Table Of ContentSome Other Pergamon Titles of Interest
1. CONSTANTINESCU&MAGYARI:
Problems in Quantum Mechanics
2. DAVYDOV:
Quantum Mechanics, 2nd Edition
3. FARINA:
Quantum Theory of Scattering Processes
4. FARINA:
Quantum Theory of Scattering Processes:
General Principles and Advanced Topics
5. LANDAU & LIFSHITZ:
Course of Theoretical Physics (9 Volumes)
6. LANDAU & LIFSHITZ:
A Shorter Course of Theoretical Physics (2 Volumes)
7. MITTRA:
Computer Techniques for Electromagnetics
A full list of other titles in the International Series in Natural Philosophy
follows the Index.
WAVE SCATTERING
FROM STATISTICALLY
ROUGH SURFACES
BY
F. G.BASS and I. M.FUKS
Institute of Radiophysics and Electronics
Kharkov, USSR
TRANSLATED AND EDITED BY
CAROL B.VESECKY
Palo Alto, California
and
JOHN F. VESECKY
Stanford Center for Radar Astronomy
Stanford, California
PERGAMON PRESS
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Copyright ©1979 Pergamon Press Ltd.
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted
in any form or by any means: electronic, electrostatic,
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otherwise, without permission in writing from the
publishers.
First Edition 1979
British Library Cataloguing in Publication Data
Bass, Fridrikh Gershonovich.
Wave scattering from statistically rough
surfaces.
(International series in natural philosophy; 93)
Translation of Rasseianie voln nat statisticheski
nerovnoi poverkhnosti.
Bibliography: p.
Includes index.
1. Electromagnetic waves—Scattering.
2. Sound-waves—Scattering. I. Fuks, Iosif
Moiseevich, joint author. II. Title.
QC676.7.S3B3712 537 77-23113
ISBN 0-08-019896-1
In order to make this volume available as economically
and as rapidly as possible the typescript has been
reproduced in its original form. This method unfor-
tunately has its typographical limitations but it is hoped
that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
Preface to the Russian
Edition
The phenomenon of wave scattering from a statistically rough surface is
unavoidably encountered in the solution of a large number of physical
problems. Here we refer primarily to problems in radio physics and hydro-
acoustics related to wave propagation under conditions found in nature: for
example, radar and sonar, the effect of irregularities in topographic relief
or ocean waves on the propagation of radio and sound waves, the reflection
of radio waves from the lower layers of the ionosphere, fluctuations of
signals within the Earth-ionosphere waveguide, and so on. Alongside these
problems, which are traditional research topics in the statistical theory of
wave propagation, methods of statistical diffraction theory have recently
been applied successfully in the solution of problems which, at first glance,
seem completely unrelated: for example, radar astronomy (the reflection of
radio waves from the Sun and planets) and solid state physics (the
scattering of various types of waves and particles at discontinuities),
seismology (the propagation of seismic waves in the Earth's interior) and
radio engineering (the investigation of statistically irregular waveguides
and quasi-optical lines of communication), etc.
Despite the existence of a broad range of physical phenomena related to
the scattering of waves from a statistically rough surface, and the abundance
of journal articles on this subject, monographs devoted to the question are
lacking in the literature. Basic derivations of the distribution laws for
random wave fields are set forth in the theoretical portion of a book by
Beckmann and Spizzichino (P. Beckmann and A. Spizzichino, The Scattering of
Electromagnetic Waves from Rough Surfaces, Pergamon Press, Oxford, 1963).
However, this treatment and the corresponding chapters (VII, VIII) of
Feinberg's book (Ye. L. Feinberg, Rasprostranenie radiovoln vdoT zemnoi
poverkhnosti [Radio wave propagation along the Earth's surface], Izd-vo
AN SSSR, Moscow, 1961) do not nearly exhaust the content of the problem.
Both monographs were published quite a long time ago and do not reflect the
current state of this field of science. Thus, a substantial gap exists in
the literature on statistical diffraction theory. The present book is
WSSRS—A* IX
X Preface
intended to fill this gap. This monograph represents a collective effort.
All the data included in the book have been considered by the authors
jointly. The introduction and Chapters I and III were written by F. G. Bass,
Chapters IV and V by both authors, and Chapter XI by F. G. Bass, I. M. Fuks
and V. D. Freilikher. The remaining chapters were written by I. M. Fuks.
The authors wish to thank S. Ya. Braude and E. L. Feinberg, who in the
course of 15 years' common effort contributed to the formulation of most of
the basic ideas found in the present monograph. We are also extremely
grateful to I. E. Ostrovskii, A. D. Rozenberg and A. I. Kalmykov for the
physical substance of the problems considered in the book and their
descriptive interpretation. L. M. Brekhovskii examined the final version of
the manuscript and made a number of useful notes, for which the authors are
sincerely grateful. We consider it our pleasant duty to thank I. A.
Urusovskii, who took on the great effort of the preliminary editing of the
manuscript, for a great many important notes and much valuable advice. We
are grateful to V. D. Freilikher for his participation in the writing of the
final chapter and his aid in the shaping of the manuscript.
F. G. Bass
I. M. Fuks
Preface to the English
Edition
The need for a comprehensive monograph covering both basic theory and
some current applications of statistical diffraction theory has become
increasingly evident in recent years. F. G. Bass and I. M. Fuks have
filled this need admirably. Our aim during the translation and editing of the
book has been to make this English edition as useful and up-to-date as
possible. To further this end we have given English translations in place
of the original Russian references whenever such translations are widely
available, e.g. cover to cover translations of Russian journals. Some
fifty additional references have been added to tie the book more closely
to the English language literature and direct the reader to work which has
occurred since the publication of the Russian edition. These supplementary
references are denoted by an asterisk following the year of publication.
An index has also been added. We have adopted the Library of Congress
system, commonly used in the natural sciences for transliteration of
Cyrillic to Roman letters, when rendering Russian authors' names into
English. A number of typographical errors have been corrected including an
errata list kindly supplied by Bass and Fuks.
We wish to express our thanks to Professor Allen Peterson of Stanford
University for bringing the book to our attention and suggesting its
translation into English. Our thanks also go to Professor Derek ter Haar
of Oxford University for timely advice and encouragement. We gratefully
acknowledge the expertise of Judith Windley in Leicester for typing the
draft manuscript and Michal Plume at Stanford for typing the entire
camera-ready copy. Finally, we thank Brenda Solomon, David Everitt, and
Peggy Braasch of Leicester and Peter Henn, William Buchanan, Lawrence
Walton and Neil Warnock-Smith at Pergamon Press for their help during the
sometimes tortuous production of this book.
Carol B. Vesecky
John F. Vesecky
xi
Introduction
Wave scattering from real surfaces is of interest in various fields of
modern physics. In radio physics and acoustics, these include the
scattering of radio and sound waves by topographic irregularities (Feinberg,
1961 and Barrick, 1970a*), by the disturbed surface of the ocean (Braude,
1962; Wright, 1968* and Matthews, ed., 1975*), and by other objects. In
solid state physics, the interaction of quasiparticles (electrons, phonons,
etc., treated as waves), with boundaries (Zaiman, 1962; Kaner, et al., 1968)
is considered. In optics, scattering is relevant to the diffraction of light
from the surface fluctuations at the interface between two media (see Kaner,
et al., 1968; Mandel'shtam, 1913; Andronov and Leontovich, 1926 and Gans,
1924 and 1926) and from various mat surfaces (Gorodinskii and Galkina, 1966;
Karp et al., 1966; Brandenberg and Neu, 1966; Torrance, et al., 1966 and
Pedersen, 1975*). In radar astronomy, the reflection of radio waves from
the surfaces of the Sun, Moon and planets (URSI, 1964 and 1970*; Goldstein,
1964a,b; James, 1964; Evans, 1965; Evans and Hagfors, 1966 and 1968*) is
treated. The number of such examples could easily be enlarged substantially.
A real surface is, of necessity, rough. The causes for the emergence of
irregularities can be of the most varied nature — from the surface
irregularities of a solid body which are related to the corpuscular structure
of the material and defects in processing, to the disturbed surface of the
ocean, whose shape is affected by the turbulence of the wind. In the
majority of problems which are of practical interest, the shape of the rough
surface is described by a random function of coordinates, and sometimes of
time as well. Therefore the diffraction of waves from real surfaces should
also be seen as a statistical problem consisting of finding the statistical
characteristics of the scattered field (distribution functions, moments,
correlation functions, etc.) given the statistical properties of the surface.
Thus, the theory of wave scattering from a statistically rough surface is a
synthesis of the theory of wave diffraction from a surface of arbitrary shape
and probability theory. The methodology of solving this sort of problem is
the same, regardless of the physical nature of the irregularities.
1
2 Introduction
The first mathematical investigation of the scattering of sound waves
from a rough surface was conducted by Rayleigh (1878), who discussed the
diffraction of a plane wave by a sinusoid. The scattering of electromagnetic
waves from a statistically rough surface was investigated by Mandel'shtam
with regard to the molecular scattering of light from a liquid surface
(Mandel"shtam, 1913). The exhaustive theory of this effect was given in the
almost simultaneous work of Andronov and Leontovich (1926) and Gans (1924 and
1926). In the aforementioned works, the directional intensity distribution
of the scattered field was calculated in a perturbation theory approximation.
The first work investigating the coherent component field was published by
Feinberg (1944-1946). In that work it was demonstrated that the propagation
of electromagnetic waves over a statistically rough, ideally conducting
surface is equivalent to the propagation of waves over a plane with an
effective impedance determined by the statistical parameters of the
irregularities. Feinberg mentioned the cumulative effect which plays a
fundamental role in the physics of wave propagation over a statistically
rough surface. This effect refers to the fact that the influence of small
irregularities on the process of wave propagation is cumulative and for
sufficiently long paths will substantially affect the nature of the field.
A perturbation method has been applied in the above works in one form or
another, i.e. the irregularity height was assumed to be small in some sense.
In the early 'fifties Antokol'skii (1948), Brekhovskikh (1951a,b and 1952)
and Isakovich (1952) considered the other limiting case of diffraction on
high, but gently sloping irregularities, using the tangent plane method.
[In the literature the tangent plane method is mentioned frequently as "the
Kirchhoff method". This is related to the fact that the field on the
scattering surface is assigned according to a local law, analogous to the way
in which this was done by Kirchhoff in the problem of the diffraction of
light by a slot (see, for example, Kirchhoff, 1891). We will use this
designation in view of its general acceptance, even though it cannot be
considered to be correct, since the method, in application to the problem of
diffraction on a periodic surface, was first elaborated in the cited works
(Antokol'skii, 1948; Brekhovskikh, 1951a,b and 1952).] An analogous
examination in the area of solid state physics and rarefied gas hydro-
dynamics has been conducted in the works of Zaiman (1962), Barantsev (1963),
Barantsev and Alekseeva (1963), Barantsev and Miroshin (1963) and Gurzhi and
Shevchenko (1967 and 1968).
Introduction 3
Further development of the theory went along the lines of the small
perturbation approximation and the Kirchhoff approximation. Here we should
note the introduction of nonlocal boundary conditions (Bass, 1960), the
calculation of shadowing by the Kirchhoff method (Bass and Fuks, 1964;
Smith, 1967), the concept of resonant scattering (Crombie, 1955; Braude, et
al., 1962; Bass, 1961), and scattering from a surface with two types of
irregularities (Kur'yanov, 1962; Fuks, 1968; Barrick and Peake, 1967 and
1968*). The perturbation method and the Kirchhoff method are now the most
widespread in the theory of wave scattering by statistically rough surfaces.
Basically their development can be considered finished. For this reason the
exposition here is based on these two methods and their derivatives.
The first two chapters are of an introductory nature. The basic concepts
of wave propagation theory and the theory of random processes as applied to
rough surfaces and to wave fields are formulated here. In the third chapter,
the average fields of sound and electromagnetic waves scattered by a surface
with random irregularities are considered. It is shown that the mathematical
description of an average field scattered by a statistically rough surface is
equivalent to the description of a field scattered by a determinant surface
with effective boundary conditions defined by the statistical characteristics
of the random irregularities. Using the effective conditions, a boundary
problem for a point source and plane wave is solved.
The fourth chapter is devoted to the calculation of the characteristics of
the fluctuation field: second moments and phase fluctuations. The
investigation is conducted for both the Fraunhofer zone and the near zone.
In this chapter we formulate the principle of selective scattering. This
principle states that only one harmonic component of the entire energy
spectrum of the rough surface takes part in the scattering of waves. This
component is determined by the wave vectors of the incident and scattered
fields. The principle holds true in a perturbation theory approximation if
the source and observation point are located in the spectral partition
zone. However, it also makes possible the analysis of more complex cases.
Wave scattering from random moving surfaces is investigated in the fifth
chapter. The frequency spectrum of the scattered field is enriched in this
case due to modulation by the motion of the surface. In the far zone it
consists of two monochromatic lines corresponding to combination scattering
of waves from a random surface. In the near zone, the spectrum of the
4 Introduction
scattered field possesses a substantially more complex shape and sometimes
coincides with the spatial frequency spectrum of the scattering surface,
which is essential in the solution of the inverse problem.
Spatial correlation characteristics in the approximation of small
perturbations are studied in the sixth chapter. The correlation functions
are of fundamental interest as a source of information on a moving surface.
Thus much attention is paid to the inverse problem in this chapter. In the
chapter's conclusion, we point out the relation between the time and spatial
correlation functions of a random surface represented in the form of a linear
superposition of surface waves. This relation is transferred to the wave
field and is also utilized for a solution of the inverse problem.
The seventh chapter is devoted to scattering from large-scale
irregularities (Kirchhoff method). In this approximation, the reflection of
an electromagnetic wave from a rough surface is seen at each point as
reflection from a tangent conducting plane at that point; averaging is then
carried out over the directions of the tangent plane normals. In this
chapter the scattered intensity of acoustic and electromagnetic waves is
calculated, a calculation of shadowing is performed, and effective
distribution functions of heights and slopes are also discussed. Results
are obtained for the Fraunhofer zone and for a finite surface.
In the eighth chapter, using the tangent plane method, we investigate the
mean field and scattered intensity over an infinite rough surface, the
correlation functions, and other statistical characteristics. In the
chapter's conclusion the frequency spectrum of the waves scattered by a
moving large-scale surface is considered. It should be noted that in this
case the principle of selective scattering, generally speaking, has no place,
and the entire spectrum of surface irregularities participates in the
scattering process. This leads to a broadening of the spectral density
maxima of the scattered field, although its maxima in a number of limiting cases
fall at the frequencies defined by the principle of selective scattering.
The scattering of waves on bodies bounded by a random surface is of
practical interest. This sort of scattering is normally characterized by the
so-called effective cross section, the statistical properties of which are
investigated in the ninth chapter.
An analysis of the experimental data conducted in the tenth chapter shows
that neither perturbation theory nor the Kirchhoff method fully reflects the
