Low and High Frequency Asymptotics: Acoustic, Electromagnetic and Elastic Wave Scattering

527 Pages·1986·6.16 MB·English

by  V.K. VARADAN and V.V. VARADAN (Eds.)

MECHANICS AND
MATHEMATICAL METHODS
A SERIES OF HANDBOOKS
General Editor
J.D. ACHENBACH
Northwestern University, Ε vans ton, Illinois, USA
Third Series
ACOUSTIC, ELECTROMAGNETIC AND ELASTIC
WAVE SCATTERING
Series Editors
V.K. VARADAN AND W. VARADAN
The Pennsylvania State. University, University Park, Pennsylvania, USA
NORTH-HOLLAND
AMSTERDAM · NEW YORK · OXFORD · TOKYO
LOW AND HIGH FREQUENCY
ASYMPTOTICS
V o l u me 2 in
Acoustic, Electromagnetic and Elastic Wave Scattering
Edited by
V.K. VARADAN AND V.V. VARADAN
The Pennsylvania State University
University Park, Pennsylvania, USA
NORTH-HOLLAND
AMSTERDAM · NEW YORK · OXFORD · TOKYO
© Elsevier Science Publishers B.V., 1986
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, mechanical, photocopying, recording or otherwise,
without the prior permission of the copyright owner.
ISBN 0444 87726 6
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Library of Congress Cataloging in Publication Data
Main entry under title:
Low and high frequency asymptotics.
(Acoustic, electromagnetic, and elastic wave
scattering; v. 2)
Includes bibliographies and indexes.
1. Waves. 2. Asymptotic expansions. I. Varadan,
V.K., 1943- . II. Varadan, V.V., 1948- .
III. Series.
QC157.A25 vol. 2 531M133s 86-18271
(531M133)
ISBN 0-444-87726-6 (U.S.)
PRINTED IN THE NETHERLANDS
Preface
This is the second volume of the Handbook on Acoustic, Electromagnetic and
Elastic Wave Scattering and it deals exclusively with high and low frequency
methods for scattering problems. This volume although numbered two is the
first to be published. The idea for a handbook of this type germinated after the
conference on "Recent Developments in Acoustic, Electromagnetic and Elastic
Wave Scattering-Focus on the T-matrix Approach", that was organized by us at
The Ohio State University in 1979. This was the first conference of its kind that
brought together acousticians, electrical engineers and solid mechanists. From all
reports, it was a useful experiment and lines of communication opened up between
scientists who would otherwise not look at each others' work. At the suggestion of
Dr. Nicholas Basdekas of the US Office of Naval Research an introductory chapter
was written for the proceedings of that conference that surveyed the field equations
in differential and integral form, and presented Green's functions, boundary
conditions, etc., for acoustic, electromagnetic and elastic fields. The proceedings
contained contributions from several well known scientists who have pioneered
very general numerical/analytical techniques for the solution of wave propagation
and scattering problems.
At that time, it was felt that many of the techniques had come to a stage where it
would be useful to the scientific community and to graduate students to have a
reference that could guide them very quickly through the basic principles of a
certain technique as it is applied to any of the three fields and lead them to the
relevant references. It was felt that collecting pedagogical treatments of the same
technique as it is applied to all three wavefields would open up a whole new source
of ideas to solid mechanists who would otherwise not look at the electromagnetic
literature, and vice versa. Further, it was felt that a major part of the work was
scattered in journal articles, conference proceedings, review articles and special-
ized monographs. This made it very difficult for a researcher just entering the field,
or a more established one looking for new ideas, to have quick access to a major
reference source. Finally it was felt that the development of paralld techniques for
all three fields and the mathematical unity underlying the field theories was
sufficient justification to come out with a unified presentation for acoustic,
electromagnetic and elastic wave scattering problems. This will also, perhaps,
encourage university faculty to offer courses to graduate students that would
expose them to all three wavefields at the same time.
vi Preface
A major project of this type cannot be undertaken without the help, encourage-
ment and cooperation of several professional colleagues, technical monitors at
funding agencies, secretaries and graduate students. We are extremely grateful to
Professor Jan D. Achenbach who first listened to our half gelled ideas and thought
that there may be something worthwhile in them. As a General Editor for
North-Holland and, more importantly, as a well known scholar in the area of waves
in elastic solids, he agreed to support our proposal to North-Holland to publish a
multiple volume Handbook on Acoustic, Electromagnetic and Elastic Wave Scatter-
ing. This was back in 1982. Soon after, Dr. Werner Neubauer, who was then at the
Naval Research Laboratory, Washington DC, provided us with a year's funding to
start work on the handbook volumes. We also benefitted greatly from discussions
with Professor Staffan Strφm who spent four months with us while we were at The
Ohio State University. An agreement was signed with North-Holland to edit a four-
volume handbook series. That was 1982 and this is 1984. In the interval there have
been several unexpected delays, contributors who had other commitments and
needed to be coaxed and nudged to hurry up with their valuable contributions, and
delays that were sometimes beyond our control. We hope that at least some of the
delays have resulted in a better Handbook.
The scope of the Handbook is very wide. We want the Handbook to serve as a
useful reference source for general descriptions of the various field theories, a
complete set of results for the scattering of all types of waves from penetrable and
impenetrable spheres and cylinders, experimental techniques for measuring scat-
tered fields for all three types of waves, high and low frequency asymptotic
methods, all general numerical techniques that have a broad applicability and the
final volume is expected to be devoted to inverse problems which is still at an
evolving stage at the time of this writing.
Contributions for pedagogical treatments of the above topics were requested
from scientists in universities and industry, who pioneered the techniques and are
well known by their publications on the subject. Every effort was made to find the
best authors for a particular topic. Omissions, if any, are only due to limitations on
our contacts and personal relationships with several other equally well-respected
scientists who have not been requested to contribute to the series.
Our primary thanks are to the contributors of the various chapters who have
been so generous with their time and ideas. This Handbook would not have been
possible without their cooperation.
We want to thank Judy Gerber of The Ohio State University, who initially
assisted us in streamlining procedures, corresponded with the contributors and got
several of the manuscripts ready for North-Holland. Even after our move to The
Pennsylvania State University, Judy has continued to type and assist us via
telephone. Lastly, we want to thank The Ohio State University and the Pennsyl-
vania State University for moral and material support of our efforts and our
graduate students and post-doctoral fellows who have assisted us in several ways in
preparing these volumes.
Preface vii
In an undertaking of this kind, it is impossible to preserve any type of uniformity
in style and convention even for two chapters in the same volume that deal with the
same wavefield. However, some care was taken to prevent major overlaps and
preserve a uniform format. We hope that our readers will forgive us any small
errors and still find the series as a whole useful in leading them on to new ideas for
research and new ways of thinking about problems in wave phenomena.
University Park, Pennsylvania Vi jay K. Varadan
October 1984 Vasundara V. Varadan
Introduction
This is Volume II of the Handbook on Acoustic, Electromagnetic and Elastic
Wave Scattering and is titled Low and High Frequency Asymptotics.
This volume is devoted to asymptotic methods for solving scattering problems in
the high and low frequency regimes. By high frequency, we refer to wavelengths of
the excitation that are short compared with the dimensions of the scatterer; and the
low frequency limit refers to wavelengths long compared with the size of the
scatterer. Both of these limits are of interest not only from a theoretical point of
view but also for practical applications. They allow us to solve problems analytically
that would not be possible otherwise at intermediate wavelengths. Further, the
so-called canonical problems solved at the high or low frequency regimes can be
used to solve other problems that are of practical interest. For example, using the
Geometrical Theory of Diffraction (GTD), the canonical problem of edge
diffraction is first solved and then used in solving the problem of diffraction by a
finite crack. In recent times, the crack problem has been of much interest for its
applications to Non-Destructive Evaluation (NDE) of flaws in structural materials.
The study of wave scattering by finite objects and extended surfaces at long
wavelengths or low frequencies was pioneered by Rayleigh very early in this
century and generally scattering in this frequency regime is called Rayleigh
scattering. Rayleigh scattering from finite bodies depends on the volume of the
body and Kleinman and Senior in this volume have chosen to define Rayleigh
scattering more formally as the first or dominant term in an expansion of the
scattered field in powers of the frequency. Lord Rayleigh made important
contributions to our understanding of scattering by a single object such as a sphere,
collections of spheres and periodic rough surfaces in the long wavelength limit. The
approach of Kleinman and Senior is to solve the integral representation of the
scattered field by an iterative procedure. The advantage of their approach is that
they have systematized the calculation of the dominant term in the long wavelength
limit for acoustic and electromagnetic problems involving impenetrable, penetr-
able, non-lossy and lossy obstacles. For those interested just in applications, their
final expressions either in analytical or integral form can be easily programmed on a
minicomputer. The next step is to extend this systematic procedure to the
elastodynamic case.
An alternative method used to study Rayleigh scattering is the method of
"Matched Asymptotic Expansions" as described by the contribution of Datta and
Introduction
χ
Sabina. Their treatment is quite extensive for elastodynamic problems involving 2D
and 3D geometries with and without boundaries. Owing to the mathematical
analogy between 2D antiplane strain problems in elastodynamics and scalar or
acoustic problems as well as electromagnetic 2D problems for TE or TM
polarization, the results presented by them for SH waves should be useful in
acoustics and electromagnetics. The method of Matched Asymptotic Expansions
(MAE) as used by Datta and Sabina relies heavily on a convenient analytical
solution for the corresponding static problem. Two expressions for the field outside
the scatterer are introduced, an inner one based on the static solution and an outer
solution that satisfies radiation conditions. The Van Dyke matching principle is
then used to match the two solutions and evaluate the unknown coefficients. This
particular contribution is noteworthy for the extensive applications considered.
Most of the solutions depend heavily on the famous solutions of Eshelby for static
problems involving elliptic cylinders and ellipsoids. One of the disadvantages of the
method especially in the 3D elastodynamic case is the complexity of the algebra and
the resulting danger of algebraic errors. Computer techniques such as MACSYMA
may be used successfully in the future to avoid such problems.
Interest in high frequency scattering or high frequency wave propagation, dates
back to the time of Fermβt in 1654 who made the first formal statements of ray
optics or geometrical optics which was successfully used to explain many known
features of the propagation and scattering of light. It was also recognized fairly
early that geometrical optics encountered difficulties at so-called shadow boun-
daries.
The necessary extensions to geometrical optics were first introduced by J.B.
Keller in 1958. This extension called the Geometrical Theory of Diffraction (GTD)
generalizes the results of canonical problems of straight edges and plane wavefronts
to curved edges and curved wavefronts. However, GTD solutions break down at
shadow boundaries, at caustics and at boundaries of zones of reflected waves.
Lewis and Boersma, /. Math. Phys. 10, 2291 (1969) corrected GTD for acoustic
and electromagnetic problems by developing the Uniform Theory of Diffraction
(UTD).
The chapter by Kouyoumjian and Pathak in this volume, on uniform GTD for
EM problems gives a simple formulation and several practical examples are
presented in a very lucid manner. Achenbach and Gautesen extended GTD and
UTD to elastodynamic problems and their contribution to this handbook summar-
izes recent results for 2D and 3D scattering of elastic waves by cracks and slits.
They have also developed a uniform Crack Opening Displacement (COD) method
in the high frequency limit using the COD of a semi-infinite straight edge as a
canonical solution to consider more realistic problems of finite curved slits and
cracks.
Finally, Felsen and Heyman have described a hybrid ray-mode technique for
acoustic and electromagnetic problems involving smooth convex bodies. Unlike
GTD, they have chosen to describe surface diffracted ray fields in terms of surface
Introduction xi
guided modes which shed energy due to the curvature of the body. Thus, these
modal fields are characterized by complex rays. They have also developed uniform
asymptotics to correct the failure of ordinary geometric theory. The diffracted field
is represented in terms of various wave types - leaky, creeping, surface trapped,
whispering gallery, etc. One of the special features of their formulation is that they
also consider penetrable scatterers unlike the GTD. They also indicate that there is
great promise in a hybrid ray-mode representation using the singularity expansion
method (SEM) in which the scattered field is represented in terms of damped
oscillations, each term being a particular resonance mode associated with the
scatterer. But the SEM is poorly convergent at short times when the ray-mode
description is expected to be at its best. A single self-consistent framework
incorporating the best features of both methods is also explored in their contrib-
ution although applications and further development are still to come.
As a general observation, it can be stated with certainty that asymptotic solutions
to scattering problems provide us with important physical insight into the types of
processes that dominate the scattered field. This understanding is essential if we are
to have any confidence in the results provided by even the most accurate numerical
technique implemented on a supercomputer. In many cases the asymptotic
solutions are the only ones that can be obtained analytically. Lastly in many
important practical applications, these are all that are required or physically
significant.
Contributors to this volume need no introduction from us. Their names are quite
familiar to all those interested in wave phenomena. They have all made rich
contributions to the literature in the form of journal articles, reviews and surveys as
well as specialized monographs and books. Although all chapters are confined to
asymptotic methods, the scope of the subject matter is so wide that it is impossible
to achieve any degree of uniformity in notation or style. Some effort was made to
make the format uniform and all contributors were asked to be generous in the use
of examples and references to original works as well as the display of results in the
form of graphs and tables. We hope that these features will make this volume and
the other volumes truly a Handbook on scattering that will be useful to the
specialist and the novice alike.
CHAPTER 1
Rayleigh Scattering
R.E. KLEINMAN
Department of Mathematical Sciences
University of Delaware
Newark, Delaware, USA
and
T.B.A. SENIOR
Department of Electrical and Computer Engineering
University of Michigan
Ann Arbor, Michigan, USA
Low and High Frequency Asymptotics
Edited by V.K. Varadan and V.V. Varadan
© Elsevier Science Publishers B.V., 1986