Physical Acoustics

389 Pages·1993·6.6 MB·English

by  ALLAN D. PIERCE and R.N. THURSTON (Eds.)

CONTRIBUTORS TO VOLUME XXII
ROGER H. HACKMAN
ALLAN D. PIERCE
Underwater Scattering
and Radiation
Edited by ALLAN D. PIERCE
THE PENNSYLVANIA STATE UNIVERSITY
UNIVERSITY PARK, PENNSYLVANIA
R. N. THURSTON
BELLCORE (retired)
RED BANK, NEW JERSEY
RUTGERS UNIVERSITY
PISCATAWAY, NEW JERSEY
PHYSICAL ACOUSTICS
Volume XXII
ACADEMIC PRESS, INC.
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Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
ROGER H. HACKMAN (1)
Lockheed Palo Alto Research Laboratory
Palo Alto, CA 94304-1191
ALLAN D. PIERCE (195)
Graduate Program in Acoustics
The Pennsylvania State University
University Park, PA 16802-1400
vn
Preface
Radiation and scattering occupy a significant portion of the science of
acoustics, and this is also the case for its sister science, electromagnetism.
The relevant aspects of the mathematical theory date back to the middle
of the nineteenth century, to Poisson, Green, Stokes, Kirchhoff, and
Rayleigh. However, up until the middle of the twentieth century, almost
all of the literature on radiation and scattering was concerned with
relatively simple shapes, such as spheres and infinitely long cylinders.
The chief analytical tool was the separation of variables, whereby
relatively general solutions of partial differential equations could be built
up of sums of products of solutions of ordinary differential equations.
Even when this was possible, the convergence of the sums was often so
slow that the applicability was severely limited. The arduousness of the
task of extracting numerical results for even what seems to be a very
simple problem is well-illustrated in a 1904 paper by Rayleigh entitled,
'On the Acoustic Shadow of a Sphere," in which appear calculated
results for what we now term ka = 10, with the sums including 20 terms.
Anyone who has ever attempted hand calculations cannot help but feel
sympathetic with Rayleigh's poignant remark: "as will readily be under­
stood, the multiplication by P and the summations involve a good deal of
n
arithmetical labour." (Rayleigh did these calculations himself, and this
was in the same year that he received the Nobel prize.)
The digital computer has of course changed the situation described in
the previous paragraph. Asymptotic methods, such as the Watson
transformation, matched asymptotic expansions, and the geometrical
theory of diffraction, have widened the scope of problems that can be
attacked by analytical means. Characteristically, one's desire for solutions
and insights always tends to exceed the analytical and computational
tools that are available, and this tends to spur additional creative efforts.
This is particularly true in regard to underwater acoustics applications,
where the medium conveying the radiated or scattered waves is strongly
coupled to the motion of the structure. The present volume contains two
treatises that describe the relevant theoretical foundations upon which
progress in underwater scattering and radiation is currently being made.
The first article gives a comprehensive discussion of the scattering by
ix
X Preface
elastic objects. The initial discussion is about the classic idealized shapes
of spheres and infinite cylinders and serves to introduce important
concepts such as normal modes, the S-matrix, the Γ-matrix, resonances,
whispering gallery modes, Franz modes, and Stoneley waves. Subsequent
sections present powerful and wide-sweeping methods for treating
scattering by elastic bodies of more general shapes. The Γ-matrix
formalism is discussed with exceptional clarity and then applied to
spheroidal scatterers and finite cylinders. The article relates general
theory to simple models and gives simple analytical and physical
interpretations to results. The combination of theoretical foundations
and numerical results provides an opportunity for readers new to this
subject to become familiar with the basic ideas and the limitations of
current theoretical work.
The second article is an extensive discussion of how variational
principles can be used in acoustics, with the choice of topics directed
toward applications to underwater acoustic radiation and scattering.
Variational techniques in acoustics have a long history proceeding from
late eighteenth century and early nineteenth century work by Lagrange,
Euler, Hamilton, Jacobi, and Kelvin, with substantial subsequent in­
novations and applications invented by Rayleigh. There are strong
indications that variational techniques will tend to be more frequently
used as the structural acoustics and physical acoustics communities seek
to develop techniques for bridging the gap between circumstances where
finite elements techniques are applicable and where ray acoustic tech­
niques are applicable. This chapter is the first comprehensive monograph-
length discussion of variational techniques specifically directed toward
acoustical applications. It will be of interest to anyone who wishes to
explore how one can implement variational methods and to anyone
interested in learning about specific variational principles that have
emerged in acoustics research.
All of the methods discussed require computation. However, as the
tone of the articles suggests, computational results are not enough to
furnish the sort of insight that is needed for a real understanding of
underwater scattering and radiation. One needs new modes of thinking,
new analytical approaches, and new terminology to handle such prob­
lems. It is our desire that this volume will help present and future
researchers, along with those who will use the results of such research, to
acquire an in-depth understanding of the physics of underwater scattering
and radiation.
ALLAN D. PIERCE
R. N. THURSTON
December 1991
— 1 —
Acoustic Scattering from Elastic Solids
ROGER H. HACKMAN
Lockheed Palo Alto Research Laboratory, Palo Alto, CA
1. Introduction 2
2. Spherical Solids 3
2.1. Introduction 3
2.2. The Normal Mode Solution and Its Analytical Structure 4
2.3. Resonances 15
2.4. Sommerfeld-Watson Transformation 35
3. Infinite Cylindrical Solids 46
3.1. Introduction 46
3.2. Normal Mode Theory 47
3.3. Free Waves on a Cylindrical Solid in Vaccuum 50
3.4. Sommerfeld-Watson Transformation 54
3.5. Resonances and Resonance Identification 57
4. The Γ-Matrix Formalism 61
4.1. Introduction 61
4.2. Huygens' Principle 63
4.3. Applications 66
4.4. Symmetry Properties of the S-Matrix 77
5. Finite Cylinders 81
5.1. Introduction 81
5.2. Large Aspect Ratio, Rigid Cylinders 82
5.3. Large Aspect Ratio, Elastic Cylinders 86
5.4. Axisymmetric Analysis 93
5.5. Flexural Analysis 119
5.6. Flexural Modes of Higher Circumferential Order 133
6. Prolate Spheroids 137
6.1. Introduction 137
6.2. Large Aspect Ratio, Rigid Spheroids 137
6.3. Large Aspect Ratio, Elastic Spheroids 139
6.4. The Axisymmetric Modes 150
6.5. The Flexural Modes 165
7. Surface Waves and Quasicylindrical Modes 174
7.1. Introduction 174
7.2. Transition to the SWP 176
7.3. Critical Tests 178
7.4. An "Improved SWP" 184
Acknowledgments 185
References 185
1
Copyright © 1993 by Academic Press, Inc.
AH rights of reproduction in any form reserved.
PHYSICAL ACOUSTICS, VOL. XXII ISBN 0-12-477922-0
2 Roger H. Hackman
1. Introduction
The acoustic scattering from elastic targets has been studied extensively
in the last three decades. For separable geometries, such as spheres and
infinite cylinders, formally exact solutions in the form of infinite,
eigenfunction series expansion can be obtained straightforwardly. Conse­
quently, these solutions have been studied extensively and relatively
complete numerical and analytical results are readily available in the
literature. Much of the more recent research on these geometries has
been concerned with resonance scattering theory (RST) and its implica­
tions for the inverse scattering problem. Such problems also serve as
testing grounds for the development of new methods of solution and,
often, new physical insights, as is witnessed by the development of the
high-frequency, surface wave picture (SWP) of the elastic response of
solid targets and by related developments in the generalized geometric
theory of diffraction (GTD).
By way of constrast, until recently, comparatively few results were
available for the scattering from elastic targets of more general
geometries, due to the relative analytical intractability of such problems.
For example, the extensive scattering treatise of Bowman et al (1969)
deals only with impenetrable bodies. However, with the advent of such
numerical/analytical techniques as the transition-matrix method (de­
scribed in Section 4), the state-of-the-art is that numerical solutions can
be obtained for most relatively "smooth" geometries of interest. Thus,
there have been a number of recent articles concerning the scattering by
solid elastic targets with finite cylindrical and prolate spheroidal
geometries.
This article gives an overview of the scattering from elastic targets,
with an emphasis on scatterers of nonseparable geometries. Sections 1
and 2 review the free-field, acoustic scattering from elastic spheres and
infinite cylinders with an emphasis on those aspects that play an
important role for finite cylinders and prolate spheroids. Particular
attention is paid to general properties that essentially establish the
analytic form of the solution and to the underpinings of the surface elastic
wave (SEW) picture of the high frequency elastic response of these
targets. Somewhat surprisingly, this picture is apparently valid even for
low characteristic frequencies for spherical and infinite cylindrical targets,
despite the breakdown of the formal basis for its justification. In Section
4, we present a brief review of the transition matrix method due to its
importance as a tool for the analysis of the latter class of scatterers. The
final Sections 5-7 discuss the analysis of the acoustic scattering from finite
cylinders and prolate spheroids. Of some interest here are the modifica-
1. Acoustic Scattering from Elastic Solids 3
tions of the SEW picture required for these targets, even at "relatively
high characteristic" frequencies, particularly for large length-to-diameter
ratios. This statement pertains to both the characterization of the elastic
response and to the coupling of the acoustic and elastic fields.
Although this chapter is intended as a tutorial/review of the scattering
from elastic solids, a number of new results are presented. While to some
extent the final selection of material is a reflection of the research
interests of the author, there is an extensive literature base in this general
subject area and a number of omissions (some of them important) have
been necessary due to space limitations. For example, the treatment of
RST is brief and meant to complement the more comprehensive review
of Flax et al. (1981). Likewise, recent developments in generalized GTD
and related material, such as the hybrid ray-mode theory, that pertain to
elastic solids are covered elsewhere in this volume and we do not discuss
them here. We have made an exception to this latter exclusion in the case
of applications of the Sommerfeld-Watson transformation, due to its
central role in the SWP. Finally, we should invoke a qualification at this
point. Most of the research in this area has been limited to elasic
materials whose characteristic acoustic impedances are much larger than
water; the subject material here is likewise limited. The physical picture
will change, in some cases significantly, when this condition is relaxed,
particularly concerning the effects of fluid loading on the vacuum
dynamical characteristics of the target and the coupling of the acoustic
field to the elastic fields of the target. The former point is amply
illustrated by considering the effects of fluid loading on thin shells
(Sammelmann et al., 1988).
Before concluding this introduction we note the existence of a number
of other review articles that complement this chapter and that may be of
interest, by Neubauer (1974) (see also Neubauer [1987]), Überall (1973),
and Gaunaurd (1989). A further useful collection of references on the
Γ-matrix is given by Varadan et al. (1988).
2. Spherical Solids
2.1. INTRODUCTION
The conceptual foundation for much of our intuition regarding the
acoustic scattering from elastic targets is based on studies of the
scattering from spherical and infinite cylindrical elastic solids. In this
section we consider spherical geometries; infinite cylindrical solid scat­
terei are reviewed in Section 3. The elastic sphere in a fluid is the
simplest nontrivial, three-dimensional example of a fluid-loaded, elastic
4 Roger H. Hackman
scatterer. This dynamical system is unique in that it is the only
three-dimensional geometry amenable to an exact, numerical/analytical
treatment over an extended region in the complex frequency plane, due
to the convenience of the eigenfunction series solution. By contrast, the
investigation of the scattering from more general geometries, such as
finite cylinders or prolate spheroids, are primarily numerical in nature
and essentially confined to the real frequency axis (for an exception to
this rule, see Peterson et al. [1983]). Thus, our emphasis here will be on
the physics of the dynamical scattering process and the development or
extraction of principles or results that can serve to guide the analysis of
more complicated geometries. Given the seminal nature of the physics of
this dynamical system, a careful treatment of the subject material is
merited.
In the following, we systematically develop the theoretical constructs
necessary to establish the analytic properties of the solution for the
acoustic scattering from an elastic sphere. We begin by outlining the
normal mode solution for the scattering problem and developing the
consequences of conservation of energy and causality. We then present
the results of the numerical analysis of the analytic structure and discuss
the implications of resonance scattering theory. We also discuss the
nature of the nonresonant background contribution and recent refine­
ments of this concept. The connection with the Regge pole repre­
sentation and the high frequency surface wave picture of the elastic
excitations is reviewed, as well as recent attempts to synthesize the
scattering solution from simple physical ideas.
2.2. THE NORMAL MODE SOLUTION AND ITS ANALYTICAL STRUCTURE
Theoretical solutions to the acoustic scattering by rigid, immovable
spheres and infinite cylinders were first obtained by Rayleigh in 1877
(Strutt, 1896). In the process, Rayleigh developed the method of normal
modes, known alternately as the harmonic or Rayleigh series, valid for all
targets whose surface conforms to one of the constant-coordinate surfaces
of any of the 11 separable coordinate systems. It is noteworthy that this
method of solution has served as the starting point for almost all
subsequent theoretical analyses of the scattering by spherical and infinite
cylindrical geometries.
There were a number of early theoretical studies of the acoustic
scattering by fluid cylindrical and spherical targets (Morse, et al. 1946;
y
Morse, 1936; Anderson, 1950), and Lax and Feshbach (1948) extended
Morse's analysis to the case of a sphere with a coating that had a known
acoustic impedance. The normal mode scattering solution for an elastic