JOURNAL OF RESEARCH of the National Bureau of Standards-D. Radio Propapa tion Vol. 66D, No. 1, January- February 1962

On the Diffraction of Spherical Radio Waves by a Finitely Conducting Spherical Earth 1,2

Lillie C. Walters and J. Ralph Johler

Contribution From Central Radio Propagation Laboratory, National Bureau of Sta ndards, Boulder, Colo.

(R eceived August 3, 1961)

The theory for t he diffraction of spherical electromagnetic waves by a finitely conducting spherical earth was developed from Maxwell's equations by Watson [1918] and the intricate computation details were later worked out by van del' Pol and Bremmer [1936] as the now classical seri es of residues. Two aspects of this computation present considerable difficulty, especially at low frequencies:

(1) The calcu lation of t he height-gain factor which takes account of an elevated trans­mitter and/or recciver.

(2) The evaluatio n of the spccial roots, r=r., of Riccati' differential equation,

near the circle of convergence, lo2rl =7f. These analytic difficulti es arc avoided with the aid of modern analysis techniques

applied to a large scale electronic computer. Hankel functions of t he first and second kind of order one-third and two-thirds are calculated by numerical integral methods and then used with iterat ion to solve Riccati ' differential equation. The amplitude and phase of the spherical r adio wave diffracted in the vicinity of the earth with various altitudes above t he surface of the earth , of both the transmitter and the receiver, are t hen calculated by a summation of the series of residues.

1. Introduction stant, can be evaluated as a summation of the clas ical series of residues,

The field of the diffracted spherical wave in the vicinity of a finitely conducting spherical earth has been discussed in detail as the ~ound wave by Watson [191 ], van der Pol and .tlremmer [1937], and the particular ca e of a vertically polarized H ertzian dipole source, E T , volts/meter; i.e., the electric field directed radially with respect to the center of the earth, can be written [Johler, Kellar, and Walters, 1956],

(1)

where,

where, d is the distance, meters, f=w /27r the fre­quency, cycles/second, I o!' is the dipole current moment, ampere-meters, exp (-iwt) imphes a wave varying harmonically in time, t, in a medium with

wave number, k1=": 1'}1, where c is the speed of light, c

c",3 (108 ) m /sec, 71 1 is the index of refraetion of air, 1'}1'" 1. The attenuation function or secondary factor, FT) which takes account of the earth's cmvatmo, finite conductivity, and dielectric con-

1 Sponsored by Air Force Cambridge Research Laboratories. CSO&A 58-40 (3/31/58).

• An earlier version of this paper was presented at tbe 1961 Spring meeting of the International Scientific Hadio Union (U RSI). Washington, D.C. (May 1961)

X exp {i [(1c,a)!r sa} ~+~~+~J} ' (3)

where a is the radius of the earth and a is a factor which takes account of the vertical lapse of the permittivity of the earth 's atmosphere.

The parameter, 0= 0. for vertical polarization is defined,

. k~ A ~ - a 3

kf (4)

. w [ .(JiJ.OC2J~ . . wILh k2= C E2+~-;- rachans/meter, ~ 2 the dlClec-

tric constant relative to a vacuum, (J the conductivity of the medium, mhos/meter, iJ.o the permeability of free spaee, henrys/meter, and r= r. describes the special roots of Riccati's differential equation,

101

do -202r+1=0 dr '

(5)

which define the positions of the poles at which the residues for the series 8= 0,1,2, 3 . .. are calculated. In the case of horizontal polarization, ET may be replaced by HT (vertical magnetic field) providing oe is replaced by 0", where,

(6)

The isolated factors, f s(h l ) , f s(h2) , which are in each term of the series describe the effect of elevating either the transmitter, hi , or the receiver, hz, or both, and can be evaluated without further ap­proximations [Johler et al., 1956] as follows:

[(kla)ZI3 2hoYs - 2Ts] I/Z

f s(h) = a - 2Ts

Hm{~ [(k1a)Z/3 ¥-2TsJ/2}

HI}! { ~ - 2T s) 3/2 }

(7)

It is often more convenient to use some sort of modified Hankel function, hl(z) [Furry, 1945], for which values of complex argument (z= x+ iy) have been tabulated,

hI { (D 2/ 3 [ (k1a)ZIS 2h:l/S -2TsJ1-hW ~ ) W

hi {- (2) 1/3 T s }

The roots, T., are found [Johler, Walters, Lilley, 1959] by expanding the Riccati differential equation in a power series,

_ 00 1 [dnTSJ n T s-~ -, --n Oe'

n= 1 n. dOe 0,=0

<-n U e '

T s== r s, co

WT I>t,

(9)

(10)

where the limiting roots, T s ,O and T s,oo for oe= O and 0.= 00 have been tabulated [Johler, Walters, Lilley, 1959].

The power series in on and o-n both seem to lose precision near the circle of convergence, 102TI = t, for a finite number of terms. Although a considerable number of terms have been determined [Johler, Kellar, Walters, 1956] the calculation of more terms becomes very laborious. Furthermore, the tabula­tion of the Hankel functions [Furry, 1945] does not seem to provide an adequate range of values, es­pecially at low frequencies and indeed is not especially suitable for application to large scale computers.

Since the Riccati differential equation can be expressed as a ratio of Hankel functions,

(11)

a basic technique to evaluate Hankel functions H!}a~~/s (z) would permit an iterative solution of Riccati's differential equation and at the same time could be employed t.o evaluate the Hankel functions Hm (z) in the height aain factor, j ,(h ). The development of such a technique is the prime task of this paper.

2 . Evaluation of Hankel Functions by Gaussian Quadrature

Bessel functions of order v are solutions of the differential equation.

(12)

where v and z are real or complex, The Bessel func­tions of the third bnd, known as Hankel functions, are linearly independent solutions of eq (12) and are denoted by HS]) (z) and H S2l (z) where v= l or j for the purpose of this discussion. Hankel functions have complex values for a real argument, but are real for i ,+1 H, (1l (iy) an d i-(,+l) HSZl (-iy) when y is positive. These functions are important in applications because they are t.he only Bessel func­tions that vanish for an infinite complex argmnent, H~!) (z) if the imaginary part is positive and H~2l (z) if it is negative [Jahnke and Emde, 1945].

J ntegral representations for H~I) (z) and H~2l (z) are as follows [Jeffreys and Jeffreys, 1956]:-

H(ll (Z)=-: exp - z A-- - , 1 1-00 [1 ( I)J dA , 7r1 Od 2 A AV+ I (13)

1 f O [1 ( I)J dA H S2l (Z)= -: exp -z A- - ,+ 1 7r1 - oo,-i 2 A A

(14)

where the expression of the limits in eq (13) means that the path is taken from 0 to - (X) by way of i .

Integrals with real limits are derived by selecting a path as shown in figure 1.

FIGURE 1. Contour oj integration oj Hankel j1tnctions, H l!)(Z).

102

If X= u, the part of tbe integral for H~I) (z) from 0 to 1 becomes,

~ ( I exp [~ (u-.!)] u - v- Idu. (15) 7r~Jo 2 u

On the semicircle from + 1 to-I, if X= exp (iO), this part is,

.! ( " exp (i z sinO) exp (- viO)dO. 7r J o

From- l to - <Xl, if X= u - Ie i ", this part is,

and together,

H~!)(z)=.! ( " exp (iz sin O- vio)do 7r J 0

+ :i 11 exp [~ ( u-~) ] [u- v - I

(16)

+ u v- I exp (- v7ri)]du (18) [or R e (z) > o.

As H~I) (z)=[H~~ (z*) ]* 3 [El'delyi, 19531.

H~2) (z)= .! ( " exp (-iz sin O+ viO)do 7r J 0

+~ 11 exp [~( u - t)] (u- v- l+ u v- 1 exp V7ri) du

(19)

for R e (z) > O. Note that this method is applicablc not only for v=t and j, but also for complex v.

The integrals in (18) and (19) may be evaluated by Gaussian quadratme ]Kopal, 1955]. In this method a finite integral is expressed as a sum:

(20)

where f (J.1!J) is an error term which may, in general, be made arbitrarily small by increasing M where 171 = 1, 2, 3, . . . kI.

(21)

The xlI.'s are the Gaussian abscissas and M determines the number of values of x to be used in the quad­ratme. The Gaussian weights and abscissas may be determined from the following:

(22)

(23)

3 The ' denotes tbe con jugate of the quantity.

The xm'S are the roots of the Legendre polynomials defined by

Po(x) = 1

PI(x) = x

P2(X) =~X2_.! 2 2

P3(X) =~X3_~X

p4(x) = 35x4_ 15x2+~ 8 4 8

(24)

Polynomials of higher degree are determined by use of the recmsion formula :

( 171+ I )P rn+l(x) + mPm_l(x) = (2m+ l )xP m(x). (25 )

Upon determination of the roots, the weight co­efficients, H m, of the corresponding quadrature formulas are evaluated as follows :

(26)

Forty-eight weights and abscissas were used in this quadratme [Davis and Rabinowitz , 1956] approxi­mating the integrand in this case with a polynomial of ninety-fifth degree [Kopal, 1955].

Equations (18 and 19) are valid for R e(z» O, but the values of the Hankel functions in the second and third quadrants may be found from the equations:

and

H~j) (z) = [H~~) (z*)]*,

sin ( ~ - m)v7r H~I)(z) s ln V7r

_ _ p"i S ill 17!V7r 7:1 (2) ( ) e . LL V z , s In V'/r

H~2) (zemrri ) sin (~ +m) v'/r H ~2) (z) SIn V7r

with m=- 1 in this instance [Watson, 1948].

3. Method of Iteration

(27)

(28)

(29)

An iterative method with the above procedure for finding Hankel functions may be used in \11) to find the roots ot (5). The iterative method described by Muller [1956], although it was developed to solve polynomial equations, may be used for any function which is analytic in the neighborhood of the roots.

103

The functions for T are from (11) :

F(T) -H{}Htc _2T)3/2] exp i7r -0 10 1<1 .J-2TH~)§[tc-2T)3/ 2] 3" e

and (30)

- .J-2d:1~}Htc -2T) 3/ 2] - i7r 1 H{)Htc- 2T)3/2] exp - 3- -5".' 10e l>1.

(31)

Alternate expressions for F(T) are used so that oe does not completely dominate the function and prevent convergence.

Each iteration in the process of finding one root is obtained from the calculation of the nearer root which passes through the last th ree points of the function, j eT). The quadratic equation is, in general, complex; i.e., it possesses complex roots and complex coefficients. The function j( T) has the same ordinates as F(T) for n + 1 distinct abscissas as Tn, Tn- I . .. TO. The Lagrange interpolation formula through the three points Ti-2 . F(T;_2), T;- I, F(Ti-l), and T;, F(Ti ) of F(T) is,

j(T) = A 2T2+AIT+ Ao (32)

where A 2, AI, and Ao are evaluated from the re­quirement: j(T t-2) = F(T t- 2), jh- l) = F(Tt- l) and j(T.) = F(Tt). Upon solving for A 2, Ail Ao and using the quantities

(3 (3; (3= T- Ti, (3i = Tt- T;- I, (3,-I= T;-I- T/- 2, A= (3- ' Ai=(3- '

t ; - 1

and Ot= 1 + Ai the Lagrange interpolation formula becomes a quadratic in A:

j (T) = A20i l[F(T i-2)A~-F(T ;-1) AiOi

+ Fh)At]+ A8 - I [F(T; -2)A1-Fh-I)o~

+ F(T;) (A;+o;)] + F(T;). (33)

TtH is found for the condition,j(T) = O, solving for A and employing the relationship,

(34)

Rationalizing the numerator of the standard quad­ratic formula, and solving for A;+l>

A= - 2Fh)0; gi ±.Jg~-4Fh)OiAi [F(T;- 2)Ai-F(Ti- l)Ot + F(T;)]

(35) where,

gt=Fh-2)A~- (FT i- l)oi+ F(T;) (A; + Oi) . (36)

The sign choice in the denominator of AH I (35) is resolved by selecting the value for the larger denomi­nator. This choice of A;+I gives T ;+I the root of the LaGrange interpolation formula (32) which is nearer T;, TO, TI, and T2, the initial value of T are estimated

in this instance from the limiting roots T8I o(Oe= O) and T.,oo (0. = CD) for any desired 8 [Johler et al., 1956], with the corresponding FH evaluated from (30 or 31) .

The actual values of T 8,0 and T 8,00 are used for 8

< 4 while approxin1ations for other 8 are determined from the following [Miller, 1946]:

(37)

(38)

where ,

37r Y2=S (48+ 1). (40)

The iterative process is terminated with Ti, the desired root when,

where € is a predetermined number. With procedures developed for finding Hankel

functions and T'S, Er may be calculated from (1) and (3) .

4. Discussion of Results

In figures 2 and 4, describing effects of land and sea water, respectively, the transmitter on the sur­face, the receiver at varying heights above the sur-

face in wavelengths, A= ] and for any distances along

the surface of the earth, the ratio of the field aloft . IE(w d h) 1 relatIve to that at the surface (h= O), E(~, ;z) ,ap-

proaches a value of one as the height of the receiver approaches zero. For increasing heights of the re­ceiver, this ratio first exhibits a minimum less than one, although slight in some instances, and then in­creases in an exponential manner. The analytical behavior of the height-gain function has been dis­cussed in considerable detail by Wait [1956J.

Characteristics of the amplitude ratio described above, however. with some differences, are present if both transmitter and receiver are elevated (fig. 6) . For a transmitter height of 1 wavelength and vary­ing receiver height , for any distance along the sur­face of the earth, the amplitude ratio approaches a value less than 1 as the receiver height approaches zero. For a transmitter of height 10 wavelengths and varying receiver height , the amplitude ratio ap­proaches values greater than one which varies with the distance along the surface as the receiver height approaches zero. The amplitude ratios for these elevated transmitters increase very rapidly as ex­pected because there are two height gain factors increasing in an exponential manner.

For the phase relationships (figs. 3, 5, and 7) cor­responding to the amplitude ratios (figs. 2, 4, and 6)

104

10'

-"0 21_ "0_ 3- 1000 3 _ WW

~ ~ 100 a: w o ::::J f-

a: 10 ::;; <l:

[:- 100kcls ---- 1000 --- 10000

a ": 0.75 E2: 15

CT '" 0.005

I

~o' 4,dI0-') I

100

10-' L:-_..LJ..l---,---'-l-L' ...J'..L' :C" 'L--'--'-..l-LLL __ L-...l..L'-;:_--L..L..L.Lill

10-2 10-' 10 102 10'

h ALTITUDE, ~, WAVELENGTHS

FIGURE 2, Amplitude of the vertical electric field, E" relative to surface value at various distances over land, with eithel' transmitter aI' receiver elevated,

In thiS figure as wel] as those which follow h, and h2 can be interchanged

102 .

0 VERTICAL ELECTRIC DIPOLE SOURCE

II 1.4>(~-6) p'sec <'oj

.c 10 II

I =~~:D I 5 III 0 Z W 0 ::::J U ..J W a : 0 .75

§! III tO z : 15

~~~~OO 0 W ex: ". 0.005 u u <l: :E 1-'0= 417"(10 -7 ) 200 ,- 200 lL.

/:;;/ 10000 kcAl 100 a: 10-' -

::::J " III

i 0 100 kcAl /~,OOO mi f- , / // 500

W N

:::: .c 10-2 // / 200 f- "- / / 100 <l: / / ..J ~ // /'--1000 kcAl W a: /

/ /'

W 10-' / (f)

<l: I a.

h ALTITUDE, f ' WAVELENGTHS

FIGURE 3, Phase of the vertical elec17'ic field, E" relative to surface value at vaI'i ous distances over land with either trans­mitter aI' receiver elevated,

N_ .L:~ "0_

" 3 i w W

~ !:i a: W 10 0 ::::J ':::: ..J a. ::;; <l:

DIPOLE SOURCE

! , f 10000 kc/s

lWm~~/ 500 mi

100 kc/s IOO ml

h ALTITUDE, ~ ,WAVELENGTHS

/

FIGURE 4, Amplitude of the vertical electl ic field, E" relative to surface value at various dis tances ovel' sea watel', with either tmnsmitter or receiver elevated,

0 II N

.c II £ III

0 W Z ::::J 0 ..J U

W <l: III > 0 W ex: u u <l: :E lL. a:

'0 ::::J (f)

i 0 f-

~ W > .c f- ,,-<l: :3 ..J W a: W III <l: I a.

h ALT IT UDE 'T ,WAVELENGTHS

FIG URE 5. Phase of the vel,tical electr'ic field, E " relative to surface value at various distances over sea water with either tmns mittel' 0 1' I-eceiver elevated,

respectively, the difference of the phase aloft and that at the surface (h = O), t (w,d ,h )-t (w,d) < < 0 with a very slight minimum at low increasing heights. To illustrate details of the phase in this region, a logarithmic scale was used with t (w,d,h) - t (w,d)<O graphed as lead with dashed lines, and t (w,d,h) - t (w,d) > 0 designated as lag with solid lines (figs.

105

- -0 .c: _

"0 3 j W 102 w

o >-­<t cr w o ::> >-­-' Q

::;; <t

10

f = 100 kc/s

- -- - -/"'/

/ /

/

I I

/ /

tOO mi

1200 I I I

: /1500 L",,// 11000

==::-:-:-==~~~~_=-=::::-:-=-===:--.::: :J ... / 500 "

100

10 - 2 10- 1 I 10 10 2

h ALTITUOE, -f, WAV EL ENGT HS

FIGURE 6. Amplitude oj the vertical electric field, E " relative to surjace value at various distances over land, with both transmi tter and receiver elevated.

3, 5). The phase lag may be ambiguous by 271" r adians in any caso, which in microseconds is 10 . 1, or 0.1 for 100, 1,000. and 10,000 kc/s, respectively. In figure 7, the lower solid lines represent lead for the transmitter at 1 wavelength. while the upper solid lines represent lag for the same case. For 10-wavelength height of transmitter, the dotted lines represent lead.

5. Conclusions

All of the amplitude ratio curves have a minimum., even though slight in some cases, and then rise ex­ponentially. For transmitter and receiver both elevated this ratio is very pronounced. The phase curves also exhibit a minimum, sometimes very slight, with t (w,d,h)- t (w,d) changing from fl, nega­tive to a positive quantity or from a phase lead to a phase lag relative to the surface.

In applying the m ethods used in this paper , it may be concluded that the iterative method described is applicable for roots of Riccati's differential eq (5) to any desired accuracy consistent with the capacity of the electronic computer used . Also, the method for evaluating Hankel functions, as it is applicable for real or complex order, may be used for the many other applications of these functions.

In particular, the height gain functions and the ground wave field can be evaluated at most any dis­tance or altitude of transmitter and/or receiver with the aid of the t echniques presented, the ultimate limitation being the computer capacity and speed , and tho failure of the approximation in (3).

a II N

.c:

100 0

II 100 - (/) .c: 0 - z W 0 ::> u -' w ~ ~ w a: 10 u U <t ~ lL. cr --;::: => -0 (/)

a ] >-- I w > .c: t:i -o~ -' ] w _ cr -1 0-1 W (/)

<t I Q

I 100 mi /

I I

I 200

I i f = 10 0 kc/s / I a = 0.75 / /

"2 =15 / I 200

(J = 0.00 5 /! 100 J, jl -VIO OO

- ---- -----------' / _ __________ ___________ / ....... ....-/, 500

-------------------- -------

: ___________________ 1000

~IOOO '\ 500

200

10-3 10-2 10 - 1 10 100

h ALTITUDE , ;. WAV EL ENGTHS

FIGURE 7. Phase oj the vertical electric field, E" relative to s1trface value at various distances over land with both trans­mitter and receiver elevated.

6 . References Bremmer, H. , T errestia l radio waves; th eory of propagation

(Else vier Publ. Co. , New York, N.Y., 1949). D avis, P. and P. Rabinowitz, Abscissas and weights for

Gaussian quadratures of high order, J . R esearch NBS 56, 35- 37. (Jan. 1956) RP2645.

Erdely i, A ., H igher transcendentalfunctions, Vol. 2 (McGraw­Hill Book Co., Inc., New York, N.Y. , 1953) .

Furry, W., Tables of modified Hankel function s of order one­third and t heir derivatives (Harvard Univ. Press, Cam­bridge, Mass., XVII, 1945).

J ahnke, E., and F. Emde, Tab les of function s with formulae a nd curves (Dover Publication s, New York, N.Y., 1945).

J effreys, H. , and B. J effreys, Methods of mathematical physics (Univers ity P ress, Cambridge, 1956).

Johler, J . R , W. J. Kellar, and L . C. Walters, Phase of the low radiofrequency ground wave, NBS Circ. 573 (March 14, 1956).

Johler, J . R , L. C. Walters, and C. M. Lilley, Low and very low-frequency tables of gro und wave parameters for t h e spherical earth theory: t he roots of Riccati's differential equation, N BS T ech. Note 7 (1959).

Kopal, F., Numerical analysis (J ohn Wiley & Sons, Inc., New York, N .Y. , 1955).

Miller, J . C. P ., The Airy integral, Math. T ables, P ar t- vol. B. (University Press, Cambridge, B-48, 1946).

Muller, D . E., A method for solving algebraic equations usin g a n a uto matic computer, Math. T a bles and oth er Aids to Computation , vol. 10, 208 (1956).

van del' Pol, B ., and H . Bremmer, The diffraction of electro­magnetic waves from an electrical point source round a fin itely conduct ing sphere, with applications to radio­telegraph y and the t heory of t he rainbow, Natuurkundig Laboratorium del' N. V. Philips Gloeilampenfabrieken, EindllO ven, Holla nd, Phil. :vlag., Ser . 7, 24" pt. 1, 141 (July 1937); p t. 2, 825 (Nov. 1937); Supp. J . Science 25, 817 (June 1938).

'Wait, J . R. , R adiation from a vertical antenna over a curved stratified gro und, J. Resemch N BS 56, 237- 244 (April 1956) RP2671.

IYatson, G. N., The diffraction of electric waves by the ear th and th e t ransmission of electric waves round t he ear th , Proc. Roy. Soc. (London) [AJ 95, 83 (1918).

IYatson, G. N., Besselfunctions (University Press, Cambridge, 1948).

(Paper 66D 1- 177)

106