Waves in Layered Media

566 Pages·1960·7.162 MB·1-561\566

by  LEONID M. BREKHOVSKIKH (Eds.)

APPLIED MATHEMATICS
AND MECHANICS
An International Series of Monographs
EDITORS
F. N. FRENKIEL G. TEMPLE
University of Minnesota Mathematical Institute
Minneapolis, Minnesota Oxford University
Oxford, England
Volume 1. K. OSWATITSCH: Gas Dynamics,
English version by G. Kuerti (1956)
Volume 2. G. BIRKHOFF and Ε. H. ZARANTONELLO:
Jet, Wakes, and Cavities (1957)
Volume 3. R. VON MISES: Mathematical Theory of Compressible Fluid Flow,
Revised and Completed by Hilda Geiringer and G. S. S. Ludford
(1958)
Volume 4. F. L. ALT: Electronic Digital Computers—Their Use in Science
and Engineering (1958)
Volume 5. W. D. HAYES and R. F. PROBSTEIN:
Hypersonic Flow Theory (1959)
Volume 6. L. M. BREKHOVSKIKH: Waves in Layered Media, Translated from
the Russian by D. Lieberman (1960)
Volume 7. S. FRED SINGER (ed.):
Torques and Attitude Sensing in Earth Satellites (1964)
Volume 8. MILTON VAN DYKE: Perturbation Methods in Fluid Mechanics
(1964)
In Preparation
ANGELO MIELE (ed.) : Theory of Optimum Aerodynamic Shapes
WAVES IN LAYERED MEDIA
BY
LEONID M. BREKHOVSKIKH
Director, Acoustics Institute, Academy of Sciences, USSR
TRANSLATED FROM THE RUSSIAN BY
DAVID LIEBERMAN
Under the direction of the American Institute of Physics and with
the support of the National Science Foundation
TRANSLATION EDITED BY
ROBERT T. BEYER
Department of Physics, Brown University, Providence, Rhode Island
I960
ACADEMIC PRESS · PUBLISHERS · NEW YORK · LONDON
COPYRIGHT© 1960 BY ACADEMIC PRESS INC.
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WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS
ACADEMIC PRESS INC.
Ill FIFTH AVENUE
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United Kingdom Edition
Published by
ACADEMIC PRESS INC. (LONDON) LTD.
BERKELEY SQUARE HOUSE, LONDON W. 1
Library of Congress Catalog Card Number: 00-8046
First Printing, 1960
Second Printing, 1965
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
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 <idisharmony'\ Furthermore
as one of my primary tasks, I attempted to give the reader a clear
physical picture of the phenomena under investigation. As regards the
mathematical rigor, it is possible that it was not attained to a sufficient
degree everywhere, partly due to the fear of 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 the common 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 comparatively
complete, but is far from exhaustive.
I would like to express my deep gratitude to V. A. Polianskii and
I. F. Treshchetenkovii, who were of great help in checking the equations
and in preparing the manuscript for publication.
February 1, 1956 L. BREKHOVSKIKH
TRANSLATION EDITOR'S PREFACE
The appearance of this work by Professor Brekhovskikh
climaxed a great deal of research by him on various aspects of
wave propagation through laminated media. The book presents
a thorough survey of such wave propagation—both acoustic
and electromagnetic. While it is by no means restricted to
Russian work in the field, it does present a complete picture of
Soviet researches on wave propagation through layered media.
Because of the significance of the subject matter, and because
of the insight the text provides to Russian wave propagation
studies, the American Institute of Physics undertook the
translation, operating with the aid of a grant from the National
Science Foundation.
The Editor is joined by the translator in expressing gratitude
to Professor Brekhovskikh for his cooperation in the translation,
and especially for supplying a list of corrections. They are also
grateful to Professor Arnold Schoch, now of C.E.R.N., Geneva,
Switzerland, for furnishing original photographs of Figs. 32, 33
and 35.
ROBERT T. BEYER
March 10, 1960.
vii
CHAPTER I
PLANE WAVES IN LAYERS
THE theory of wave reflection from interfaces and from layers will be
developed in this chapter. Principal attention will be given to plane
harmonic waves. The behavior of beams bounded in space and pulses
bounded in time are not considered before § 8. In all cases, the media
in which the waves propagate are assumed to be homogeneous and to
be bounded by parallel planes.
For completeness of presentation, the first sections of the chapter are
devoted to relatively simple questions such as plane waves in homo­
geneous media, the reflection and refraction of waves at an interface,
etc. However, even in these sections, the reader will find some com­
paratively new problems, such as the theory of inhomogeneous plane
waves and their refraction and reflection, an analysis of Leontovich's
approximate boundary conditions (which are satisfied in cases of so-
called "locally reacting" surfaces), and others.
Acoustic and electromagnetic waves will be considered simul­
taneously.
§ 1. PLANE WAVES IN HOMOGENEOUS UNBOUNDED MEDIA
1. Fundamental concepts and definitions
The plane wave is the simplest form of wave motion. The most
general analytic expression for a plane wave is the function
where n, n, and n are three numbers which satisfy the condition
x y z
n2 + nl + n* = 1,
x
and are the projections on the coordinate axes of the unit vector normal
to the wave front, i.e. normal to planes of constant phase.
The function (1.1) is a solution of the wave equation
d2F d2F d2F _ 1 d2F
(1.2)
1
2 PLANE WAVES IN LAYERS
It describes a disturbance which is propagated through the medium
with the velocity c. The form of the wave, which is determined by the
form of the function F, remains unchanged as the wave propagates.
We shall use the so-called spectral method for the study of wave and
vibrational phenomena, which is quite widely used in physics and
engineering. When the principle of superposition holds, the analysis of
the behavior of waves of any form can be reduced to the analysis of the
behavior of the simplest"harmonic" waves by using the spectral method.
We let
nx + ny + nz
x y z
in Eq. 1.1, and represent the function F(£) as the real part of a Fourier
integral _
J'(f) = Re J Φ(ω)^άω (1.3)
where the symbol Re denotes the real part.
Since the real part of any complex number a can be written in the
form Re a = \(a + a*), the last expression can also be written in the form
l
F(£) = Γφ(ω)β^άω + ]- Γ°Φ*(ω)β-^<Ζω. (1.3a)
We multiply this expression by β~ίω'£άξ and integrate over ξ from
— oo to +oo. We then easily obtain
Φ(ω) = -f+"iP(f)e-<*df (1.4)
for the spectral density function, j
The integrand in Eq. 1.3, corresponding to a definite value of ω
/(ω, χ, y, ζ, t) = Φ(ω) exp (ίωξ)
» . Γ. inx + ny + nz Υ] .
= Φ( ωx)θχρ wo!-x- ^v - *z —-t\ , (1.5e)
represents a plane harmonic wave.
f The derivation of Eq. 1.4 becomes especially simple if we use the Dirac
function f+oo
2πδ(χ) = J eix*d£
and take account of its fundamental property
f+co
Φ(χ) 6{x)dx = Φ(0),
>
where Φ(χ) is a continuous function at χ = 0.
PLANE WAVES IN HOMOGENEOUS MEDIA 3
The Fourier integral and the expression for an individual harmonic
wave are written in a complex form. As has already been mentioned,
only the real parts of these expressions have physical meaning. There­
fore, in the end, a harmonic plane wave must be written as the real part
of Eq. 1.5, i.e.
A(OJ)COS\OJ— Y~ —-αΛ + </>(ω) I , (1.6)
where, in going from Eq. 1.5 to Eq. 1.6, we represent the (generally
complex) function Φ(ω) in the form
Φ(ω) = A(co)eW<»\
We use the usual notation
— = k = — k u = k, kfiy = k, ku — k,
x x y z z
where k, k k k are the modulus of the propagation vector and its
x) yi z
components along the coordinate axes, respectively, and λ is the wave­
length. Then Eq. 1.5 can be written in the form
f = φ()βχρ [i(kx + ky + kz — ojt)] = Φ(ω)βχρ [i(k-r — ωί)].
ω x y z
(1.7)
Since the time differentiation of a function of this form reduces
simply to multiplication by — iw, the wave equation for / can be written
in the form
«V w av
The spectral approach to wave phenomena has attained widespread
application as a result of the following features:
1. The comparatively simple analysis of the behavior of each of the
harmonic waves.
2. The possibility of the expansion of any wave process into harmonic
waves, when the principle of superposition holds.
3. The extremely high monochromaticity of many of the radiators
used in practice. As a result, the radiated waves are close to harmonic.
The expansion of a complex wave process into harmonic waves and
the reduction of the problem to Eq. 1.8, where the frequency is con­
sidered as given, is the most convenient method of analysis when
dispersion is present. In this case, even Eq. 1.2 has no meaning because
of the vagueness of the meaning of the quantity c.
4 PLANE WAVES IN LAYERS
In what follows we shall consider harmonic waves (Eq. 1.7) almost
exclusively; when necessary, we shall use them to construct wave
disturbances of more complex forms.
2. Inhomogeneous plane waves
The expression for a plane harmonic wave (1.7) admits of an inter­
esting generalization, which will be of importance in the future. It was
indicated above that k, k, k are the components of the propagation
x y z
vector along the coordinate axes, and it was assumed that these quanti­
ties could be any triplet of real numbers, satisfying the relation
k% + kl + kl = k\ (1.9)
We now abandon the graphical description in the treatment of these
numbers, and assume that the set k,k,k is a triplet of complex
x y z
numbers
k = k'+ik", k = k'+ik;, k. = v,+w (l.io)
x x x y v r
We again require that Eq. 1.9 be satisfied for a real value of k = ω/c.
Then, as previously, Eq. 1.7 satisfies the wave equation (1.8).
Let us see what is represented by a wave described by Eq. 1.7 with
complex values of k, k, k. Substituting Eq. 1.10 into Eq. 1.7, we obtain
x y z
/=Φ(ω)βχρ[^> + λ> + ^- ω Ο - ( ^> + λ> + λ>)]. (l.H)
2
This expression describes a wave with varying amplitude. As is easily
seen, planes of constant amplitude of this wave are given by the
equation
kx + k;y + k:z = c (1.12)
x v
and planes of constant phase by the equation
k'x + k'y + kz = c, (1.13)
x y z 2
where c and c are constants. It can be shown that the planes of equal
x 2
phase are orthogonal to the planes of equal amplitude. In fact, substi­
tuting Eq. 1.10 into Eq. 1.9, and equating the imaginary parts on both
sides of the equation, we obtain.
^x ~^kyky~^ ^z = 0·
This equation expresses the condition of orthogonality of the families
of planes (1.12) and (1.13).
A wave of the form (1.11) is usually called an inhomogeneous plane
wave. This wave propagates in the direction given by the vector
k' (k', k', k'), and its amplitude falls off in the perpendicular direction.
x y z
PLANE WAVES IN HOMOGENEOUS MEDIA 5
The coordinate system can always be chosen in such a way that
k'y = ky = 0. Then, just as with ordinary plane waves, we can introduce
the angle ft and set
k = ksinft, k = kcosft. (1.14)
x z
However, since, according to (1.10), k and k are complex, ft will be a
x z
complex angle.
Let us consider, for example, the case ft = (π/2) — ioc, where α is real.
From (1.14) we obtain k = & cosh a, k = i&sinha, and expression (1.7)
x z
for a plane wave is written
/= Φ(ω)βχρ(ί&cosh or a; — Asinhorz — ίωί). (1-15)
Thus, we obtain a wave which is propagated in the ^-direction and
has an amplitude which falls off exponentially in the ^-direction. As
follows from (1.15), the velocity of propagation is
ω c
k cosh OL cosh OL 9
i.e. it is always less than the velocity of propagation of the ordinary
plane wave c The corresponding wavelength is (27r/fc cosh α) = λ/ cosh a,
i.e. it is less than the wavelength of the ordinary wave at the same
frequency. The greater a, the smaller the wavelength and the greater
the damping coefficient of the wave in the z-direction.
All that has been said here refers to the case in which there is no
absorption in the medium (real k). The introduction of the concept of
inhomogeneous waves in the presence of absorption involves no funda­
mental difficulties. Here, the planes of equal phase and of equal ampli­
tude will naturally no longer be perpendicular to one another.
Running somewhat ahead of ourselves, we shall show that upon
refraction of plane waves at an interface, inhomogeneous plane waves
can be transformed into ordinary homogeneous waves, and vice versa.
This is immediately evident from the law of refraction
nsinft = sinft, (1.16)
1
where η is the index of refraction, ft is the angle of incidence and ft is
1
the angle of refraction.
If n< 1 and sin#>n, then it follows from (1.16) that sini?> 1, i.e.
1
ft is complex, and the refracted wave is inhomogeneous. This is a well
x
known occurrence and is realized in the case of the total internal reflec­
tion of waves.
If, on the contrary, sin#> 1, i.e. the incident wave is inhomogeneous,
but sinft<n (in this case, of course, n> 1), then we obtain sini?< 1,
1