Some 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 OXFORD · NEW YORK · TORONTO · SYDNEY · PARIS · FRANKFURT U.K. Pergamon Press Ltd., Headington Hill Hall, Oxford OX3 OBW, England U.S.A. Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. CANADA Pergamon of Canada, Suite 104, 150 Consumers Road, Willowdale, Ontario M2J 1P9, Canada AUSTRALIA Pergamon Press (Aust.) Pty. Ltd., P.O. Box 544, Potts Point, N.S.W. 2011, Australia FRANCE Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus, OF GERMANY Pferdstrasse 1, Federal Republic of Germany 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, magnetic tape, mechanical, photocopying, recording or 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