Ocean Acoustic Propagation by Finite Difference Methods

121 Pages·1988·6.42 MB·English

by  D. Lee and S. T. McDaniel (Auth.)

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OCEAN ACOUSTIC PROPAGATION
BY FINITE DIFFERENCE METHODS
by
D. LEE
Naval Underwater Systems Center, New London, CT06320, U.S.A.
S. T. McDANIEL
Applied Research Laboratory, The Pennsylvania State University,
University Park, PA 16802, U.S.A.
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Published as a special issue of the journal Computers &
Mathematics with Applications, Volume 14 Number 5 and
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Also available to non-subscribers.
In the interests of economy and rapid publication this
edition has not been re-paginated.
Printed in Great Britain by A. Wheaton & Co. Ltd, Exeter
FOREWORD
The monograph presented here to readers of Computers & Mathematics with Applications
represents, on the one hand, a new departure; and, on the other, a classical adherence to our
announced aims.
The nature of the new departure is that this is the first time that we are publishing a single work
of monograph length. We haven't been doing this, nor do most other scientific journals, for obvious
reasons of limitations of space, desire to publish works of many authors, and several other
considerations.
The principal argument that made us change our minds (for now, on just an experimental basis)
is that it is very difficult to fulfill everything that we promised to do in our Aims and Scope, and
implied in our title: to present works which start out by modelling mathematically some important
physical phenomenon, and then implement it on a computer.
The work presented in this volume, however, fulfills all of these requirements.
Briefly, it consists of a nontrivial mathematical description of ocean acoustic fields, and the
solution of the model equations using interesting and very efficient numerical techniques. What
makes the work even more worthwhile is that detailed computer listings accompany the text. These
are proven programs; they are currently in use by many ocean acoustic facilities, which are listed
in the Preface.
An additional feature of this monograph is that it combines not only mathematics, physics,
engineering and computer science, but it also draws on the most recent published sources for
material. Thus, from this point of view, we have a monograph here which contains some (as yet)
unpublished material by the authors, together with results of theirs that were published earlier; and,
of course, properly referenced usage of the works of others.
We hope that our readers will find this volume as useful and interesting, as our Board of Editors
did.
E. Y. RODIN
Department of Systems Science and Mathematics
Box 1040, Washington University
St Louis, MO 63130
U.S.A.
Vll
PREFACE
Over the past several years, intensive research has been devoted to solving ocean acoustic wave
propagation problems. However, until recently, direct numerical techniques for solving these
problems received little interest or attention. Now, with the introduction of finite difference
schemes, numerical solution of wave propagation problems has proved successful. Other numerical
methods, such as ordinary differential equation methods and finite element techniques, have also
been applied to wave propagation problems. The finite difference approach, however, offers distinct
advantages for treating realistic ocean acoustic propagation problems.
The application of finite difference schemes to provide efficient solutions to a class of propagation
problems forms an important and useful contribution to the ocean acoustics community. The
particular technique that is used is an implicit finite difference (IFD) scheme of the Crank-Nicolson
type. The computer code (called the IFD model) that implements the IFD scheme was designed
mainly for research purposes and has been in existence since 1982. Because of its capability,
accuracy and reliability, the IFD model has attracted a large number of user institutions worldwide
including various universities and research laboratories that use the model for research studies as
well as for production runs. Notably, the Naval Postgraduate School (U.S.A.), University of Texas
at Austin (U.S.A.), Naval Ocean Research and Development Activity (U.S.A.), Massachusetts
Institute of Technology (U.S.A.), Yale University (U.S.A.), NATO SACLANT ASW Research
Center (Italy), Admiralty Underwater Weapons Establishment (England), Weapon Systems
Research Laboratory (South Australia), and many others find applications for the IFD model.
The increasing use of the IFD model suggests the need for a clear and detailed documentation
of the model. This monograph presents an intensive development of the finite difference schemes
for the solution of acoustic propagation problems in realistic ocean environments. The method used
is also applicable for solving many of the differential equations that arise in different scientific and
engineering applications. Such applications include atmospheric acoustics, plasma physics, quan-
tum mechanics, optics, seismology, electromagnetics, etc.
This monograph is aimed at ocean acoustic scientists and engineers at all levels. No specialized
background or knowledge is required to understand this work: the material is basic, but advances
the state-of-the-art. The volume may be used along with the IFD model to readily perform research
computations. It is thus, a very practical volume for readers, particularly graduate students,
involved in basic research.
Major portions of this monograph are based on a series of research articles published in the
Journal of the Acoustical Society of America, the Journal of Mathematical Modelling, the Journal
of Computers & Mathematics with Applications and the Journal of Computational Physics.
Without the sponsorship of the Naval Underwater Systems Center (New London, Connecticut),
Office of Naval Research and Naval Sea Systems Command this monograph would not have been
possible. We express our deep gratitude to our scientific managers who played an important role
in continuously supporting our research efforts: Drs W. A. Von Winkie and K. M. Lima, both
of the Naval Underwater Systems Center, and Dr R. L. Lau of the Office of Naval Research
deserves our special thanks. We are extremely grateful to Professor M. H. Schultz of Yale
University, Professor W. F. Ames of the Georgia Institute of Technology and Professor D. F. St.
Mary of the University of Massachusetts for their valuable discussions during the preparation of
this monograph. G. Botseas also deserves our special thanks for his diligent assistance in developing
the computer code.
D. LEE
S. T. MCDANIEL
IX
Comput. Math. Applic. Vol. 14, No. 5, pp. 305-423, 1987 0097-4943/87 $3.00 + 0.00
Printed in Great Britain. All rights reserved Copyright © 1987 Pergamon Journals Ltd
CHAPTER 1
INTRODUCTION
Ocean acoustic wave propagation problems deal with the solution of representative partial
differential equations. These equations, which govern realistic physical ocean acoustic phenomena,
are all regarded as wave equations. Because of the complex nature of the ocean, the various wave
equations can be very complicated in nature and permit a closed form solution only in very simple
cases. This motivated ocean acousticians as well as ocean scientists and engineers to consider
specialized approximations to these problems for simplicity. Thus, a number of wave equations are
in existence in different forms; each has its own advantages due to special approximations or
treatments. The approximations made generally result in a loss of accuracy or limit the range of
validity of the solution. Even when these approximations permit a closed form solution; it usually
involves special functions, integrals, etc., which are often not convenient to evaluate. In such
situations, direct numerical solution of the wave equation may offer significant advantages, not
only by permitting treatment of more realistic environments but also from a computational
viewpoint. This monograph demonstrates the applicability of numerical methods for the efficient
solution of complex ocean acoustic wave propagation problems.
Numerical methods which have useful applications to ocean acoustic wave propagation problems
did not receive much attention or interest until numerical ordinary differential equation methods
as well as finite difference schemes were introduced for solving these problems. Due to the rapid
growth of supercomputers and modern numerical techniques, solutions for complicated scientific
problems are now possible. Ocean scientists have, however, not yet taken full advantage of modern
numerical techniques and advanced supercomputer technology. Among these numerical methods
the finite difference is well-known, straightforward and stands out as being universally applicable
to most scientific problems because of its generality and unconditional stability. Moreover, it is
not difficult to implement into computer codes. However, the formulation of efficient finite
difference schemes is an art; the precise formulation determines the size of the computational grid,
and thus affects the computation speed.
The monograph begins with a description of general ocean acoustic wave propagation problems
which are mathematically represented by a partial differential equation, the wave equation.
Appropriate initial and boundary conditions need to be prescribed for the problem to be
well-posed. The wave equation, conventionally regarded as the reduced wave equation, is a scalar
elliptic equation whose solution consists of transmitted and reflected fields in three dimensions. It
is convenient to express the wave equation in cylindrical coordinates in a three-dimensional ocean,
since in most cases the wave field has a very small azimuthal angular dependence. This reduces
the three-dimensional problem to a two-dimensional one which is a little easier to handle. Thus,
a solution of the two-dimensional wave equation is sought which still consists of a transmitted field
and a reflected field. A class of ocean acoustic propagation problems which typically fall into the
two-dimensional category is long range propagation at low frequencies. This class of problems can
best be represented by a pseudopartial differential equation with complex coefficients. In the second
chapter, "Ocean acoustic wave propagation problems", we derive this type of pseudopartial
differential equation and discuss how this equation reduces to a parabolic wave equation of the
Schrödinger type. Approaches that yield both transmitted and reflected fields using an operator
splitting technique are discussed. Due to the nature of this class of long range, low-frequency
propagation problems, we generally require only the solution for the transmitted field. The
representative pseudopartial differential wave equation sets the stage for us to apply numerical
finite difference methods to obtain the transmitted field by a marching process.
The third chapter, "Finite difference schemes", is devoted to fully describing the development
305
306 D. LEE and S. T. MCDANIEL
of an implicit finite difference scheme applicable to the problem addressed. Finite difference
techniques are described in detail from the development of fundamental concepts to the analysis
of discretization errors in a style particularly suitable for readers who have little knowledge of finite
difference concepts. To solve the representative wave equation, both explicit and implicit finite
difference schemes are developed. When the conventional explicit schemes is used to solve the
parabolic equation with real coefficients, e.g. the Euler scheme, it is conditionally stable. We show
that if the same conventional explicit technique is applied to solve the parabolic wave equation with
imaginary coefficients, the resulting computational scheme is unconditionally unstable. This
provides a strong motivation for applying an implicit scheme to solve the parabolic wave equation.
The technique that we develop is an implicit finite difference scheme of the Crank-Nicolson type
which will be referred to as the IFD scheme. The basic formulation of this scheme and the theory
regarding its consistency, stability and convergence are described in detail.
Implementation of the finite difference scheme to solve the wave equation, requires the
application of appropriate initial and boundary conditions. Boundary conditions for the reduced
elliptic wave equation involve not only both surface and bottom boundaries, but also the "wall"
boundary that restricts the solution to the computational range of interest: solutions of the elliptic
wave equation are solutions of a pure boundary value problem. The special class of problems
discussed in the previous chapter can be solved by a marching process and do not require a wall
boundary condition, thus, a pure boundary value problem is transformed into an initial boundary
value problem. However, a different kind of boundary condition known as the interface boundary
occurs and needs to be treated realistically and accurately. In the ocean, temperature, salinity and
pressure affect the sound speed as well as the density structure of the water mass creating a layered
medium. The sediments that compose the ocean floor are layered due to periodic deposition of
sedimentary material. These phenomena stratify the ocean environment into a layered medium,
thus, forming interfaces. At each interface, "continuity conditions" must hold, i.e. the pressure and
normal component of particle velocity are continuous at the interface. This type of interface can
be handled elegantly by a finite difference technique which is new to ocean acoustics. Numerical
treatment of the interface conditions by the finite difference technique not only produces accurate,
realistic results but also advances the state-of-the-art in numerical solution of ocean acoustic wave
propagation problems. The treatment of interface conditions by the finite difference technique is
a major portion of Chapter 4, "Initial and boundary conditions". An extended treatment to the
case of irregular interfaces is also presented. Although the mathematics involved is very
complicated, the procedure is clear and is not difficult to follow. To demonstrate the effects of
interface conditions we include two benchmark test problems to show the validity of the numerical
treatment. The selection of the time step size for the heat equation presents no problem for the
Crank-Nicolson scheme but it presents difficulty for the wave equation of the Schrödinger type.
Following Chapter 4, is an analysis of range step size selection.
Long range acoustic propagation in the ocean is dominated by energy that propagates over a
narrow angular regime with respect to the horizontal direction. This occurs because energy at high
angles interacts strongly with the boundaries and is rapidly attenuated. The parabolic wave
equation we discussed in an earlier chapter is an equation of the Schrödinger type which can
accommodate only a narrow angle of propagation; for this reason the equation which we
introduced as the "standard" parabolic wave equation is also called the "narrow angle" parabolic
equation (PE). For many cases, however, the narrow angle PE is not accurate enough to represent
the wave field. To accommodate wide angle propagation, wide angle wave equations have been
developed which, mathematically speaking, are again pseudopartial differential equations. Chapter
6 is devoted to the fundamental mathematical development of the wide angle wave equation and
its solution. The basic questions which are addressed in this chapter are the manner in which the
development accommodates wide angle propagation and how the propagation angle is measured.
The key to this development is the derivation of a satisfactory mathematical representation of a
square root operator which we choose to represent by a rational function approximation to the
desired order of accuracy. In this chapter, rational function approximations are derived to handle
angles of propagation in two different ranges: 10° and 40° angular range.
In this monograph we show that the IFD scheme is an effective solution to the parabolic wave
equation. In fact, there are a number of other solutions to the parabolic wave equation as well
Introduction 307
which are worthwhile studying. Among these methods is a special class of explicit finite difference
schemes. Remember that the motivation for applying the IFD schemes to solve the parabolic wave
equation was its advantageous unconditional stability over conventional explicit schemes. We have
already proved the instability of conventional explicit schemes when used to solve equations of the
Schrödinger type. However, it is known that explicit finite difference schemes have a number of
desirable advantages: they are easy to implement and require less storage. These advantages are
specially desirable for solving multi-dimensional problems. In addition, it is often easy to vectorize
an explicit scheme on the many pipeline-oriented computers available today. Then, it is natural
to ask the question whether there exist alternative stable explicit schemes for the Schrödinger
equation. This question has been answered affirmatively for a case where an appropriate dissipative
term was introduced into an explicit scheme which was conditionally stable. A class of new explicit
schemes have been developed based on this approach. We include these new explicit schemes in
Chapter 7, "Applicable solution methods other than the implicit finite difference scheme".
Formulations as well as theoretical developments are included with proofs. These developments
represent a significant advance in numerical techniques. These new explicit schemes can also be
used effectively to solve problems in other areas of physics.
We have developed representative wave equations for ocean acoustics, and introduced an
efficient numerical implicit finite difference method to solve these wave equations. In addition, we
developed numerical procedures to treat realistic ocean environmental effects. To combine all these
effectively together, we constructed a computer code, the IFD computer code. We have made this
code as reliable as possible, easy to use, accurate, general purpose and easy to modify. Not only
can this code be used for research purposes, but it can also be used to make production runs. At
the date of publication of this monograph, there are a large number of satisfied IFD code users
among universities and research laboratories nationwide as well as worldwide. For the IFD code
user, we present a chapter entitled "Representative test examples". Most of them are benchmark
test problems. Users can exercise these problems to gain experience in solving realistic problems
and also to get a "feel" for the computer code. We have carefully selected these test examples for
the sole purpose of exhibiting different ocean environmental effects, to examine the computational
accuracy, and to demonstrate the versatility of the model. It should be pointed out that this IFD
model is not limited to only the solution of the class of ocean acoustic wave propagation problems
addressed earlier in this monograph. It also provides practical applications to other scientific and
engineering problems frequently arising in the fields of atmospheric acoustics, plasma physics,
quantum mechanics, optics, seismology, electromagnetics, etc.
This monograph concludes with a comprehensive listing of the IFD computer code designed in
FORTRAN language. This code represents a self-contained efficient numerical treatment of a class
of computational ocean acoustic wave propagation problems.
CHAPTER 2
OCEAN ACOUSTIC WAVE PROPAGATION PROBLEMS
In this chapter, we formulate the general underwater acoustic propagation problem and introduce
the approximations that will be used to reduce this problem to a computationally tractable form.
The limitations that these approximations impose on our solution and the conditions under which
they are valid are discussed in some detail.
2.1. THE GOVERNING WAVE EQUATION
Acoustic wave propagation in an ocean medium for a harmonic point source is governed by the
reduced wave equation, a homogeneous Helmholtz equation
V2/>+Ä:2p=0, (2.1)
where
V2 = Laplacian operator;
p =p(x,y z) the wave field in Cartesian coordinates;
9 9
k =2nf/c(x y z), the wavenumber;
9 9
/= source frequency;
c = c(x y z), the sound-speed profile.
9 9
Consider a simple acoustic wave propagation problem in a rectangular region, as shown in
Fig. 2.1.
The region shown in Fig. 2.1 is two-dimensional. Although the effects of horizontal variations
in the sound-speed profile can be observed [1], they are usually small. Furthermore, rarely is
sufficient oceanographic information available to warrant a full three-dimensional solution of the
wave equation. Hence, our first simplification will be to consider a two-dimensional wave equation.
(Solution of the three-dimensional wave equation is discussed by Weinberg and Burridge [2] and
Perkins and Baer [3].) To include the radial spreading experienced by a point source, this
two-dimensional wave equation will be expressed in cylindrical coordinates.
In Cartesian coordinates, the V2 operator takes the form
d2p d2p d2p
l2 Λ
(2.2)
SURFACE
INITIAL RADIATION
FIELD CONDITION
BOTTOM
Fig. 2.1. A region of sound propagation.
309
310 D. LEE and S. T. MCDANIEL
If we use cylindrical coordinates for the operator V2, equation (2.2) becomes
d2p I dp δ2ρ 1 82p ,,
Here we make use of the following:
x —r cos (9,
y = r sin Θ
and
z = z.
Further, as we mentioned earlier, the horizontal or azimuthal variation is often negligible in
practice, then, equation (2.2) can be written as
d2p 1 dp d2p , , ,
In equation (2.4), we defined
k = k n,
0
where
k = 2 nf/c , c is a reference sound speed
0 Q 0
and
w(r, z) = c /c(r, z) is the index of refraction.
0
The associated boundary conditions with equation (2.4) can be described by formulas (2.5)-(2.8)
with the use of Fig. 2.2:
/7(r ,z)=/7 (z), (2.5)
0 1
P(r,z )=p (r) (2.6)
0 2 9
P(r,z )=p (r) (2.7)
B B
and
lim/7(r ,z) = 0, (2.8)
m
where
z = the depth of the ocean bottom,
B
r = the maximum range of propagation,
m
r = range variable
and
z = depth variable.
The solution of equation (2.4) satisfying the conditions (2.5)-(2.8) requires certain environ-
mental information. Noticeably important environmental information involves the sound-velocity
profile (SVP), and surface and bottom conditions.
P(r, z0) rm
I 9λ22pη 11 ddpo d2p 9 .
P(r0, z) dr2 r dr dz2
P(r, z )
B
Fig. 2.2. A well-posed acoustic wave propagation problem.