i(Z2-*i)ti2)

n

x Σ S™n e-*(«8-*2)«™Sö», e-*<*o-*3>«(* (3) P

which, of course, does not involve the locations of the source A and the receiver B except to the extent that they must straddle the transition region.

4. ALTERNATE FORM

A form of (2), which is more convenient, is

Em = ^ ~ â e-^οί^ [A«>F 2 W 8 ^ » e-*<*4-*o>C, (4)

where

£rf = [AM'V (5)

This form, involving S*nm, is more meaningful than (2) since the excitation

factors for the source and receiver occur in a symmetrical fashion. For example, if mode conversion could be neglected we would simply replace Sf^m by 1 for q = m and by zero for q Φ m. Such a result is consistent with the approximate W. K. B. forms [Wait, 1964] that neglect mode conversion at the outset. Also, it is of some interest to see that the scattering coefficient S£n for propagation from night-to-day, is the same as S™q for propagation from day-to-night. This interesting reciprocity allows us to restrict our calculations for propagation in the direction of increasing width of the waveguide.

The generalization of the form for Sf^m or for SfJm, to any number of steps, is obtained by induction. First of all, we note (3) may be compacted to


n =-·■ 1 n = 1

^!m-S^mt-^m-^tn{}+1) (7)

^ J / ^ may be regarded as the (n, m) element of a matrix S?V\ while S%}m is the (n, m) element of a matrix S^K

The generalization of the scattering coefficient for the N step transition is thus

where P^, m is the (#, w) element of the matrix product P = 5(^0^(^-1)^(^-2) # # β ^ ( 1 ) .

The corresponding generalization of (5) is

°ff.w ~~ A ( i V + 1 ) vi ff.»»'

(8)

(9)

(10)

5. SOME NUMERICAL EXAMPLES

A sample of the calculations, using the approach described above, is given in Table 1. As indicated, the frequency is 20 kHz and the ionosphere conductivity, ωΓ, for the assumed sharply bounded isotropie ionosphere is 2 X 105

everywhere. The number of equally spaced steps varies from 2 to 6, and the distance Ad between the first and last step takes the values 200, 500, 1000, and 2000 km. The initial and final reflection heights are Ai = 70 km and hf = 90 km, respectively. The complex value of S^[ shown in the table is computed from (8) with XQ = (XN — *i)2 (i.e., the scattering coefficient corresponds to the effective value for a single step in the middle of the transition region).

In carrying out the calculations of the type indicated in Table 1, modes of order 4 and above were consistently neglected. This is justified mainly on the

TABLE 1. Complex values ofS^\ [hi = 70 km, hf = 90 km, ωΓ = 2 x 10 5 , /= 20 kHz]

Ad

2 3 4 5 6

200 km

0.449/2.3° 0.458/2.6° 0.461/2.7° 0.462/2.6° 0.462/2.6°

500 km

0.320/-3.20

0.357/ 1.0° 0.370/-1.9° 0.377/ 2.3° 0.381/ 2.5°

1000 km

0.099/-105.4° 0.136/-020.50

0.178/-009.50

0.200/-006.20

0.213/-004.70

2000 km

0.435/-152.1° 0.138/-139.40

0.133/—133.8° 0.114/-128.10

0.107/-123.1°


grounds that the attenuation rate of the higher order modes is sufficiently large that they do not contribute significantly.

In the case where Ad = 200 km, we see from the first column in the table that S^i is approaching some sort of a limit as N increases. The value 0.462/2.6° for N = 6 is reasonably close to the computed value 0.5108/ — 1.15°, which corresponds to a single step [Wait, 1968] between two uniform sections (i.e., the case Ad = 0). Thus, if the transition length Ad is anything less than about 200 or 300 km, the conversion coefficient for Sf[ is effectively the same as that for an abrupt height change. This statement is in accord with a previous perturbation analysis [Wait, 1968] of a closely related problem.

1.00

0.96

Q) 1-1

— 0.92 c ω o

g 0.88 u Di C

'ω 0.84 o υ

C/)

0.80

0.76 14 18 22 26 30

Frequency, kHz

FIG. 2. Scattering coefficient for a six-step transition for m = 1, and n = 1.

Some further insight into the nature of the mode conversion is shown in Figs. 2 and 3, where the coefficients p i , i| and \S2, i| are plotted as a function of frequency. The model and the parameters are the same as in the above example. As indicated, the mode conversion is more significant as the frequency increases toward the upper end of the VLF band. (The portions of the curves shown dashed in Figs. 2 and 3 refer to extrapolated data.)

J 1 I ! ! I L


0.71 1 1 1 r

22 26

Frequency, kHz

F I G . 3. Scattering coefficient for a six-step transition for m — 1, and n = 2.


No attempt will be made here to make detailed comparisons with experimental data such as obtained by Crombie [1967]. However, it is worth mentioning that the magnitude of the conversion coefficients shown in Fig. 3 suggests that the effective value of Ad is of the order of 500 km for a sunrise transition.

Extensive calculations of mode conversion based on the multistep model have been carried out. The results will be made available in a published report. Hopefully, such results will be of use in carrying out interpretations in future experimental studies of mode conversion phenomena in the nonuniform earth-ionosphere waveguide.

We thank Mrs. Eileen Brackett for her help in preparing the manuscript.

7. APPENDIX. DERIVATION OF Sn, m FOR A SINGLE STEP The basic building block for the scattering from a graded transition is the

single step as indicated in Fig. Al. The central idea is to equate the field of a

FIG. Al. The basic single-step prototype model.

& , m — (A2)

mode m in region (1) to a sum of modes in region (2). Thus, over the aperture plane (0 < y < ^2)

G™(y) = Σ Sn. mGf(y)9 (Al) n

where G™ and G^2) are height-gain functions in regions (1) and (2), respectively.

We now multiply both sides of (Al) by G{*){y) and integrate over the interval 0 to y%. This yields

ftG$(y)GfWy Syo2lG%Xy)?dy '

where we have utilized the orthogonality property

JZ» Gf(y)Gf{y)ày = 0Ίΐηφ η\ (A3 appropriate for surface impedance-type boundaries. Also, we observe that the denominator in (A2) is

11* [G{M?dy = (W2) [Λ<2,]-ι, where Α{„] is the excitation function appropriate for region (2).

To obtain an explicit formula for Sn, m, we note that, for any y,

d2G«'(j) d j 2

while G«»(0) = 1, and

-lt^-y]G^(y), y = l , 2 , (A4)

[f'-^L 0,

where qj is a normalized wall impedance (Wait, 1964) for waveguide region (j). Thus,

2 G(1)(vii

"m "n where

2Λι2) f 2 (A6)

Here, AUt m is the correction accounting for the integration over the vertical face of the step in the aperture plane. In previous work (e.g., Wait, 1968), this contribution was neglected. Clearly, this is justified when the boundary is a magnetic wall whence Gf(y) must vanish in the aperture plane over the interval yi <y < j>2.

In order to provide an estimate of the smallness of Δη, m, we will argue that the field over the whole aperture plane is the same as if no reflection occurred. Thus, if in region (1) the upper medium is a homogeneous plasma,

Gii'C) * GU'Cyi) exp [-am(j - yi)], (A7)

where y > yi where am is a complex quantity whose real part is greater than zero. Thus,

2 P2'2

Δ», m = v ΛΜ>0>ι ) e-«m(y-yi)G%Xy)dy. (A8) y 2 Jyi

To facilitate the integration, we can use the expansion (Wait, 1964)

Gf{y) = Gf{y2) 14«>(ya - y}r9 (A9) /7 = 0

where A™ = 1, A[^ = -q2, A^ = (if - y2)/29 A3 = [1 - (t™ - j 2 )^ ] /6 , and so on. Thus, it readily follows that

Δ,, m = A^G^XyùGfiyz) Σ Ρ™Α£\ (AIO)

where

2 f 2

p(m) = __ e-am(y-2/i) (y2 - y)vay. (All) J2 Ji/i

The integral above is readily evaluated in closed form. For example,

pirn) = __ e - am^2-2/l) J2 \ am /

and

pirn) = A e-am(2/2-2/l) (eam^2-2/l) l ^ ~ ^ - ~] + - ^ l .

When |a m ( j 2 — yi)\ < 1,

P i * > ^ ( 2 / W ) ( y a - w ) (A12) and, thus,

Δ„, w ~ 2{y*~n) ATG«\yi)Gf(.yz). fA13)

Under most conditions, |AW> m| is small compared with \Sn, m\, so we can neglect it considering that other restrictive assumptions are already inherent in the analysis.


In obtaining the explicit formula for Sn, m given by (A5), we have assumed that the propagation is perpendicular to the junction. However, for an oblique path, the mode conversion at a single step is not appreciably changed. The effects of obliquity are considered more fully elsewhere (Wait, 1968).

8. REFERENCES BAHAR, E. (1966) Propagation of radio waves in a model nonuniform terrestrial waveguide,

Proc. IEE 113, No. 11, 1741-50. CROMBIE, D. D. (1964) Periodic fading of VLF signals received over long paths at sunrise

and sunset, Radio Sci. J. Res. NBS 68D, No. 1, 27-34. CROMBIE, D. D. (1967) The waveguide mode propagation of VLF radio waves to great

distances, IEE (London), Conference Publication No. 36. RUGG, D. E. (1967) Theoretical investigations of the diurnal phase and amplitude varia

tions of VLF signals, Radio Sci. 2 (New Series), No. 6, 551-6. WAIT, J. R. (1964) Two-dimensional treatment of mode theory of the propagation of

VLF radio waves, Radio Sci. J. Res. NBS 68D, No. 1, 81-93. WAIT, J. R. (1968) Mode conversion and refraction effects in the earth-ionosphere wave

guide for VLF radio waves, / . Geophys. Res. 73, No. 11, 3537-48.


BAHAR, E. (1968) Scattering of VLF radio waves in the curved earth-ionosphere waveguide. Radio Sci. 3 (New Series), No. 2, 145-54.

WAIT, J. R. (1968) On the theory of VLF propagation for a step model of nonuniform waveguide (including the influence of reflections), Can. J. Phys. 46, No. 17, 1979-83.

Errata and Addenda

THE following is a list of the errata and addenda to the first edition, which also relates to this edition unless noted otherwise.

page 3, Eq. (9). For +ηΒ Q~rz read +η~1Β Q~rz (first edition only.) page 19, Eq. (36). For Eoy read EQX

Eq. (37). For E\y read E\x

page 26, line 12. For electric read magnetic page 30, line 11. Should read . . . Eo and E% in the plane . . . page 31, line 11. For by Eq. (48) read by Eq. (21) page 32, line 5. For Hx = —Ex read Hx-+ —Ex

page 34, line 20. For (Ca + Cb) read (Ca + Ce) page 36, Eq. (110). For Δ2 read V2

Eq. (111). For φο' read φ08

page 37, line 1. For φο' read ψοδ

Eq. (117). Right-hand side reads fôaM(X) e ~ ^ z J0(Xp) dX page 38, line 23. For ΕΜφ read Hm4>

page 39, Eq. (130). For à\ read

page 40, line 2. For |Δ| < 1 read |Δ|2 <ξ 1 last line. For {γιγζ) read (71/72)

page 41, line 8. For y\h\ <ξ 1 read \y\h\\ < 1 page 52, Eq. (173). For e^o^cosAo reacj e<*0r2Acose

line 27. For The firm read The first page 63, Additional Reference:

KING, R. J., and SCHLAK, G. A. (1967) Ground wave attenuation function for propagation over a highly inductive earth, Radio Science 2 (new series), 687-93.

page 68, last line. For Lim read Lim z—»-0 z - > 00

page 71, line 25. For (0/2) read (2/ß) ΠΛ v nn r 2koC A lk«C

page 74, Eq. (73). For read — a a

Eq. (75). For exp (ilkCzi) read exp (ilkoCzi) line 21. For exp (ilkCzi) read exp (HkoCzx)

593

page 76, Eq. (87). For exp (f i2/3) read exp (fi3/2) page 77, Eq. (97). For k\ read ko (in two places)

Eq. (99). For k\ read ko page 79, line 15. For right-hand side read left-hand side page 83, Eq. (142). For E = read Ey = page 88, Eq. (8), final integral. For dx read dz page 89, line 20. For on read of

Eq. (20). For p(z) read P(z) Eq. (21). For q(z) read Q(z)

page 90, Eqs. (24) to (30). For p(z) read P(z) and for q(z) read Q{z) page 92, Eq. (41). For <λ 0 read ^ λ " 1

Eq. (42). For <ξ λ0 read \<λ~1

page 94, Eq. (49). For Q-^h^ read Q-MWZ dz gh SN

page 95, Eq. (52). For — 2 read —2

SN cos% Eq. (53). Should read 2C2 1 - (Ixlrrf

h h On line above : for 0 < z < h read ~ < z < ~

gh 8N Eq. (54). For —2 read — 2

On line above: for 0 < z < oo read — oo < z < oo page 98, Eq. (71). Should read

E - EovCtyVyqW line 18. Sentence beginning In fact should read

However, this is only strictly true when q2(z) is a linear function of z in the transition region. Also, it is important. . .

line 25. Addition'. . . . and possibly others (e.g. Heading, 1962). An important

extension has been made recently by Baker (1967). page 105, Additional References:

HEADING, J. (1962) An Introduction to Phase Integral Methods, Methuen, London.

BAHAR, E. (1967) Generalized WKB method with applications to problems of propagation in non-homogeneous media, / . Math. Phys. 8, 1735-46.

page 112, Eq. (23). Right-hand side should start

ts q* Eq. (25). For e-<*/8 read Q~2i7r^

page 113, Eq. (28). For &xt read Q~ixt

Errata and Addenda 595

ά2Υ d2Y page 119, Eq. (62). For — read ^

Eq. (67). For ( - i)2/3 read ( - 03/2

Eq. (68). For ( - 02 / 3 reerf ( - 03/2

page 128, Eq. (122). For s" reads'" page 137, Fig. 2. For μο read μ (three times)

For co read e Eq. (3.1). For Hq = read Ηφ =

page 141, line 16. For n > 2 read n > 0 page 143, Fig. 4. För/io read μ (three times)

i w eo read € **s^ ,* Λ^ π ilàsk , j'/dsfc page 146, Eq. (3.41). For -— read —^

Eq. (3.43). For —y— read ^j~

Eq. (3.44). For 2gn(z0) read 2gn(z)

page 149, For So = 1 — ι ^ττ retfrf So = 11 — i! ττ I

encü ect> page 150, line 18. For G = — rajrfG = — page 151, Second equation. For (Cn)2 read (Cn)2

page 153, Fig. 5. For μο read μ (three times) For eo read e

page 155. First element in determinant should have subscript a to square brackets

[ ]a page 159, Eq. (6.9). Should read

on

page 162, Eq. (7.1). Right-hand side starts i /ds^ Okhä2 β ' β

Eq. (7.2). Concludes Ξ (kaSn)2

page 164, Eq. (8.4). Right-hand side starts ildsv(i>+ 1)

4πα2 eooA page 166, Eq. (8.8). Left-hand side should read

(φ)2 v{y + 1}

lines 18 and 19. Should read Ias = (Jàs)oS(t) (8.9)

where 8(t) is the unit impulse function at t = 0. Eq. (8.10). Right-hand side should read

(Ids)0 line 24. For (a/c)2 read {cja)2

page 171, Fig. 8. For μο read μ (three times) For co read e

page 172, Eq. (9.29). For 3PV read 3P-page 173, line 3. For (30) and (31) read (31) and (32)

Eq. (9.38). Right-hand side starts - i fo /λ) . . .

page 175, line 3. Should start more, k9 = ( e ^ ) 1 / 2 (1 — /σ^/^ω)1/2 ω and

ki = {ειμι)112 (1 — iai/eto))1/2 ω are . . .

Eq. (10.3). For -r- read -r-

page 185, Eqs. (11.13) to (11.15). For iNl read iNkl (six times) line 27. For I read kl (twice)

page 186, line 16. For I read kl (twice) page 199, Eq. (11). Right-hand side reads

— v α(θ sin 0)i/2 v

page 214, line 10. Third equation should read qt = qtS-M(clä)

page 235, line 2. For exp [ißPy] read exp [ißpky] where ßp = [(MVNV)~^ - S2j*

Eq. (47). For exp [—i2ßp-\lp-{\ read exp [—ilßp-iklp-ι] (twice)

Eq. (48). For exp [—i2ßplp] read exp [—ilßpklp] (twice) page 238, Eq. (62). ± { } should read ± { }* page 240, Eq. (71). For Hy = read ηΗυ = page 248, Eq. (115). New version reads

[F(C)r = (e^(Ä-zo> + [A]")(l - [Ag*i]" Q-2ikhC\-l ÎQikCz _|_ [Rg]" Q-ikCz\ Q-ikCh

page 249, Eq. (124). Should start „Λ,, Eq. (125). Should start LRX

page 252, line 19. For 2kh read 2kCh page 256, Eq. (159). Left-hand side should read

1 + ί€(Ω)ω = page 261, line 1. For Im D > 0 read Im D < 0 page 266, Eq. (6). First relation should read

2kh + iai

Errata and Addenda 597

page 288, Additional References: BAYBULATOV, R. B. (1967) Correlation of the reciprocity principle

in the reflection of ultralong radio waves from the ionosphere, Geomagnetism and Aeronomy, No. 5, 906-8.

BAYBULATOV, R. B. and KRASNHUSKIN, P. YE. (1968) Determination of the daytime electron concentration profile of the C and D layers of the ionosphere, Geomagnetism and Aeronomy, No. 6, 1051-60.

WATT, A. D. (1967) VLF Radio Engineering, Pergamon Press, Oxford.

page 292, line 8. For kSnp read kSna page 309, first paragraph. In connection with the series which Johler and

Berry program, L. A. Berry has indicated in a private communication that "the summation of the series is cumbersome and expensive, except below 500 c/s, and is not easy to interpret physically".

page 312, Add sentence to bottom of page. In comparing these impulse responses with experiment, one should regard a positive current moment as pointing downward.

page 325, line 18. For ±ikz read ±ikCz page 326, last line. For replacing z by ZQ read interchanging z and ZQ page 336, Eq. (62). For aS5 read dS page 337, line 6. For N'1^)5 read Ν-λ(ζ)

/ 2 \ 1 / 3 / 2 \ - 2 / 3

page 342, Eq. (9). For I— I read 1^1 (twice) page 344, Eq. (22). For/i(0, t) readf^O, t) page 348, Eq. (50). In left-hand side, lower limit of integration is y± page 349, Eq. (55). Should read

(dy\ 1/2

page 353, Eq. (86). Lower limit of integration is yi page 355, line under Table 2. For (py) readp(y) page 357, Eq. (100). For [t - p(y)] read [t - p(y)]1^

line 20. For q = 0 read q = oo Eq. (105). For [p(y) - t] read [p(y) - Z]1/2

page 361, Fig. 3 (centre). For s = 0 read s = 1 (first edition only) Eq. (131). Equation should be in the form (first edition only)

I l = o

AUTHOR INDEX

Aarons, J. 289, 315, 321 Alfvén, H. 254, 262 Allan, A. H. 287 Allis, W. P. 256,262 Al'pert, la. L. 6, 134, 157, 193, 195, 256,

262, 288 Anderson, L. V. 131 Arndt, D. 257,258,262 Atwood, S. 9, 20, 62 Austin, L. W. 195

Bain, W. C. 288 Banos, A. 6 Barber, N. F. 233,262 Barlow, H. E. M. 9, 10, 62 Barron, D. W. 195, 288 Bateman, H. 70, 84 Bean, B. R. 124, 130 339 340 Belkina M. G. 6, 341, 352, 354, 362 Benoit, R. 289, 321 Berry, L. A. 309, 321 X Bickel, J. E. 277, 278 287 Blair, W. E. 6 Booker, H. G. 242,243,260,262,301,302,

357, 362 Borodina, S. V. 195 Bossy, L. 85 Bouwkamp, C. J. 8, 62 Bowe, P. W. A. 286, 287 Brekhovskikh, L. M. 6, 85, 93, 102, 105,

189, 193, 330, 340 Bremmer, H. 6,67,84, 88,93,98,105,112,

113,115,118,124,128,129,130, 133, 173,193,195,196,197,225, 239, 241, 242, 243, 260, 262, 334, 340, 358, 362

Brick, D. B. 9, 62 Budden, K. G. 6, 74, 84, 95, 98, 100, 102,

105,133,147,191,193,195,238, 239, 244, 247, 249, 251, 262, 301, 321

Campbell G. 22, 62 Carroll, T. J. 129,130 Carter, N. F. 186, 194, 233, 263, 297, 322 Casselman, C. J. 281, 282, 287 Chapman, F. W. 133, 193, 286, 287, 289,

307, 308, 321 Chapman, J. 286, 287 Chapman, S. 226, 262 Clemmow, P. C. 50, 62

Colin, R. E. 6 Compton, K. T. 226, 262 Conda, A. M. 123, 129, 130 Cowling, T. G. 226, 254, 262 Cravath, A. M. 226, 262 Crombie, D. D. 233, 239, 262, 285, 287,

288, 322 Crawford, A. B. 105 Croom, D. L. 288 Cullen, A. L. 9, 10, 33, 62

Deal, O. E. 322, 323 De Vuyst, A. 85 Dingle, R. B. 257, 258, 262 Doherty, L. H. 330, 339, 340 Du Castel, F. 93, 94, 105 Duffus, H. J. 322 Dungey, J. W. 315, 321

Eckersley, T. L. 6, 77, 84, 115, 130, 195, 278, 286, 287, 288, 341, 357, 362

Elias, G. J. 74, 84 Eliassen, K. E. 45, 46, 47, 62 Elliott, R. S. 9, 62 Emde. F. 30, 62, 190, 193 Epstein, P. S. 63 Ewing, M. 6

Fainberg, Ya. B. 229,262 Feinberg, E. F. 262 Feinberg, E. L. 6 Fernando, W. M. G. 9, 62 Feshback, H. 108, 130 Field, E. C. 262 Fock, V. A. 6,112,113,115,116,118,119,

130, 196, 225, 341, 350, 352, 354, 362 Försterling, K. 63, 85 Foster, R. 22, 62 Franz, W. 6 Freehafer, J. 360, 362 Friedman, B. F. 134, 193, 195 Friis, H. T. 93, 105 Furry, W. 6, 129, 341, 360, 361, 362 Furutsu, K. 52, 62

Galejs, J. 79, 84, 186, 193, 308, 321 Gallet, R. M. 322 Ginzburg, V. L. 6, 229, 262 Goldberg, P. A. 289, 321

QQ

600 Author Index

Gorbatenko, M. F. 229, 262 Goubau, G. 9, 62 Gray, Marion 85 Grosskopf, J. 45, 62 Gurevich, A. V. 263

Hack, F. 45,62 Hales, A. L. 314, 321 Harper, J. D., Jr. 85, 243, 256, 262, 263 Harrington, R. F. 6 Harris, F. B. 308, 321 Henissart, M. 289, 321 Hepburn, F. 289, 313, 314, 321 Herd, J. F. 323 Heritage, J. L. 277, 278, 279,281, 282, 287 Hines, C. O. 227, 254, 262, 323 Hogg, D. C. 105 Hölzer, R. E. 323 Hönl, H. 6 Howe, H. H. 134, 147, 191, 193, 194, 195

Jahnke, E. 30, 62, 190, 193 Jardetsky, W. 6 Jean, A. G. 284, 287, 288, 323 Jeffreys, Sir Harold 87, 105 Johler, J. R. 85, 239, 242, 243, 256, 262,

263, 269, 287, 309, 321

Kaden, H. 133, 193 Karbowiak, A. E. 140, 194 Katzin, M. 292,321 Kendrick, G. W. 195 Kerr, D. E. 360, 362 Knudsen, E. 315, 317, 322 Koo, B. Y. C. 292, 321

Lahiri, B. N. 82, 84 Lange, L. J. 133, 194, 286, 288 Langer, R. E. 95,98,105,111,130 Langmuir, I. 226, 262 Leontovich, M. A. 118, 119, 130, 131 Lieberman, L. 134,194,289, 309, 314, 321,

322 Logan, N. A. 6, 349, 362 Lorenz, L. V. Ill, 130 Lösch, F. 190, 193 Lunnon, F. C. 278, 288 Lutkin, F. E. 323

Macario, R. C. V. 133, 193, 286, 287, 307, 308, 321

Magnus, F. 162, 163, 164, 194, 291, 322 Marcuvitz, N. 6, 10, 62 Mason, R. L. 349, 362 Matthews, W. D. 289,321 Mave, A. W. 6

Maxwell, E. L. 192, 194, 323 Mentzer, J. R. 6 Miller, J. C. P. 75, 76, 84 Miller, W. E. 118, 128, 130 Millington, G. 115,129,130,131, 357, 362 Misme, P. 105 Molmud, P. 256,263 Monteath, G. D. 131 Moore, R. K. 6 Morrison, R. B. 286,287 Morse, P. M. 108, 130 Murphy, A. C. 195, 288, 323

Namba, S. 133, 194 Nasmuth, P. W. 322 Nath, S. C. 284,288 Newman, M. 287 Messen, K. F. 8 Norinder, H. 315, 317, 322 Northover, F. H. 106, 128, 130 Norton, K. A. 8, 9, 13, 40, 41, 45, 62, 128,

130

Oberhettinger, F. 162, 163, 164, 194, 291, 322

Officer, C. B. 6 Oliner, A. A. 6

Pearcey, T. 337, 340 Pekeris, C. L. 85, 98, 105, 292, 322, 359,

362 Phelps, A. V. 256, 257, 263 Pierce, B. O. 128, 130 Pierce, E. T. 286, 287, 289, 314, 317, 321,

322 Pierce, J. A. 278, 281, 282, 284, 285, 287,

288 Poeverlein, H. 195, 288 Press, F. 6 Price, A. T. 82, 84

Ratcliffe, J. A. 226,238,263, 301, 302, 322 Rice, S. O. 63 Ring, R. M. 129 Rolf, B. 8, 62 Rotman, W. 9, 62 Round, H. J. T. 5, 264, 278,279, 280, 287,

288 Roy,S.K. 257,258,262 Rydbeck, O. E. H. 63, 106, 133, 157, 174,

194 Rytov, S. M. 66, 84

Schelkunoff, S. A. 6, 93,105,233, 263, 318 Schmelovsky, K. H. 195 Schumann, W. O. 133, 148, 164, 166, 168,

174, 194, 289, 309, 322, 323 Sen, H. K. 256, 257, 263, 290

Author Index 601

Shand, J. A. 322 Shmoys, J. 85 Smith, E. J. 323 Sommerfield, A. N. 8, 9, 35, 41, 46, 49, 62,

63, 138, 139, 144, 154, 156, 157, 160, 179, 194

Spies, K. P. 134, 191, 192, 194, 201, 206, 225, 263, 264, 288

Spitzer, L., Jr. 226, 254, 263 Stanley, G. M. 47, 62 Stanley, J. P. 85 Storey, L. R. O. 323 Suhl, H. 263 Surtees, W. J. 131

Tai, C. T. 9,62 Tamarkin, P. 262 Tanner, R. L. 308, 321 Taylor, L. S. 85 Taylor, W. L. 5, 133, 194, 264, 286, 287,

288 Tepley, L. R. 309, 317, 322 Thayer, G. D. 124, 130 Tibbals, M. L. 281,282,287 Tremellen, K. 278, 288

Van de Hülst, H. C. 6 Van der Pol, B. 8, 112, 118, 130, 196, 225 Van der Waerden, B. L. 50, 62 Van Trier, A. A. 263 Voge, J. 105 Vogt, K. 45,62 Vvedensky, B. 131,357,362

Wait, J. R. 7, 21, 22, 28, 35, 39, 46, 47, 63, 65, 72, 84, 85,100,104,105,113,114,

123,129,130,131,134,168,182,184, 186,190,191,192,193,194,195,196, 198,206, 225,226,233, 243, 244,263, 278, 287,288,289, 290, 293, 297, 298, 308,309,314,315,317,322,323

Walker, L. R. 263 Walkinshaw, W. 357, 362 Walters, L. 239, 242, 262, 263 Wasmundt, D. F. 323 Watson, G. N. 76, 84, 107, 110, 111, 130,

132, 133, 156, 168, 174,194, 349, 362 Watson-Watt, R. A. 323 Watt, A. D. 192, 194, 323 Watts, J. M. 289, 322 Weinstein, L. A. 6, 341, 352, 354, 362 Weisbrod, S. 277, 278, 287 Westfall, K. 6 Westfall, W. D. 284, 288 Weil, H. 8 Whitmer, R. F. 226, 263 Whitson, A. L. 288 Whittaker, E. T. 349, 362 Wieder, B. 263 Willis, H. F. 289, 322 Wise, W. H. 8, 63 Wong, M. S. 339, 340 Wright, C.S. 322 Wyller, A. A. 256, 257, 263

Yabroff, I. 239,242,263 Yee, K. S. 6 Yudkevitch, F. S. 66, 84

Zenneck, J. 8, 9, 19, 20, 45, 63 Zucker, F. J. 7, 9, 63, 329, 340

SUBJECT INDEX

Acoustic analogy to electromagnetic reflection 101

Analogy in transmission line theory 12 Anisotropie curved ionosphere, v.l.f. mode

problem for 391 Anisotropie ionosphere, mode representa

tion 246 Antipodal effects in e.l.f. propagation 291 Appleton-Hartree formula 238 Asymptotic approximation to the spherical

wave functions 156 Asymptotic development for guided wave

propagation 324 Asymptotic expansion for the Legendre

function 161 Asymptotic solution for inhomogeneous

atmosphere 122, 343, 350 Atmospheric wave forms 285, 312 Attenuation curves for spherical earth-

ionosphere waveguide 205, 209, 210, 212, 217, 307

Attenuation of dominant mode in v.l.f. propagation for flat earth 266, 267, 270

Attenuation of e.l.f. modes for exponential conductivity variation

308 influence of magnetic field 301

Attenuation of v.l.f. modes, influence of magnetic field 271

Attenuation of waveguide modes for finite conductivity 148

Attenuation rate calculated from sferics 286

Attenuation rate for e.l.f. waves when exponential model is used 186, 306

Average decay laws 187 Averaged intensity compared with actual

intensity at v.l.f. 191 Averaging the mean square field in the wave

guide 189 Axial focusing 164 Azimuthal-type travelling waves 164

Bilinear profile 360 Booker quartic 242, 260 Boundary conditions

at air-plasma interface 232 at an interface 11, 326

Brewster angle 19 Caustic of the ray system 78, 336 Cavity-resonator

modes in a cylindrically stratified magnetoplasma 545

oscillations 166 type modes 164

Collision frequency effective (complex) form taking account

of heavy ions 229 energy dependent case 256

Complex angles of incidence for waveguide modes 137

Complex resonance condition 137, 139 Concentric cylindrical waveguide, fields in

the 448 Concentric spherical capacitor 165 Conductivity 2

of ionized gas 257 parameter 430

Continuously stratified media 65, 326 Continuously varying conductivity profile

64 Contour integral representation

for field in the spherical earth model 177 for ground wave 197 for guided wave 325 for sky wave 199

Convergence of rays in stratified media 335 Coupled differential equations 89, 91 Critical reflection 68, 72 Curvature corrected field for short distances

113 Curvature corrected flat earth formulae

115,292 Curvature, its effect on surface wave propa

gation 107 Curvature of earth, influence on modal

characteristics 251 Curved ionosphere, anisotropie, v.l.f. mode

problem for 391 Cylindrically stratified magnetoplasma,

cavity resonator modes in a 545 Cylindrically stratified magnetoplasma,

transverse propagation of waveguide modes in a 529

Cylindrical waveguide, concentric, fields in the 448

603

604 Subject Index

Damping of the cavity modes 167 Debye-Watson representation of Hankel

functions 156 Dielectric constant 2 Dielectric tensor 227, 228, 254, 256

inverse form 230 Dipole, line, source 446 Dipoles, radiation from, in an idealized

jungle environment 575 Distance to the horizon 124 Diurnal phase shifts at v.l.f. 284

Earth crust waveguide, idealized, electromag

netic propagation in an 551 curvature influence on the modes 251 detached mode in earth-ionosphere wave

guide 224 flattening approximation 291, 359 inhomogeneous spherical, illumination

of by an l.f. plane electromagnetic wave 565

ionosphere condenser 166 Earth-ionosphere waveguide at v.l.f. mode

conversion at a graded height change in the 583

Earth-ionosphere waveguide, influence of an inhomogeneous ground on the propagation of v.l.f. radio waves in the 491

Effective earth radius 118 Effective numerical distance 41 Eigen values, determination for curved

waveguide 155 Electric Hertz vector 247 Electrical constants of the ground 45 Electromagnetic waves, reflection of, from

a lossy magnetoplasma 463 Electromagnetic waves, reflection of, from

an inhomogeneous lossy plasma 481

Energy dependence of collision frequency 256

Equivalent earth radius concept 117 Error function of complex argument 51 Exact and approximate solutions of modal

equation 275 Excitation by horizontal dipoles for curved

earth 168 Excitation factor 381 Excitation factor of waveguide modes as a

function of frequency 222, 223 Excitation of waveguide modes

by horizontal dipole 141 by vertical magnetic dipole 143, 293

Experimental measurements on ground wave propagation 45

Exponential conductivity profiles and mode theory 184

Exponential integral Ei(-ig) 30 Exponential layer 374 Exponential profile for vertical polariza

tion 79 Exponential profile, step approximation

to 406 Exponentially increasing propagation con

stant 70 Exponentially varying medium 80, 306 Extraordinary ray in plasma 240 Extremely low frequency propagation 289

Field of a line source in the interface between two homogeneous media 25

Fields in the concentric cylindrical waveguide 448

Field strength vs distance data at e.l.f. 294 atv.h.f. 356; at v.l.f. 277

Fourier-Bessel transforms 37 Fourier integral representation for a line

source 22 Fresnel reflection coefficient 13, 94, 181

for curved boundaries 123 for spherical boundaries 200

Geometrical interpretation of guided waves 333; of reflected fields 24

Geometrical optics, for stratified media 92, 332, 347

Ground, inhomogeneous, influence on the propagation of v.l.f. radio waves in the earth-ionosphere waveguide 491

Ground wave attenuation 40

Hankel approximation to spherical wave func

tions 179 or third order approximation for the

spherical wave functions 180 Harmonic series representation 179 Heavy ion effects 229 Height dependence of the mode fields 146,

353 Height-gain for v.l.f. radio waves 379 Height-gain functions for the modes 160;

in earth-ionosphere waveguide 219 Hertz vectors, matrix form 248 Higher order approximations to the curved

earth theory 174 Horizontal currents in lightning discharge

315

Subject Index 605

Horizontal dipole between two plane interfaces 143 excitation for curved earth 168 fields compared with vertical dipole fields

173, 293, 316 Horizontal layers in the troposphere 93 Hyperbolic layer, reflection from 95 Hyper-geometric series representation 154

Illumination of an inhomogeneous spherical earth by a l.f. plane electromagnetic wave 565

Image approach to waveguide problem 134 Impedance matching in stratified media 17 Impedance type boundary conditions 176,

318 Inhomogeneity of the lower ionosphere

182 Inhomogeneous atmosphere treated in

spherical earth problem 115,318 Inhomogeneous ionosphere, reflection of

v.l.f. radio waves from 403 Inhomogeneous lossy plasma, reflection of

electromagnetic waves from 481 Integral representation

for fields of horizontal dipole in waveguide 144

for fields of vertical dipole in waveguide 139, 325

for reflected wave 22 Integration contour for a line source over

a coated conductor 34 Internal reflections in stratified media 93 Iterative solution for reflection coefficient

of inhomogeneous media 102 Ionized gas

conductivity of 257 theory of electrical conduction 254

Ionosphere anisotropie curved, v.l.f. mode problem

for 391 inhomogeneous, reflection of v.l.f. radio

waves from 403 lower, influence of the, on propagation

of v.l.f. waves to great distances 367

Ionospheric reflection coefficient for stratified model 183

Jacobi theta function 190

Laplace transform approach to cavity-resonator oscillations 166 to e.l.f. pulses 309

Line dipole source 446 Line of images 135

Line source 134 excitation for stratified media 21 magnetic, over a dielectric coated con

ductor 33 magnetic, over a stratified medium 32 on a homogeneous medium 25 over a stratified half-space 31 over a thin layer 27

Linear layer, reflection from 95 Longitudinal propagation in plasma 239 Lossy magnetoplasma, reflection of electro

magnetic waves from a 463 Lossy plasma, inhomogeneous, reflection

of electromagnetic waves from 481 Lower ionosphere, influence of the, on

propagation of v.l.f. waves to great distances 367

Magnetic conductor 136 Magnetic field, influence on attenuation

of e.l.f. modes 301 of v.l.f. modes 271

Magnetic Hertz vector 248 Magnetic line source

over a dielectric coated conductor 33 over a stratified medium 32

Magnetoplasma cylindrically stratified, cavity resonator

modes in a 545 cylindrically stratified, transverse propa

gation of waveguide modes in a 529 lossy, reflection of electromagnetic waves

from a 463 Magnetoplasma media 226 Matching conditions for isotropie wave

guide 138 Matrix form of Hertz vector 248 Matrix reflection coefficient 246 Maximum value of attenuation for wave

guide modes 149 Maxwell's equations 2

for anisotropie media 230 for electron plasma 261 for tensor dielectric properties 230

Measured attenuation of ground waves 47 Modal equation, approximated for spheri

cal earth-ionosphere waveguide 202 exact and appropriate methods com

pared 275 for spherical earth-ionosphere wave

guide, more accurate form 213 for spherical waveguides 201 in coupled form 250, 252 in limiting form of perfectly reflecting

spherical boundaries 203 in tropospheric propagation 351, 361 in uncoupled form 249

606 Subject Index

Mode conversion at a graded height change in the earth-ionosphere waveguide atv.l.f. 583

Model terrestrial waveguide of non-uniform height, propagation in 505

Mode properties for flat earth case 147 Mode representation for anisotropie iono

sphere 246 Modes, transverse electric 247 ; transverse

magnetic 247 Monotonie variation of refractive index of

atmosphere 124, 344 Multiple reflections between ground and

ionosphere 198 Mutual impedance between dipoles 170

Natural oscillations in stratified media 17 Near field behavior at e.l.f. 298 Newton's method applied to solution of

mode equation 216 Non-linear atmosphere 118,342 Normalization factors for modes 159 Norton surface wave 44 Numerical results for surface impedance of

a stratified conductor 53

Oblique incidence, reflection coefficients in simplified form 245

Ordinary ray in plasma 240 Orthogonality properties of the modes 158

Parabolic cylinder functions 349 Parabolic layer, reflection from 95 Permeability 2 Perpendicular incidence 15 Phase characteristics of v.l.f. carriers 281 Phase integral

its physical significance 99, 332 for vertical polarization 99

Phase integral method 74, 330, 357 generalized 98

Phase shifts, diurnal variation at v.l.f. 284

Phase velocity for e.l.f. modes 304 for v.l.f. modes 275

Phase velocity curves for spherical earth-ionosphere waveguide 204, 207, 208,211

Plane wave incidence 10 propagation 3

Plasma inhomogeneous lossy, reflection of

electromagnetic waves from 481 reflection from plane boundary 231,239 strongly ionized 228

weakly ionized 227 Pole of reflection coefficient 18 Power intensity of the modes 189 Primary excitation as a spectrum of plane

waves 138 Profile which leads to solutions in terms of

Bessel functions 64 Profile with an exponential transition 68 Propagation

around a sphere 107 between concentric spherical surfaces

196 factor for stratified media 41 of v.l.f. radio waves in the earth-ionos

phere waveguide, influence of an inhomogeneous ground on the 491

of v.l.f. radio waves, two-dimensional treatment of mode theory of the 445

over actual stratified media 45 transverse, of waveguide modes in a

cylindrically stratified magneto-plasma 529

v.l.f. approximations 581 v.l.f. radio, mode and ray theories of

441 Properties of the modes for horizontal

polarization 151

Quadratic variation of atmospheric refractive index 126

Quartic, Booker 242, 260 Quasi-longitudinal (Q.L.) approximation

243, 302 Radiation from dipoles in an idealized

jungle environment 575 Radio twilight theory of Carroll 129 Radio v.l.f. propagation, mode and ray

theories of 441 Radio waves, v.l.f. reflection of, from an

inhomogeneous ionosphere 403 Ray geometry for waveguide 135, 332 Ray theory for normal atmosphere 346 Reciprocity theorem applied to horizontal

dipole problem 168 Reflection

from a linear varying layer 77 from plasma interface 231, 243 from stratified plasma 233 of electromagnetic waves from a lossy

magnetoplasma 463 of electromagnetic waves from an

inhomogeneous lossy plasma 481 of horizontally polarized waves 15 of vl.f. radio waves from an inhomo

geneous ionosphere 403

Subject Index 607

v.h.f. from a tropospheric layer 477 Reflection coefficient

exponential representation for ionosphere 264

for rapidly varying transition region 100

for reflection from an inhomogeneous ionosphere 408

for tropospheric layer 94 matrix 246 simplified for oblique incidence 245

Refractive index 477 Resonance condition 19, 329

for waveguide modes 135, 136, 329 Resonance equation

for earth-ionosphere waveguide 181 for spherical waveguide in W.K.B.

approximation 157 Resonator modes, cavity, in a cylindrically

stratified magnetoplasma 545 Residue series formula

for field in the presence of a sphere 111 representations 132, 178, 351

Riemann surface of two sheets for the planar waveguide problem 140

Saddle-point method conventional 52 of integration 48, 50, 52, 331

Seasonal changes in the effective ground constants 46

Sferics 285 calculation of attenuation rates 286

Sinusoidal layer, reflection from 95 "Slow tail" in atmospherics 312 Smoothly tapered profile, reflection from

129 SnelPs law for anisotropie media 241 Sommerfeld attenuation function 41 Space wave 44 Stationary phase method of integration

331 Step approximation to an exponential

profile 406 Stratification at the lower edge of the iono

sphere 182 Stratified magnetoplasma

cylindrically, cavity resonator modes in a 545

cylindrically, transverse propagation of waveguide modes in a 529

Stratified plasma, reflection coefficient 233 Super refraction 341 Surface admittance of a stratified medium

17

Surface impedance applied to the sphere problem 109, 318 of a stratified conducting medium 53,

318 Surface wave

of Norton 44 of Zenneck 20

Tensor, dielectric 237, 256 Terrestrial waveguide, model, of non-

uniform height, propagation in 505 Transmission line

analogy for stratified medium 12 theory, non-uniform 234

Transverse electric modes 247 Transverse magnetic modes 247 Transverse propagation in plasma 239 Transverse propagation of waveguide

modes in a cylindrically stratified magnetoplasma 529

Transverse resonance condition 329 Trapped surface wave 41,43,44 Tropospheric ducting 341, 354 Tropospheric layer, v.h.f. reflection from

477 Tropospheric propagation of radio waves

93, 338, 341 Two-layer ground 14

Vertical dipole in space between ground and ionosphere 182, 290

Vertical electric dipole in a parallel-plate region 137 over a stratified half space 35

v.h.f. reflection from a tropospheric layer 477

v.l.f. mode problem for an anisotropie curved ionosphere 391

v.l.f. propagation approximations 581 v.l.f. radio propagation, mode and ray

theories of 441 v.l.f. radio waves, propagation of, in the

earth-ionosphere waveguide, influence of an inhomogeneous ground on the 491

v.l.f. radio waves, reflection of, from an inhomogeneous ionosphere 403

v.l.f. radio waves, two-dimensional treatment of mode theory of the propagation of 445

Voltage gradient in the atmosphere 166

Watson transformation 110 application of 538

Wave admittance for exponentially varying medium 71,

80

608 Subject Index

for power law variation of conductivity 83

for inhomogeneous media 326 Wave equation, one dimensional form 86 Waveguide

concentric cylindrical, fields in the 448 earth-ionosphere, at v.l.f. mode con

version at a graded height change in the 583

earth-ionosphere, influence of an in-homogeneous ground on the propagation of v.l.f. radio waves in the 491

idealized earth crust, electromagnetic propagation in an 551

junctions, generalized reciprocity theorem for 525

model terrestrial, of non-uniform height, propagation in 505

Waveguide modes beyond "cutoff" 140 complex angles of incidence 137 near "cut-off" 148 transverse propagation of, in a

cylindrically stratified magneto-plasma 529

Waveguide resonance condition and its relation to poles in the complex C plane 141

Waveguide with localized obstruction 456 Wave impedance

for plasma media 235 for waveguide 296

Wave tilt 13 correction factor 14, 16 for stratified ground 46 measurements 46

Weber parabolic cylinder functions 349 Whispering gallery phenomenon 224 WKB method and its extensions 88, 353

generalized form for oblique incidence 87 for vertical polarization 90

WKB or second order approximation for the spherical wave functions 154, 344

WKB solution for slowly varying media 87, 329, 344

Zenneck surface wave 19, 20, 45 Zero of reflection coefficient 18