ishizuka chap one

8/13/2019 Ishizuka Chap One

1/44

Scattering of Acoustic and Electromagnetic

Waves by Small Impedance Bodies of

Arbitrary Shapes


2/44


3/44

Scattering of Acoustic and Electromagnetic

Waves by Small Impedance Bodies of

Arbitrary Shapes

Applications to Creating New Engineered

Materials

Alexander G. Ramm

Department of Mathematics

Kansas State University, Manhattan, KS 66506-2602, USA

[email protected]

MOMENTUM PRESS, LLC, NEW YORK


4/44

Scattering of Acoustic and Electromagnetic Waves by Small Impedance Bodies of Arbitrary Shapes

Copyright Momentum Press, LLC, 2013.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or

transmitted in any form or by any meanselectronic, mechanical, photocopy, recording or any

other except for brief quotations, not to exceed 400 words, without the prior permission of the

publisher.

First published by Momentum Press, LLC

222 East 46th Street, New York, NY 10017

www.momentumpress.net

ISBN-13: 978-1-60650-621-9 (hard cover, case bound)

ISBN-10: 1-60650-621-8 (hard cover, case bound)

ISBN-13: 978-1-60650-622-6 (e-book)

ISBN-10: 1-60650-622-6 (e-book)

DOI: 10.5643/9781606506226

Cover Design by Jonathan Pennell

10 9 8 7 6 5 4 3 2 1

Printed in the United States of America.


5/44

Contents

Contents v

Preface ix

Introduction xv

1 Scalar wave scattering by one small body of an arbitrary shape 1

1.1 Impedance bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Acoustically soft bodies (the Dirichlet boundary condition). . . 11

1.3 Acoustically hard bodies (the Neumann boundary condition). . 14

1.4 The interface (transmission) boundary condition. . . . . . . . . 18

1.5 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Scalar wave scattering by many small bodies of an arbitrary shape 27

2.1 Impedance bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 The Dirichlet boundary condition. . . . . . . . . . . . . . . . . . 36

2.3 The Neumann boundary condition. . . . . . . . . . . . . . . . . 39

2.4 The transmission boundary condition. . . . . . . . . . . . . . . . 42

2.5 Wave scattering in an inhomogeneous medium. . . . . . . . . . 44

v
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


6/44

vi A. G. Ramm

2.6 Summary of the results. . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Creating materials with a desired refraction coefficient 51

3.1 Scalar wave scattering. Formula for the refraction coefficient. . 51

3.2 A recipe for creating materials with a desired refraction

coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 A discussion of the practical implementation of the recipe. . . . 55


4 Wave-focusing materials 59

4.1 What is a wave-focusing material? . . . . . . . . . . . . . . . . . 59

4.2 Creating wave-focusing materials. . . . . . . . . . . . . . . . . . 62

4.3 Computational aspects of the problem. . . . . . . . . . . . . . . 73

4.4 Open problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


5 Electromagnetic wave scattering by a single small body of an

arbitrary shape 79

5.1 The impedance boundary condition. . . . . . . . . . . . . . . . 79

5.2 Perfectly conducting bodies. . . . . . . . . . . . . . . . . . . . . . 84

5.3 Formulas for the scattered field in the case of EM wave

scattering by one impedance small body of an arbitrary shape. 85


6 Many-body scattering problem in the case of small scatterers 95

6.1 Reduction of the problem to linear algebraic system. . . . . . . 95

6.2 Derivation of the integral equation for the effective field. . . . . 100

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


7/44

Scattering of Acoustic and Electromagnetic Waves and Applications vii

7 Creating materials with a desired refraction coefficient 103

7.1 A formula for the refraction coefficient. . . . . . . . . . . . . . . 103

7.2 Formula for the magnetic permeability. . . . . . . . . . . . . . . 105


8 Electromagnetic wave scattering by many nanowires 107

8.1 Statement of the problem. . . . . . . . . . . . . . . . . . . . . . . 107

8.2 Asymptotic solution of the problem. . . . . . . . . . . . . . . . . 112

8.3 Many-body scattering problem equation for the effective field. 116

8.4 Physical properties of the limiting medium. . . . . . . . . . . . . 120


9 Heat transfer in a medium in which many small bodies are

embedded 125

9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.2 Derivation of the equation for the limiting temperature. . . . . 127

9.3 Various results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132


10 Quantum-mechanical wave scattering by many potentials with

small support 137

10.1 Problem formulation. . . . . . . . . . . . . . . . . . . . . . . . . . 137

10.2 Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


11 Some results from the potential theory 147

11.1 Potentials of the simple and double layers. . . . . . . . . . . . . 147

11.2 Replacement of the surface potentials. . . . . . . . . . . . . . . . 158
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


8/44

viii A. G. Ramm

11.3 Asymptotic behavior of the solution to the Helmholtz equation

under the impedance boundary condition. . . . . . . . . . . . . 173

11.4 Some properties of the electrical capacitance. . . . . . . . . . . 177


12 Collocation method 185

12.1 Convergence of the collocation method. . . . . . . . . . . . . . . 185

12.2 Collocation method and homogenization.. . . . . . . . . . . . . 192


13 Some inverse problems related to small scatterers 195

13.1 Finding the position and size of a small body from the

scattering data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

13.2 Finding small subsurface inhomogeneities. . . . . . . . . . . . . 202

13.3 Inverse radiomeasurements problem. . . . . . . . . . . . . . . . 20713.4 Summary of the results. . . . . . . . . . . . . . . . . . . . . . . . . 210

Appendix 211

A1. Banach and Hilbert spaces.. . . . . . . . . . . . . . . . . . . . . . . 211

A2. A result from perturbation theory. . . . . . . . . . . . . . . . . . . . 213

A3. The Fredholm alternative. . . . . . . . . . . . . . . . . . . . . . . . 214

Bibliographical Notes 223

Bibliography 229

Index 239
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


9/44

Preface

The author wishes to let the reader know what this book is about and what

practical conclusions engineers can find in this book.

In this book the author presents systematically his theory of scalar wave

scattering as well as electromagnetic (EM) wave scattering by one and many

small bodies of arbitrary shapes. If the characteristic size of a body is a, then

smallness of the body means that ka 1, where kis the wave number, k= 2,

and is the wavelength. In a homogeneous medium =v T, v is the wave

velocity and T is the period of the wave,= 2T is its frequency and k=2 =

v.

The boundary conditions on the boundarySof a bodyD, that we impose,

include the impedance boundary condition, the Dirichlet, the Neumann, and

the transmission (interface) boundary conditions for scalar wave scattering

and the impedance boundary condition for EM wave scattering.

In all cases, we give explicitly analytical formulas for the field scattered by

one small body of anarbitrary shape. These results are new.

The theory of wave scattering by small bodies was originated by Rayleigh

(1871), who understood that the main term of the scattered field under his

assumptions is given by the dipole radiation. Later it was found that the mag-

netic dipole radiation for perfectly conducting body is of the same order of

ix


10/44

x A. G. Ramm

magnitude as the electric dipole radiation. Rayleigh and his followers did not

give analytic formulas for calculating the induced dipole moment on a small

body of an arbitrary shape. This was done nearly 100 years later in Ramm

(1969a,b,1970,1971a,b,c) and presented in the booksRamm(1980b,1982,

2005b). The new results on wave scattering by small bodies of arbitrary shapes,

presented in this book, are based on the papersRamm(2007a,b,c,d,e,f,g,i,j,

2008a,b,d,2009a,b,e,2010a,b,c,2011a,b,2013a,b,c,d,f). These results include

an analytical and numerical solution of wave scattering by many small bodies

and a derivation of the integral equation for the effective field in the medium

in which many small bodies (particles) are embedded. The important point

of our theory is the reduction of the many-body scattering problem to finding

some numbers rather than the boundary functions. This simplifies the prob-

lem drastically and allows one to solve it. On the other hand, this reduction is

asymptotically exact asa 0.

Our physical assumptions include the case when the distance dbetween

neighboring particles is much smaller than the wavelength,d , although

a d. These assumptions imply that the "multiple scattering effects" are cru-

cial. By these effects we understand that the influence of the field scattered by

all particles, except one, on this particle. It is not possible to use the Borns

approximation when these effects are crucial.

Although the theory of wave scattering by one and many small bodies is of

great interest by itself, we illustrate its possible applications by showing how

one can create a material with a desired refraction coefficient by embedding in

a given material many small impedance particles. A clear recipe is formulated

for doing this. Practical implementation of this recipe is discussed.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


11/44

Scattering of Acoustic and Electromagnetic Waves and Applications xi

Another problem of interest is creating materials which have a desired

wave-focusing property. This means that their refraction coefficient is cho-

sen so that the corresponding radiation pattern approximates well a desired

pattern, that is, a desired function on a unit sphere.

So far we have discussed mostly the scattering of scalar waves. Similar

results are obtained for electromagnetic (EM) wave scattering. We derive an-

alytical formulas for the EM field scattered by one impedance (or perfectly

conducting) small body of an arbitrary shape, solve many-body EM wave

scattering problems when the scatterers are small impedance bodies of an

arbitrary shape, derive the equation for the effective field in the medium in

which very many such bodies are embedded, and calculate the refraction

coefficient in this medium.

No such results were obtained earlier, to our knowledge. These formulas

are used for developing a numerical method for solving many-body scatter-

ing problems in the case of small impedance bodies of an arbitrary shape. An

equation for the effective field is studied. This equation leads to new physical

effects in the new medium, created by the embedding many small particles.

These effects include change of the refraction coefficient and of magnetic

permeability. One may use these results in practice in order to change the

refraction coefficient in the desired direction.

The author discussed also some physical problems of interest. For exam-

ple, wave scattering by many nanowires (thin cylinders) is discussed. Heat

propagation in the medium, in which many small bodies are embedded, is

investigated. Theory of quantum-mechanical scattering by many potentials

with small supports is developed.


12/44

xii A. G. Ramm

Collocation method is developed for solving the equation for the effective

field. Its convergence is proved and the rate of convergence is given. This col-

location method is used for a justification of a homogenization-type theory.

There are many books and papers dealing with homogenization theory. Quite

often it is assumed in these books that periodicity condition with a parameter

is satisfied, that the spectrum of the related problem is discrete, and that the

corresponding operator is selfadjoint. None of these assumptions are used in

our theory. We develop a version of the theory that fits our assumptions.

Several inverse problems are studied: finding the position and size of a

small body from the scattering data for two scatterers, one of which is large

and known; finding small subsurface inhomogeneities from the scattering

data measured on the surface; inverse radiomeasurements problem: finding

the EM field distribution in an aperture of a mirror antenna in radio-wave

diapason from the measurements of the field scattered by a small probe.

We also derive some results in potential theory that are used in this

book. Among these results some are new. For example, necessary and suf-

ficient conditions are given for a double-layer potential to be representable

as a single-layer potential, and vice versa; asymptotic of the solution to the

Helmholtz equation satisfying the impedance boundary condition is given as

the impedance tends to infinity; a variational principle is derived for pos-

itive quadratic forms, and some classical variational principles for electric

capacitance are obtained from this abstract principle.

This book is written for a wide audience which includes mathematicians,

specialists in numerical analysis, engineers, and physicists. The author hopes

that his theory will be implemented practically, in materials science and other

areas. Some results on such a numerical implementation the reader can find


13/44

Scattering of Acoustic and Electromagnetic Waves and Applications xiii

inAndriychuk and Ramm(2010),Andriychuk and Ramm(2011), and Ramm

and Andriychuk (2014b).

The book is essentially self-contained: the results necessary for under-

standing the details of the derivations are provided with proofs.

Keywords

Acoustic waves, electromagnetic eaves, eave scattering, small impedancebodies of an arbitrary shape, wave scattering by small impedance bodies

of arbitrary shapes, creating materials with a desired refraction coefficient,

meta-materials, nanowires, radio measurements, inverse problems
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


14/44


15/44

Introduction

What is this book about? Why was it written?

This book is about wave scattering by one and many small scatters of an

arbitrary shape. The interest to this question goes back to Rayleigh (1871), who

understood that the main term in the scattered field will be the dipole radia-

tion. Under his assumptions the smallness of the body means k a 1, where

k= 2

is the wave number, ais the characteristic size of the body, and the or-

der of magnitude of the scattering amplitude is O(a3) at a large distance from

the small body. Rayleigh did not give analytical formulas for calculating the

field scattered by a small body of an arbitrary shape. Such formulas were ob-

tained for balls (of an arbitrary shape) in 1908 by Mie. He did not assume that

the scatterer, the ball, is small. However, his method is based on the separation

of variables and cannot be used for bodies of an arbitrary shape.

For small bodies of an arbitrary shape, analytical formulas allowing one

to calculate scattered field with any desired accuracy were obtained by the

authorRamm(1969b,1970) about 100 years after Rayleighs work of 1871.

There is a very large literature on wave scattering by small bodies. The rea-

son is simple: this problem is of interest in many applications. Let us mention

light scattering by atmosphere dust, by cosmic dust, by colloidal particles in

xv
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


16/44

xvi A. G. Ramm

water, finding the location and size of the small subsurface inhomogeneities

from the observation of the field, scattered by these inhomogeneities and

measured on the surface.

In this book much attention is given to creating materials with a desired

refraction coefficient by embedding many small particles into a given ma-

terial so that these particles have prescribed boundary impedances and are

distributed in the given material with a prescribed density.

It turns out that a wide range of the desired refraction coefficients can be

obtained in this way. A recipe for creating a desired refraction coefficient in a

given material is formulated. Namely, if an arbitrary fixed bounded domain

is given, the refraction coefficient of the material in this domain is known to

be n0(x), outside of n0 =const, and if one wishes to create in a desired

refraction coefficient n(x), then the author gives a recipe for doing this. He

tells how many small particles of a characteristic size ashould be distributedin , and with what density (number of particles per unit volume) and with

what boundary impedances these small particles should be distributed in or-

der that the resulting medium will have the refraction coefficient that differs

from the desiredn(x) as little as one wishes.

In particular, one can create materials with negative refraction coefficient.

This is of interest in connection with metamaterials.

A refraction coefficient can be created so that the resulting material will

have wave-focusing property.

This means that the plane wave, scattered by the body, will have the radia-

tion pattern which approximates well a desired function on the unit sphere.

For example, the scattered field will mostly be scattered in a desired solid

angle.


17/44

Scattering of Acoustic and Electromagnetic Waves and Applications xvii

This book was written so that the novel methods for solving one- and

many-body wave scattering problems be available for a wide audiences in-

cluding mathematicians, engineers, physicists, specialists in computational

mathematics and materials science, and graduate students in these areas. The

material presented is interdisciplinary, and the author tries to present this

material in a self-contained way and so that it will be understandable to a

reader from the above broad audience.


18/44


19/44

CHAPTER1

SCAL AR WAVE SCATTERING BY ONE

SMALL BODY OF AN ARBITRARY

SHAPE

1.1 Impedance bodies.

Consider the following wave scattering problem

(2 + k2)u= 0 inD := R3 \ D, (1.1.1)

u

N= u onS, (1.1.2)

u= u0 + v, (1.1.3)

v

|x| i kv= o

1

|x|

, |x| . (1.1.4)

Here k> 0 is a wave number, k= const, Dis a bounded domain with a smooth

boundaryS, N is the unit normal to Spointing out ofD, uNis the limiting

value of the normal derivative ofuon S from D, is a constant which we

1


20/44

2 A. G. Ramm

call the boundary impedance,u0 = ei kx is the incident field, the plane wave,

S2, is a unit vector, S2 is the unit sphere, vis the scattered field, condi-

tion (1.1.4) is the radiation condition at infinity. It is assumed to be satisfied

uniformly with respect to the unit vector x0 = x|x|

, that is, with respect to the di-

rection along whichxtends to infinity. Smoothness ofSthroughout this book

means that the equation of the surface Sin local coordinates is aC1, func-

tion. The local coordinates are defined as follows. Fix a point s Sand let it

be the origin of the coordinate system, the z-axis of which is directed along

the normal Ns to Sat the point s, and the plane x,y is tangent to Sat this

point. We often write x1, x2, x3 in place ofx,y, z, and then x= (x1, x2, x3). Let

x3 = f(x1, x2) be the equation ofSin this coordinate system. Then, by con-

struction, f(0,0) = 0, f(0,0)

xj= 0,j = 1,2. The assumption S C1,, 0 < 1,

means that |f(x1, x2) f(x1, x

2)| C[(x1 x

1)

2 + (x2 x2)

2]1/2, where C is a

constant that does not depend ons, xandx, and = e1

x1+ e

2

x2, {e

j}3

j=1is

the Cartesian basis ofR3.

The scattering problem (1.1.1)(1.1.4) can be considered, for example, as

the scattering of an acoustic wave. In this case uhas the physical mean-

ing of the pressure or acoustic potential. The Dirichlet boundary condition

u|S= 0 describes acoustically soft body (zero pressure on the boundary) and

the Neumann boundary conditionuN|S= 0 describes acoustically hard body

(the normal component of the velocity uvanishes on S). The impedance

boundary condition describes a linear relation between the pressure and the

normal component of the velocity onS.

The first task is to prove that the scattering problem has a solution and

this solution is unique. In this case we say that the scattering problem has a


21/44

Scattering of Acoustic and Electromagnetic Waves and Applications 3

unique solution. If we want to state that there is at most one solution, but the

existence of the solution is not asserted, then we say there exists at most one

solution.

Theorem 1.1.1. Assume that Im 0. Then problem(1.1.1)(1.1.4) has at most

one solution.

Proof. Since the problem is linear it is sufficient to prove that the correspond-

ing homogeneous problem, that is, the problem with u0 = 0, has only the

trivial solution u= 0. To prove this, multiply equation (1.1.1) byu, the complex

conjugate ofu, subtract the complex conjugate equation(1.1.1) multiplied by

uand integrate over the region D BR:= DR, whereR> 0 is a large number

that we take to infinity andBR = {x: |x| R} is a ball of radius Rcentered at

the origin. The origin we take insideDarbitrarily. The result is

0 =

D

R

[u(2 +k2)u u(2 + k2)u]d x=

SR

(uuN uuN)d s

S

(uuN uuN)d s

(1.1.5)

Here the Greens formula was used.

From the radiation condition(1.1.4) one gets:

SR

(uuN uuN)d s= 2i k

SR

|u|2d s+ o(1), R . (1.1.6)

From the impedance boundary condition (1.1.2) one gets:

S

(uuN uuN)d s=

S

(|u|2 +|u|2)d s= 2iIm

S

|u|2d s. (1.1.7)

From equations (1.1.5)(1.1.7) it follows that

limRkSR |u|

2d s ImS|u|2d s = 0. (1.1.8)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


22/44

4 A. G. Ramm

If Im 0, then relation (1.1.8) implies

limR

SR

|u|2d s= 0. (1.1.9)

This and the radiation condition (1.1.4)imply thatu= 0 outside any ballBR

D. This claim is proved inRamm (1986), p.25. It is known as Rellichs lemma.

Theorem1.1.1is now a consequence of the unique continuation principle for

a solution to homogeneous Helmholtz equation: if such a solution vanishes on

an open subset in the domain D where it solves the homogeneous Helmholtz

equation, then this solution vanishes everywhere inD.

Theorem1.1.1is proved.

Remark 1.1.1. Physically the impedance can be any constant satisfying the

condition Im 0 which guarantees the uniqueness of the solution to the

scattering problem(1.1.1)(1.1.4).

Let us now prove the existence of the solution to the scattering problem

(1.1.1)(1.1.4).

Theorem 1.1.2. Problem(1.1.1)(1.1.4)has a (unique) solution. This solution

can be found of the form

u(x) = u0(x) +

Sg(x, t)(t)d t, (1.1.10)

g(x,y) :=ei k|xy|

4|xy|. (1.1.11)

The function(t)in(1.1.10)is uniquely determined.

Proof. For any function (1.1.10) solves equation (1.1.1) and satisfies condi-

tions (1.1.3)(1.1.4). Therefore, it solves problem (1.1.1)(1.1.4) if and only if
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


23/44


can be found so that condition (1.1.2) is satisfied, that is,

u0N u0 +A

2T= 0, (1.1.12)

where

A := 2

S

g(s, t)

NS(t)d t, T :=

S

g(s, t)(t)d t. (1.1.13)

Formula

NSSg(x, t)(t)d txD,xs =A

2(1.1.14)

is proved in Section 11.1, see also Gnter (1967). Equation (1.1.12) is of

Fredholm type because operatorsAand Tare compact in C(S), see Appendix.

Therefore, the existence of the solution to equation (1.1.12)follows from the

Fredholm alternative (see Appendix) if one proves that the homogeneous

equation (1.1.12) has only the trivial solution. The homogeneous equation

(1.1.12) is of the form

A2

T= 0, (1.1.15)

which is

uN u= 0 onS. (1.1.16)

By Theorem 1.1.1 the corresponding function u(x) = 0 in D because it

solves equation (1.1.1)in D, satisfies the radiation condition (1.1.4) and the

boundary condition (1.1.2).

Ifu =0 in D, then u = 0 on S because potentials of a single layer are

continuous in R3, see Section11.1.Thus,usolves equation (1.1.1) inDand

satisfies condition (1.1.2). This implies thatu= 0 in D. Indeed, multiply equa-

tion (1.1.1) in Dbyu, use Greens formula and the boundary condition (1.1.2),

and get

2iImS

|u|2d s= 0. (1.1.17)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


24/44

6 A. G. Ramm

If Im = 0, thenu = 0 onS, souN = 0 onSandu = 0 inDby the uniqueness

of the solution to the Cauchy problem for the Helmholtz equation. Ifu = 0 in

DD then= u+NuN= 0, and Theorem 1.1.2 is proved. If Im= 0, then one

deals with the real number in (1.1.2).

If, as we assume, the bodyDis sufficiently small, thenk2 is not an eigen-

value of the Laplacian inDalso in the case Im = 0. So, in this case alsou= 0

inD D and = 0.


The existence and uniqueness of the solution to the scattering problem

does not depend on the size of the body D. This size we measure by the num-

bera:= 12 diamD. We assume that the body is small in the sensek a 1. This

assumption allows us to derive analytic, closed form formulas for the field,

scattered byD, at the distancesd a, in the "far zone."

Our next task is to derive an analytical formula for the scattered field at the

distance |x| a. To do this, take into account the following formula

ei k|xt|

4|x t|=

ei k|x|i kx0t

4|x|

1 + O

|t|

|x|

, |t| a. (1.1.18)

Herex0 = x|x|

. Formula (1.1.18) can be established easily. One has:

|x t| = |x|

1

2x0 t

|x|+

|t|2

|x|2 = |x|

1 + O

|t|

|x|

(1.1.19)

and

ei k|xt| = ei k|x|i kx0t+O

|t|2

|x|

= ei k|x|i kx

0t

1 + O

|t|2

|x|

. (1.1.20)

If|t| aanda is small, then O

|t|2

|x|

O

|t||x|

, s o (1.1.18) holds. In what follows

we denotex0 = and |x| = r. One hasS

g(x, t)(t)d t=ei kr

4r

S

ei kt(t)d t

1 + Oa

r

. (1.1.21)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


25/44


Letg(r) = ei kr

4r and

Q:=

S(t)d t. (1.1.22)

If

|Q|

St(t)d t

, (1.1.23)then one can write

S

ei kt(t)d t S

(t)d t= Q, (1.1.24)

wherethe sign stands for the asymptotic equality as a 0.

If (1.1.23) holds then

u(x) u0(x) + g(r)Q, r= |x| a. (1.1.25)

Therefore, the solution of the scattering problem is completed if the number

Qis found.

To findQwe use the exact integral equation (1.1.12). However, we do not

solve it for, but find the main term of the asymptotic ofQas a 0. To do

this, integrate equation (1.1.12) overSand evaluate the significance of each

term asa 0.

The first term is

I1:=

S

u0Nd s=

D

2u0d x= k

2

D

u0d x= O(a3), (1.1.26)

where we took into account that

Du0d x

Dd x= |D| = O(a3), (1.1.27)

and |D| := Vis the volume ofD.


26/44

8 A. G. Ramm

The second term is

I2:=

Su0d s= O(a2), (1.1.28)

where S

d s= |S| = O(a2), (1.1.29)

and |S| is the surface area ofS.

The third term is

I3:=

S

A

2 d s=

Q

2 +

1

2

SAd s. (1.1.30)

We claim that S

Ad s

S

A0d s=

Sd s= Q, (1.1.31)

where

A0 := 2

S

g0(s, t)

NS(t)d t, g0(x, t) :=

1

4|x t|. (1.1.32)

Indeed, one has

A= 2

D

NS

ei k|st|

4|s t|(t)d t

= 2

S

NS

1 + i k|s t| + O(k2|s t|2)

4|s t|

(t)d t

=A0+

S

O(|s t|)(t)d t

A0 asa 0. (1.1.33)

Let us check that S

A0d s=

Sd t. (1.1.34)

This is an exact relation. One has

2

S

d s

S

NS

1

4|s t|(t)d t=

S

d t(t)2

S

NS

1

4|s t|d s

=

S(t)d t. (1.1.35)


27/44


Here we have used the known formula (see Section 11.1):

2

S

NS

1

4|s t|d s= 1, t S. (1.1.36)

Proof of this formula is given in Section11.1.

Thus,

I3 Q, asa 0. (1.1.37)

Finally,

I4:=

S

Td s=

S

d t(t)

S

g(s, t)d s. (1.1.38)

One has S

g(s, t)d s= O(a), a 0. (1.1.39)

Let us assume that

lima0

a= 0. (1.1.40)

Assuming(1.1.40) we allow to depend onain such a way that (1.1.40) holds.

If is a constant independent ofathen (1.1.40) is obviously valid. If (1.1.40)

holds then

I4 = o(Q), a 0, (1.1.41)

provided that Q= 0. We will prove that Q= 0 under our assumptions. Keeping

the terms I3 and I2 and neglecting the terms I1 and I4 of higher order of

smallness asa 0, one gets

Q |S|u0(x1), x1 D. (1.1.42)

Ifu0(x) = ei kx andx1 D, thenu0(x1) = 1 + O(ka), sincek a 1.


28/44

10 A. G. Ramm

Formula (1.1.42) is our basic result. From (1.1.42) and (1.1.25) one gets

an analytical formula for the solution of the wave scattering problem (1.1.1)

(1.1.4) in the case of a small body of an arbitrary shape:

u(x) u0(x) g(r)|S|, |x| a, (1.1.43)

whereg(r) = ei kr

4r,r= |x|, andu0(x1) 1 ifk a 1.

Itisassumedin(1.1.43) that the origin is inside D. If the origin is elsewhere,

andx1 D, then formula (1.1.43) takes the form

u(x) u0(x) g(|x x1|)|S|, |x x1| a. (1.1.44)

Let us define the scattering amplitudeA(,, k) by the formula

vei kr

rA(,, k) + o

1

r

, r= |x| ,

x

r:=. (1.1.45)

Comparing (1.1.43) and (1.1.45) one obtains

A(,, k) = |S|

4 =

Q

4, (1.1.46)

where it is assumed thatx1 = 0, so thatu0(x1) = 1.

The physical conclusion is: the scattering is isotropic, that is, it does not

depend on and, and in absolute value the scattering amplitude is O(|S|) =

O(a2).

Let us summarize the results in a Theorem.

Theorem 1.1.3. Assume that ka 1, Im 0, and (1.1.40) holds. Then the

scattering problem(1.1.1)(1.1.4)has a unique solution. This solution can be

calculated by formula(1.1.43)if the origin is inside D, and by formula(1.1.44)

if the origin is elsewhere and x1 D.

The scattering is isotropic and the scattering amplitude is O(a2).
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


29/44


1.2 Acoustically soft bodies (the Dirichlet

boundary condition).

In this section problem (1.1.1), (1.1.3), (1.1.4) is studied but the impedance

boundary condition (1.1.2) is replaced by the Dirichlet condition

u= 0 onS. (1.2.1)

Physically this condition in acoustic means that the body is acoustically soft,

that is, the pressure onSis vanishing. Mathematically condition (1.2.1) is the

limiting case of condition (1.1.2) when . This is proved in Section11.3.

The case 0 yields the Neumann conditionuN = 0 onS, corresponding to

acoustically hard body.

The scattering problem with the Dirichlet condition has a unique solution,

as we will prove, and the analog of Theorems1.1.11.1.3will be established.

Theorem 1.2.1. Problem(1.1.1), (1.2.1), (1.1.3), and (1.1.4) has at most one

solution.

Proof. It is sufficient to prove that the corresponding homogeneous problem,

that is, the problem with u0 =0, has only the trivial solution. The proof is

essentially the same as the proof of Theorem1.1.1.

One obtains an analog of relation (1.1.8)with the integral overSvanishing

becauseu= 0 onS. Therefore, one concludes that relation (1.1.9) holds.

The rest of the proof goes as in the proof of Theorem1.1.1.


Let us prove the existence of the solution to problem (1.1.1), (1.2.1), (1.1.3),

and (1.1.4).
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


30/44

12 A. G. Ramm

Theorem 1.2.2. The above problem has a solution.

Proof. Let us look for the solution of the form (1.1.10). As in the proof of

Theorem1.1.2it is sufficient to prove that the equation that one obtains from

the boundary condition (1.2.1) is solvable.

This equation is

S

g(s, t)(t)d t= u0(s), s S. (1.2.2)

It is proved in Section 11.2 that equation (1.2.2) is (uniquely) solvable provided

thatk2 is not an eigenvalue of the Dirichlet Laplacian in D. This condition is

satisfied ifDis sufficiently small.


Let us prove an analog of Theorem1.1.3.

As in Section1.1one gets formula (1.1.25) whereQis defined in (1.1.22).

To find an analytical expression for Qwe rewrite equation (1.2.2) as

S

g0(s, t)(t)d t= 1 + O(ka), g0(s, t) =1

4|s t|. (1.2.3)

Neglecting the small termk a 1, one gets

Sg0(s, t)(t)d t= 1, s S. (1.2.4)This is an equation for the surface charge distribution that generates the

constant potential U on S, U = 1. With this interpretation the bodyD is

a perfect conductor and the quantityQ =

S(t)d tis its total charge. There

is a well-known relationCU = Q, whereCis the capacitance of the perfect

conductorD. Thus,

Q= C, (1.2.5)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


31/44


and formula (1.1.25) takes the form

u(x) = u0(x) g(r)C, r= |x| a, (1.2.6)

while formula (1.1.46) becomes

A(,, k) = C

4. (1.2.7)

These formulas solve the scattering problem (1.1.1), (1.2.1), (1.1.3),and(1.1.4).

Let us formulate the results.

Theorem 1.2.3. If ka 1 then the solution to the above scattering prob-

lem exists, is unique, can be calculated by formula(1.2.6) and the scattering

amplitude is given by formula(1.2.7).

Remark 1.2.1. This result is very useful practically because the author has

derived explicit analytical formulas which allow one to calculate electrical ca-

pacitance C for a conductor of an arbitrary shape, seeRamm (2005b) and

Section11.4.

For example, the zero-th approximation of C is given by the following

formula

C(0) =4|S|2

S

S

d s d t |st|

, C(0) C, (1.2.8)

where|S| is the surface area of S and the electric permittivity in D is equal to 1.

Let us draw some physically interesting conclusions from Theorem1.2.3.

Note that the electrical capacitanceC = O(a). Formula (1.2.7) shows that

the scattering is isotropic and the scattering amplitude is O(a), that is, it is

much largerthan in the case of the impedance boundary condition. If, for ex-

ample, = O( 1a ), 0 < 1, so that condition (1.1.40) is satisfied, then formula

(1.1.46) yieldsA(,, k) = O(a2), and O(a2) O(a) if< 1 anda 0.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


32/44

14 A. G. Ramm

1.3 Acoustically hard bodies (the Neumann

boundary condition).

Consider now the scattering problem (1.1.1), (1.1.3), (1.1.4) and replace con-

dition (1.1.2) by the Neumann boundary condition

uN= 0 onS. (1.3.1)

Our plan of the study is unchanged: we want to prove existence of a uniquesolution and derive analytical formulas for the solution and for the scattering

amplitude.

Theorem 1.3.1. Problem (1.1.1), (1.3.1), (1.1.3), and (1.1.4) has at most one

solution.

Proof. One can use the proof of Theorem1.1.1taking= 0 in this proof.


Theorem 1.3.2. The scattering problem (1.1.1), (1.3.1), (1.1.3), (1.1.4) has a

solution.

Proof. Let us look for the solution of the form (1.1.10). As in the proof of

Theorem1.1.2it is sufficient to prove that equation (1.1.12) with = 0 has a

solution. Since equation (1.1.12) with = 0 is of Fredholm type, the existence

of its solution will be proved if one proves that equation (1.1.15) with= 0 has

only the trivial solution. Let us prove this. The equation

A

2= 0 (1.3.2)

implies that uN = 0 on S. By Theorem 1.3.1 it follows that u = 0 in D.

Therefore, u= 0 on Sand usolves the Dirichlet problem for equation (1.1.1) in
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


33/44


D. This implies thatu= 0 inDbecausek2 cannot be a Dirichlet eigenvalue of

the Laplacean ifDis sufficiently small. Ifu= 0 in D D then = u+N uN= 0.

By the Fredholm alternative the existence of the solution to equation (1.1.12)

with = 0 is proved.


Let us now prove an analog of Theorem1.1.3.

Formula (1.1.24) is now taking the form

Q1:=

S

ei kt(t)d t

S(t)d t i kp

S

tp(t)d t, (1.3.3)

wherehere and below over repeated indices summation is understood.

The novel point in a study of the scattering problem with the Neumann

boundary condition is the necessity to use both terms (1.3.3) because they are

of the same order as a 0, namely, they are both O(a3). This is in contrast with

the problem with the Dirichlet boundary condition where Q= O(a), and with

the impedance boundary condition where Q= O(a2) as a 0. Moreover, we

prove that the scattering in the problem with Neumann boundary condition

is anisotropic, in sharp contrast with the cases of the Dirichlet and impedance

boundary conditions.

Let us first estimate

Q=

S(t)d t.

We look for the solution to problem (1.1.1), (1.3.1), (1.1.3), (1.1.4) of the

form (1.1.10). The boundary condition (1.3.1) yields the integral equation

for :

= A+ 2u0N onS. (1.3.4)


34/44

16 A. G. Ramm

Integrate this equation overSand use formula (1.1.30) to get

Q

S

u0Nd s=

D

2u0d x=

2u0(x1)|D|, (1.3.5)

where x1 D is an arbitrary point inside D and we took into account that

because D is small 2u0(x1) does not depend on the choice of x1. Since

D= O(a3), it follows from (1.3.5) that Q= O(a3) as was mentioned earlier.

Let us estimate the last integral in (1.3.3). Introduce the matrix (tensor)pq

by the formula

p q:=1

|D|

S

tpq(t)d t, (1.3.6)

where qis the unique solution to the equation

q= Aq 2Nq, q= 1, 2, 3, (1.3.7)

andNq= N eq, where {eq} is an orthonormal Cartesian basis ofR3. Equation

(1.3.7) is of Fredholm type and it has a solution because the corresponding ho-

mogeneous equation =Ahas only the trivial solution = 0. Indeed, this

equation is equivalent to the relation uN = 0, and u=

Sg(x, t)(t)d t satis-

fies equation (1.1.1) and the radiation condition (1.1.4). Thereforeu= 0 inD.

Consequentlyu= 0 on S. Thus, usolves equation (1.1.1) in Dand vanishes on

S. This means that k2

is the Dirichlet eigenvalue of the Laplacian in D, which is

a contradiction because Dis assumed to be sufficiently small. Therefore u= 0

inD D, and = u+N uN = 0. This proves the existence and uniqueness of

the solutionq to equation(1.3.7). The functionin equation (1.3.3) solves

equation(1.3.4), where

u0N=u0

xqNq. (1.3.8)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


35/44


Recall that summation overqis understood. Thus

2u0N= 2Nq

u0(x1)

xq

. (1.3.9)

From (1.3.4), (1.3.9), (1.3.7), and (1.3.6) one gets

i kp

S

tp(t)d t= i kpqpu0(x1)

xq|D|, (1.3.10)

where |D| is the volume ofD, pqis the tensor defined in (1.3.6), and pis the

p-th component of the unit vector := xx1|xx1|

,x1 D.

Therefore,

Q1 |D|

2u0(x1) + i kp qpu0(x1)

xq

, x1 D, (1.3.11)

where Q1is defined in (1.3.3).

Consequently, an analog of formulas (1.1.44) a n d (1.1.45) in the case of the

Neumann boundary condition takes the form

u(x) = u0(x) + g(x, x1)

2u0(x1) + i kpqpu0(x1)xq

|D|, |x x1| a,

(1.3.12)

where

=x x1

|x x1|, x1 D, (1.3.13)

and

A(,, k) = 2u0(x1) + i kp qp

u0(x1)

xq |D|

4. (1.3.14)

Ifx1is the origin, x1 = 0, and u0 = ei kx, then formula (1.3.14) can be rewritten

as

A(,, k) = k2|D|

4

1 +p qpq

. (1.3.15)

One can see from this formula thatA= O(a3), the scattering is anisotropic and

its anisotropic part is described by the tensor p qdefined by formula (1.3.6).

Let us summarize the results.


36/44

18 A. G. Ramm

Theorem 1.3.3. The scattering problem (1.1.1), (1.3.1), (1.1.3)(1.1.4) is

uniquely solvable. Its solution can be calculated by formulas(1.3.12)(1.3.13).

The scattering amplitude A(,, k) is given by formula(1.3.14), A(,, k) =

O(a3), and the scattering is anisotropic.

1.4 The interface (transmission) boundary

condition.

Consider the following scattering problem

(2 + k2)u= 0 inD, k2 > 0, (1.4.1)

(2 + k21 )u= 0 inD, k21 > 0, (1.4.2)

u+ = u, 1u+N= u

N, 1 > 0, (1.4.3)

u= u0 + v, (1.4.4)

v

r i kv= o

1

r

, r= |x| , (1.4.5)

whereu0is the incident field,

(2 + k2)u0 = 0 in R3. (1.4.6)

In particular, the plane wave incident field u0 =ei kx, S2, is often con-

sidered. The number 1 in the transmission boundary condition (1.4.3) is

assumed positive and 1 = 1. The numberk21 in (1.4.2) is not equal tok

2.

Physically problem (1.4.1)(1.4.6) corresponds to the incident wave prop-

agating throughDand in this process the wave is scattered.

As earlier, the first task is to prove the existence and uniqueness of the

solution to the scattering problem (1.4.1)(1.4.6).
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


37/44


Theorem 1.4.1. The above problem with1 = const> 0has no more than one

solution.

Proof. It is sufficient to prove that the homogeneous problem, that is, the

problem with u0 = 0, has only the trivial solution. To prove this, multiply equa-

tion (1.4.1) and (1.4.2) byu, the bar stands for complex conjugate, integrate

the second equation overD, the first equation over DR := BR D, BR := {x:

|x| R}, then use Greens formula and condition (1.4.3) and (1.4.5)and getS

(uu+N uu+N

)d s

S

(uuN uuN

)d s+

S

(uuN uuN)d s

=

S

[uu+N(1 1) uu+N

(1 1)]d s+ 2i k

SR

|u|2d s+ o(1)

= 0, (1.4.7)

where limR o(1) = 0.

One has, using Greens formula,

(1 1)

S

(uu+N uu+N

)d s= 0, (1.4.8)

becauseuandusolve the same equation (1.4.2).

Thus, relation (1.4.7) implies

limR

SR

|u|2d s= 0. (1.4.9)

From this, equation (1.4.1) and the radiation condition (1.4.5) withu0 = 0, u=

v, it follows thatu= 0 inD. Thus,u = uN= 0.

Consequently,u+ = u+N= 0, andu= 0 inD.


Remark 1.4.1. The proof remains valid with only a slight change if one assumes

that Imk2 0.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


38/44

20 A. G. Ramm

Theorem 1.4.2. Problem(1.4.1)(1.4.6)has a solution.

Proof. Let us look for a solution of the form

u(x) = u0(x) +

S

g(x, t)(t)d t+

D

g(x,y)u(y)d y, (1.4.10)

where

:= k21 k2, g(x,y) =

ei k|xy|

4|xy|, (1.4.11)

and has to be found such that conditions (1.4.3) are satisfied.

Note that equations (1.4.1), (1.4.2), (1.4.4), (1.4.5), and (1.4.6) are satisfied.

This is obvious for equations (1.4.4)(1.4.6), and follows for equations

(1.4.1)(1.4.2) from the equation

(2 + k2)g(x,y) = (xy). (1.4.12)

The first condition (1.4.3) is satisfied because both integrals in (1.4.10) are

continuous functions in R3.

The second condition (1.4.3) is satisfied if and usolve the system of

integral equations (1.4.10) and (1.4.13), where

1A+

2

A

2+ (1 1)

NSBu+ ( 1)u0N= 0. (1.4.13)

Here

A= 2

Sg(s, t)NS

(t)d t, Bu=

Dg(x,y)u(y)d y. (1.4.14)

Let us rewrite equation (1.4.13) as follows:

=A+ 2B1u+ 2u0N, (1.4.15)

where

=1

1 +, B1u:=

(Bu)

NS. (1.4.16)
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


39/44


If 0


40/44

22 A. G. Ramm

To make this formula practically applicable one needs to calculate the main

terms of the asymptotic of the surface integral in (1.4.18)andoftheterm u(x1).

Note that

g(s, t) = g0(s, t)[1 + O(ka)], g0(s, t) =1

4|s t|, s, t S, (1.4.19)

g(s, t)

NS=g0

NS[1 + O(k2a2)], a 0, (1.4.20)

A= 2Sg(s, t)

NS

(t)d t 2Sg0(s, t)

NS

(t)d t= A0, (1.4.21)

Bu

B

g0(x,y)u(y)d y=B0u, (1.4.22)

B1u B01u, B01u=

B

NSg(s,y)u(y)d y. (1.4.23)

One has S

ei kt(t)d t

S(t)d t i kp

S

tp(t)d t. (1.4.24)

Let

Q:=

S(t)d t, Q1:= Q i kp

S

tp(t)d t. (1.4.25)

Then, by formula (1.4.18), one gets

u(x) u0(x) + g(x, x1)[Q1 +u(x1)|D|], |x x1| a. (1.4.26)

Let us derive analytical formula forQ1 andu(x1). Integrate equation (1.4.15)

overSand get

Q:=

S

Ad s+ 2

S

B1ud s+ 2

S

u0Nd s. (1.4.27)

One has, by the divergence theorem,

2

S

u0Nd s 22u0(x1)|D|. (1.4.28)


41/44


Furthermore, using formula (1.1.34), one gets:

S

Ad s

S

A0d s=

Sd s= Q, (1.4.29)

and

2

S

B1ud s= 2

D

d x2x

D

g(x,y)u(y)d y.

Equation (1.4.12) implies

D

d x2x

Dg(x,y)u(y)d y= k2

D

d x

Dg(x,y)u(y)d y

D

d xu(x)

u(x1)|D|. (1.4.30)

Thus,

2

S

B1ud s 2u(x1)|D|, a 0. (1.4.31)

Therefore, equations (1.4.27)(1.4.31) imply

Q 2

1 +u(x1)|D| +

2

1 +

2u0(x1)|D|. (1.4.32)

In order to estimate u(x1) as a 0 let us integrate equation (1.4.10) overD

and obtain

u(x1)|D| u0(x1)|D| +

D

d x

S

g(x, t)(t)d t+

D

d x

D

g(x,y)u(y)d y.

(1.4.33)

One has

D

d x

S

g(x, t)(t)d t=

S

d t(t)

D

d xg(x, t) QO(a2), (1.4.34)

and D

d x

D

g(x,y)u(y)d y= u(x1)|D|O(a2), (1.4.35)


42/44

24 A. G. Ramm

where we have used the relation

D

g(x,y)d y= O(a2), a 0, (1.4.36)

which holds if 0.5diamD a.

It follows from formulas (1.4.33)(1.4.36) that

u(x1)|D| u0(x1)|D| +QO(a2), a 0, (1.4.37)

where we have neglected the terms of higher order of smallness asa 0.

From equations (1.4.32) and (1.4.37) it follows that

Q2

1 +|D|

u0(x1) +

2u0(x1)

, a 0, (1.4.38)

and

u(x1) u0(x1), a 0. (1.4.39)

Let us derive an analytical formula for the second integral in (1.4.25).

Multiply equation (1.4.15) by tpand integrate over S. Take into account

relation (1.4.29)and formulas(1.4.31) and(1.4.32) to get

2

S

d sspB1u O(a4), a 0. (1.4.40)

Let us define now an analog of the matrix (1.3.6):

p q() := 1|D|

S

tpq(t)d t, (1.4.41)

where the function q(t) :=q(t,) solves the equation

q(t) =Aq(t) 2Nq. (1.4.42)

Since || =

11+

< 1 when > 0, and the operatorAhas no eigenvalues in the

interval (1,1), equation (1.4.42) has a unique solution.


43/44


One hasu0N(t) u0N(x1), x1 D, and

2u0N= 2Nq

u0(x1)

xq

, q

u0(x1)

xq. (1.4.43)

Therefore, neglecting the term (1.4.40), which is of higher order of smallness

asa 0, one gets S

tp(t)d t= |D|p q()u0(x1)

xq. (1.4.44)

Consequently,

i kp

S

tp(t)d t= i kpq()pu0(x1)

xq|D|, (1.4.45)

and formulas (1.4.25), (1.4.38), and (1.4.45) yield the formula for Q1:

Q1 =2

1 +|D|

2u0(x1) u0(x1)

+ i kpq()pu0(x1)

xq|D| (1.4.46)

where

2

1 + = 1 , p:=

(x x1)p

|x x1| , xp:= x ep. (1.4.47)

From formulas (1.4.18), (1.4.26), (1.4.39), and (1.4.46) it follows that

u(x) u0(x) + g(x, x1)

(1 )

2u0(x1) u0(x1)

+ i kp q()(x x1)p

|x x1|

u0(x1)

xq

+u0(x1)

|D|, |x x1| a. (1.4.48)

Furthermore,

A(,, k) =|D|

4

(1 )

2u0(x1) u0(x1)

+ i kp q()pu0(x1)

xq+u0(x1)

.

(1.4.49)

Formulas(1.4.48)and(1.4.49)give our final result.

Note that |A(,, k)| = O(a3) and the scattering is anisotropic.

Let us summarize the results.


44/44

26 A. G. Ramm

Theorem 1.4.4. If ka 1 and > 0, then the scattering problem(1.4.1)(1.4.6)

has a unique solution. This solution can be calculated by formula(1.4.48). The

scattering amplitude is calculated by formula(1.4.49).

1.5 Summary of the results

The results of this chapter can be summarized as follows.

Scattering problem (1.1.1)(1.1.4) has a unique solution for any, Im 0,including the limiting cases = 0 and = .

The solution can be calculated by formula (1.1.44) if ka 1 and the

scattering amplitude by formula (1.1.46).

One has |A(,, k)| = O(a2) and the scattering is isotropic. If = ,

thenucan be calculated by formula (1.2.6) and the scattering amplitude-by

formula (1.2.7) fork a 1. The scattering is isotropic and |A(,, k)| = O(a).

If = 0, thenucan be calculated by formula (1.3.12), the scattering am-

plitude-by formula (1.3.14) ifk a 1, |A(,, k)| = O(a3), and the scattering is

anisotropic.

The scattering problem (1.4.1)(1.4.6) with the interface (transmission)

boundary condition has a unique solution if 0, k2 > 0,k21 > 0. Ifka 1

then this solution can be calculated by formula(1.4.48), the scattering ampli-

tude-by formula (1.4.49), |A(,, k)| = O(a3), and the scattering is anisotropic.
http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

ishizuka chap one

Documents