Table Of ContentAPPLIED MATHEMATICS
AND MECHANICS
An International Series of Monographs
Ε DIT Ο RS
FRANCOIS Ν. FRENKIEL G. TEMPLE
Washington, D. C. The Queen's College
Oxford University
Oxford, England
1. Κ. OswATiTSCH : Gas Dynamics, English version by G. Kuerti (1956)
2. G. BIRKHOFF and Ε. H. ZARANTONELLO : Jet, Wakes, and Cavities ( 1957)
3. R. VON MISES: Mathematical Theory of Compressible Fluid Flow, Re
vised and completed by Hilda Geiringer and G. S. S. Ludford ( 1958)
4. F. L. ALT: Electronic Digital Computers—Their Use in Science and
Engineering (1958)
5Α. WALLACE D. HAYES and RONALD F. PROBSTEIN: Hypersonic Flow
Theory, Second Edition, Volume I, Inviscid Flows (1966)
6. L. M. BREKHOVSKIKH : Waves in Layered Media, Translated from the
Russian by D. Lieberman (1960)
7. S. FRED SINGER (ed.) : Torques and Attitude Sensing in Earth Satellites
(1964)
8. MILTON VAN DYKE: Perturbation Methods in Fluid Mechanics (1964)
9. ANGELO MIELE (ed.) : Theory of Optimum Aerodynamic Shapes (1965)
10. ROBERT BETCHOV and WILLIAM O. CRIMINALE, JR. : Stability of Parallel
Flows (1967)
11. J. M. BURGERS : Flow Equations for Composite Gases (1969)
12. JOHN L. LUMLEY: Stochastic Tools in Turbulence (1970)
13. HENRI CABANNES: Theoretical Magnetofluiddynamics (1970)
14. ROBERT E. O'MALLEY, JR. : Introduction to Singular Perturbations
(1974)
15. TUNCER CEBECI and A. M. O. SMITH : Analysis of Turbulent Boundary
Layers (1974)
16. L. M. BREKHOVSKIKH : Waves in Layered Media, Second Edition,
Translated by Robert T. Beyer (1980)
17. CHIA-SHUN YIH: Stratified Flows (1980)
WAVES IN LAYERED MEDIA
Second Edition
L. M. BREKHOVSKIKH
Acoustics Institute
Academy of Sciences of the USSR
Moscow, USSR
TRANSLATED BY
ROBERT T. BEYER
Department of Physics
Brown University
Providence, Rhode Island
1980
ACADEMIC PRESS
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COPYRIGHT © 1980, BY ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1 7DX
Library of Congress Cataloging in Publication Data
Brekhovskikh, Leonid Maksimovich.
Waves in layered media.
(Applied mathematics and mechanics ; )
Bibliography: p.
Includes index.
1. Waves. I. Title.
QC231.B8513 1980 53Γ.1133 79-51695
ISBN 0-12-130560-0
Waves in Layered Media, Second Edition. Translated from the
original Russian edition entitled Volny ν sloistykh sredakh,
Izdaniye vtoroye, dopolnennoye i pererabotannoye,
by L. M. Brekhovskikh, published by Izdatelstvo "Nauka,"
Moscow, 1973.
PRINTED IN THE UNITED STATES OF AMERICA
80 81 82 83 9 8 7 6 5 4 3 2 1
TRANSLATOR'S PREFACE TO SECOND EDITION
The translation of the first edition of this work appeared in 1960 and was
immediately acclaimed by the underwater sound community. Professor
Brekhovskikh has been recognized as an outstanding theorist in the field of
wave propagation. For his contributions he received the Rayleigh Medal of
the British Institute of Acoustics in 1976.
The second edition of the book represents an extensive revision of the
first, with considerable new material, as noted in Professor Brekhovskikh's
Preface to Second Edition. Some more recent references have been added
by the translator to the already extensive bibliography.
The translator acknowledges the assistance of the U.S. Office of Naval
Research in the preparation of the translation. He is also grateful to the
staff of Academic Press for substantial editorial assistance.
R. T. BEYER
ix
PREFACE TO SECOND EDITION
A number of new problems have been treated in the second edition. These
include the theory of a waveguide in a medium in which the wave
propagation velocity c — c(x, z) depends on two coordinates, the theory of
a surface waveguide and the waves of whispering galleries (including the
case of a solid), reference (étalon) equations and reference integrals,
diffraction rays, etc. The exposition of other questions such as the theory
of caustics, for example, has been greatly expanded. On the other hand,
part of the material pertaining to an underwater sound channel, which is
not of general interest, has been excluded. This and similar problems are
dealt with at much greater length in a special monograph on underwater
acoustics just completed by the author and his colleagues. The rest of the
material has been compressed somewhat by virtue of contemporary
methods of presentation. The bibliography has been increased consider
ably.
The author expresses his deep gratitude to I. F. Treshchetenkova for
highly valuable assistance in the preparation of the second edition.
L. BREKHOVSKIKH
xi
PREFACE TO FIRST EDITION
A systematic exposition of the theory of the propagation of elastic and
electromagnetic waves in layered media is given in this monograph. A
considerable part of the material originated with the author and has
appeared earlier in a number of journal articles. I have endeavored to
present the results of other authors in the spirit of my own, as far as
possible, in order to avoid methodological "disharmony." Furthermore, as
one of my primary tasks, I have attempted to give the reader a clear
physical picture of the phenomena under investigation. As regards the
mathematical rigor, it is possible that it has not been sustained throughout
to a sufficient degree, in order to avoid making the presentation too
cumbersome.
The simultaneous presentation of the theory of propagation of elastic
and of electromagnetic waves followed in the book is quite advantageous
since the same mathematical methods may be applied in both cases. Also,
as a result of this simultaneous presentation, each region is enriched by the
methods applied in the other. Thus, for example, the impedance method
developed in acoustics and radio engineering may be quite successfully
applied in calculations of multilayer reflection reduction of optical systems
and interference filters.
The bibliography given at the end of the book is fairly complete, but is
far from exhaustive.
I would like to express my deep gratitude to V. A. Polianskaya and I. F.
Treshchetenkova, who were of great help in checking the equations and in
preparing the manuscript for publication.
L. BREKHOVSKIKH
xiii
Chapter I
PLANE WAVES IN DISCRETELY LAYERED MEDIA
A plane harmonic wave is the simplest form of a wave process. In this
chapter we shall consider the reflection and refraction of such waves at
plane interfaces.
Wave fields of very general form can be represented by a superposition
of plane harmonic waves. For this it is only necessary that the functions
which describe temporal and spatial changes in the field be represented in
the form of the corresponding Fourier integrals. Therefore, the results
obtained in this chapter will be used extensively in the following chapters;
in particular, for the analysis of the reflection of bounded wave beams and
spherical waves.
§ 1. Plane Waves in Homogeneous Unbounded
Media
1.1. Basic concepts and definitions
A plane wave represents the simplest form of wave motion. The most
general analytic expression for such a wave is the function
/ nx + ny + nz
x y z
- t), (1.1)
V c
where n,n,n are three numbers satisfying the condition n2 + n2 + n2 =
x y z
1 and are projections on the coordinate axes of the unit vector normal to
the wave front, i.e., to the plane on which the argument of the function in
(1.1) remains a constant (planes of constant phase).
The function (1.1) is a solution of the wave equation
V ^ i ^ - ft 2 il ii ii (L2)
V s + +
c2 dt2 dx2 dy2 dz2
It describes some disturbance which is propagated with the velocity c. The
form of the wave, defined by the form of F, remains unchanged during
propagation.
The so-called spectral method of investigation of wave and oscillatory
phenomena is widely used in physics and engineering; we shall also use it
in what follows. This method allows us to reduce the analysis of the
behavior of the waves of very arbitrary form (in cases in which the
1
2 I. P L A NE W A V ES IN D I S C R E T E LY L A Y E R ED M E D IA
superposition principle holds) to the analysis of very simple, "harmonic"
waves.
In Eq. (1.1) we use the notation
£ = (nx + ny + nz)/c - t
x y z
and represent the function F(Q as the real part of the Fourier integral
F(£) = Re f °°Φ(ω)<?' ω* άω. (1.3)
Since the real part of an arbitrary complex number can be written in the
a = {a + a*)/2,
form Re then
r f
J oo oo
Φ(ω)βίωξ do3+\\ Φ*(ω)β-'ωξ άω.
(1.3a)
Jo
ο
e~loi'^ di; ξ
We multiply this expression by and integrate over from — oo
to 4- oo. It is then not difficult to obtain the following for the spectral
density function1":
Φ(ω) = - f + C°F(0e-^dè.
π J-oo ( 1 . 4)
The integrand in (1.3), which corresponds to a certain value of co
r // ηnχxχx +- r nvηy y +- r ηnz,zζ \ ι
/(co, x,y, ζ) = Φ(ω)β«* = Φ(ω) exp1 " / ^
— \ — - - ' )} 0-5)
is a plane harmonic wave.
Here we use the complex form of writing both the Fourier integral and
the expression for the individual harmonic wave. It is necessary to give
physical meaning only to the real part of the corresponding expressions.
Therefore, in the final analysis, the harmonic plane wave should be written
in the form of the real part of E q. (1.5), i.e.,
Α(ω) cos[(nx + ny + ηζ)ω/c — ωί
x y ζ + φ(ω)]. (1.6)
Here we have represented the generally complex function Φ(ω), in the
transition from (1.5) to (1.6), in the form
Φ(ω) = A(G>)ei<K"\
The following notation is standard:
ω/c = k = 277/λ, kn = k, kn = k, kn = k,
x x y y z 2
where k, k, k, k are the modulus of the wave vector (wave number) and
x y z
* The derivation of (1.4) becomes elementary if we use the Dirac function 2π δ(χ) =
S ΐηβ*χ* d£ and take into account its basic property f ί ^Φ(-χ) S(x) dx = <I>(0),where Φ(0) is
a continuous function at zero.
1. PLANE WAVES IN HOMOGENEOUS MEDIA 3
its components along the coordinate axes, respectively, and λ is the
wavelength. Then Eq. (1.5) is rewritten in the form*
/ = Φ(ω) exp i(kx + ky + kz - ωή = Φ(ω) exp /(k*r - ωή. (1.7)
x y z
Since the differentiation of a function such as / with respect to / reduces
to its multiplication by — /ω, the wave equation for / can be written in the
form
V2/+ k2f=0. (1.8)
The following features of the spectral approach have led to its
widespread application to wave phenomena:
1. the comparative simplicity of analysis of the behavior of each of the
harmonic waves;
2. the possibility of expansion of an arbitrary wave process into
harmonic waves in cases in which the superposition principle is valid;
3. the very high monochromaticity of many radiators used in practice,
as a consequence of which the waves radiated by them are close to
harmonic.
The expansion of a complicated wave process into harmonic waves and
the reduction of the problem to Eq. (1.8), where the frequency is assumed
to be already given, provide a most convenient method of analysis when
dispersion is present (the velocity c is a function of ω).
In what follows, we shall consider harmonic waves (1.7) almost exclu
sively, constructing wave excitations of more complex form from them
when necessary.
1.2. Inhomogeneous plane waves
There is an interesting generalization of the expression for the plane
harmonic wave of (1.7) that is important for further development. It was
shown above that k, k and k are the components of the wave vector
x y z
along the coordinate axes. It was assumed there that we can choose any
three real numbers as such quantities that satisfy the relation
k2 + k2 + k2 = k2. (1.9)
We now give up the graphical presentation in the treatment of these
numbers and assume that the set k,k,k is a triplet of complex numbers
x y z
k = k' + ik'\ k = k' + ikf, k = k' + ik. (1.10)
x x x y y z z z
t For simplicity, we shall write the expression exp(/<p), which is frequently encountered
below, in the form exp i<p.
4 I. P L A NE W A V ES IN D I S C R E T E LY L A Y E R ED M E D IA
Here we again require that Eq. (1.9) be satisfied for real values of k = ω/c.
Then Eq. (1.7) will again satisfy the wave equation (1.8).
Let us see what kind of a wave will be described by Eq. (1.7) with
complex k,k,k. Substituting (1.10) in (1.7), we obtain
x y z
/ = Φ(ω) exp[i(^jc + k' y + k'z - ωή - (k^x + IÇy + Kz)\ (1.11)
y 2
This expression describes a wave with varying amplitude. The planes of
constant amplitude are given by the equation
k'fx + k;y + k''z = C, (1.12)
x z
and the planes of constant phase by the equation
kx + k'y + k'z = C , (1.13)
x y z 2
where C and C are constants. We can show that the planes of constant
x 2
phase are orthogonal to the planes of constant amplitude. Actually, sub
stituting (1.10) in (1.9) and equating the imaginary parts of both sides of
the equation, we get
k k ~\~ k k k k — 0.
x x y y ~f~ z z
This equation expresses the condition of orthogonality of the planes (1.12)
and (1.13).
A wave of the form (1.11) is usually called an inhomogeneous plane wave.
This wave propagates in a direction given by the vector k'(k', k, k) and
x y z
has an amplitude that falls off in one of the perpendicular directions.
By an appropriate choice of coordinates it can always be arranged that
k = ky = 0. Then, just as for ordinary plane waves, we can introduce a
y
certain angle θ and set
k = k sine, k = kcos0. (1.14)
x z
For complex k and k the angle θ will also be complex.
x z
For example, consider the case θ = (π/2) — /α, where a is real. From
(1.14) we get k = k cosh a, k = ik sinh a, and Eq. (1.7) for a plane wave
x z
is written
/ = Φ(ω) exp [M:* (cosh a) - /cz(sinh a) — /ω/]. (1.15)
This wave propagates in the χ direction and falls off exponentially in the ζ
direction. The velocity of propagation of the wave is equal to c = co/k =
x x
c/cosh a, as follows from (1.15). That is, it is always less than the velocity
of propagation of an ordinary plane wave c. The corresponding wave
length is equal to 2π/k cosh a = λ/cosh a, i.e., it is less than the ordinary
wavelength at the same frequency. The larger a the smaller the wavelength
and the larger the attenuation coefficient of the wave in the ζ direction.
All the above refers to the case of the absence of absorption of the wave
in the medium (real k). The introduction of the concept of inhomogeneous
