Numerical ocean acoustic propagation in three dimensions

NUMERICAL OCEAN ACOUSTIC PROPAGATION IN THREE DIMENSIONS

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NUMERICAL OCEAN ACOUSTIC PROPAGATION IN THREE DIMENSIONS

Ding Lee Naval Undersea Warfare Center & Yale University, USA

Martin H. Schultz Yale University, USA

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co Pte Ltd P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Library of Congress Cataloging-in-Publication Data Lee, Ding, 1925Numerical ocean acoustic propagation in three dimensions / Ding Lee, Martin H. Schultz. p. cm. Includes bibliographical references. ISBN 981022303X 1. Underwater acoustics ~ Mathematical models. 2. Sound-waves - Transmission — Mathematical models. 3. Wave equation — Numerical solutions. I. Schultz, Martin H. II. Title. QC233.L39 1995 534\23«dc20 95-32946 CIP

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Preface

It is clear that the nature of the ocean itself is three-dimensional. Solving three-dimensional acoustic propagation problems is difficult because the ocean must be modeled by a large number of environmental parameters. Most existing ocean acoustic propagation models are aimed at providing solutions for two-dimensional (i.e. range and depth) problems because the three-dimensional effects are often weak. Since problems in various regions of interest can be solved two-dimensionally, there has been no urgency to solve three-dimensional problems. Indeed, solutions to two-dimensional problems are themselves quite complicated in nature, and the search for three-dimensional solutions will certainly increase the computational complexity. Moreover, even if a three-dimensional problem is formulated elegantly by means of mathematical and physical theory, the implementation of such a solution into the realistic computation presents a large-scale computational problem. We must face reality — the ocean variability is three-dimensional. As the ocean should be modeled by a large number of highly varying environmental parameters, solutions to the three-dimensional ocean acoustic propagation problems rely on advances that are not only physical and mathematical, but also computational. There is encouraging news: recent advancements in numerical mathematics and supercomputers make it

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possible to solve complicated three-dimensional problems at least partially. We should take advantage of these significant scientific advancements, together with our knowledge of ocean physics, to develop efficient models so that useful tools will be available to examine the actual ocean effects. We will henceforth move forward with confidence with the eventual objective of solving full three-dimensional problems within our (real) time requirement. With this objective in mind, we proceed to construct models aimed at solving numerical ocean acoustic propagation problems in three dimensions. Three-dimensional ocean effects have been studied by a number of scientists. However, most of these studies were performed for specialized environmental situations. A useful, general purpose model is one that can handle long range acoustic propagation through range-dependent, azimuth-coupled environments including irregular interface boundaries, a rough surface, and a rough bottom. This type of model has not yet been fully developed. This book provides some steps toward the solution of the complete ocean acoustic propagation problem. The purpose of this book is to introduce a basic, workable, three-dimensional ocean acoustic propagation model that will satisfy most fundamental environmental requirements and that will be practical for realistic applications. The model we have introduced to the ocean acoustic community is handled by an accurate technique which is general enough to create a wide range of capabilities to handle ocean physics. Its development relies on a combination of an ordinarydifferential-equation method, a functional operator splitting technique, a rational function approximation, and a finite difference scheme. Due to its generality of application, this basic model is expected to be modified or extended without excessive difficulty to accommodate any additional required capabilities. This book reports the complete theoretical result as well as presents the computational development of this model. While advanced knowledge of numerical methods and applied mathematics is not required to understand this book, such knowledge will be helpful. Scientists and engineers engaged in research, development, and testing and are interested in three-dimensional computational ocean acoustics should have little difficulty understanding the content of the book. It is oriented toward graduate students and research scientists who will find it useful for their research in numerical methods and computational acoustics.

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Major portions of this book are based on a series of research articles published in the Journal of the Acoustical Society of America, the Journal of Mathematical Modeling, the Journal of Computers and Mathematics with Applications, and the Journal of Computational Physics, and in books such as Ocean Acoustic Propagation by Finite Difference Methods (Pergamon Press, 1988), Computational Oceans Acoustics (Pergamon Press, 1985), Proceedings of the First, Second, and Third IMACS Symposium on Computational Acoustics (North-Holland, 1988, 1990, 1994), and other computational acoustics books. We wish to express our deep gratitude to a number of our technical managers who played a very important role in this research, without whose support this accomplishment would never have become possible. We would first like to thank Dr. Richard L. Lau of the Office of Naval Research for his understanding and continuing support of this research. His office initiated the research entitled "Computational Acoustics" and he encouraged us to interact with various academic scholars to apply powerful numerical methods for the purpose of solving ocean acoustic propagation problems — a first in the history of computational ocean acoustics. We also thank other technical managers of the Office of Naval Research who joined Dr. Lau in supporting this research after 1983. They are Drs. Raymond Fitzgerald, Robert Sternberg, Robert Obrochta, and Marshall Orr. Special thanks are also due to Drs. William A. Von Winkle and Kenneth Lima of the Naval Underwater Systems Center, whose continuous independent research support gave us the opportunity to make useful contributions to the acoustic community. In expanding scientific activity by developing useful research topics, the International Association for Mathematics and Computers in Simulation (IMACS), under the leadership of Prof. Robert Vichnevetsky of Rutgers University, initiated the establishment of the IMACS Technical Committee No. 17 on computational acoustics. This committee has held three international symposia: one at Yale University, USA, in 1984, one at Princeton University, USA, in 1989, and one at Harvard University, USA, in 1991. These three symposia encouraged our developments, and some of the contributions there are reported in this book. We are grateful to Prof. Vichnevetsky for his encouragement too.

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Other scholars who participated in the useful technical discussions for the preparation of this book include Prof. Allan R. Robinson of Harvard University, Prof. Donald F. St. Mary of the University of Massachusetts, Dr. Youcef Saad of the National Aeronautical Space Administration (NASA), Ames Research Center, and Prof. Gregory A. Kriegsmann of Northwestern University. Our special thanks also go to astronaut Paul D. Scully-Power of the Naval Underwater Systems Center, who supplied evidence of three-dimensional eddies from the photos he took on board the space shuttle Challenger in 1984. Also we thank George Botseas for his diligent and skillful help in developing the computer code. Ding Lee Martin H. Schultz

Contents

Preface

v

Chapter 1 Introduction

1

Chapter 2 Basic Mathematical Model Developments

11

Chapter 3 A Pseudopartial Differential Equation 3.1. Siegmann-Kriesmann-Lee Model 3.2. A Preconditioning Solution

25 25 26

Chapter 4 A High-Order Wave Equation 4.:. Lee-Saad-Schultz (LSS) Model 4.1.1. The LSS Mathematical Model 4.1.1.1. Stability 4.1.1.2. Accuracy 4.2. A Finite Difference Solution 4.3. Model Appraisal 4.4. Computer Code — FOR3D 4.4.1. Computer Algorithm: Two-Step Procedure 4.4.2. Computations for Step 1 4.4.3. Computations for Step 2 4.4.4. Computer Model

47 48 49 52 53 54 56 59 59 60 62 63

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4.4.4.1. 4.4.4.2. 4.4.4.3. 4.4.4.4. 4.4.4.5. 4.4.4.6.

Model capabilities Model structure Geometry of propagation Data structure of the acoustic field Main program FOR3D Subroutines

63 66 66 68 68 70 79 80 80 83 90 91

Chapter 5 Enhancement of the High-Order Wave Equation 5.1. Density Variation Capability 5.1.1. Mathematical Formulation 5.1.2. A Numerical Solution 5.1.3. Theoretical Justification 5.1.3.1. Interface conditions 5.1.3.2. The numerical treatment of discontinuities 5.1.3.3. Theoretical justification of the numerical treatment 5.1.3.4. Enhanced computer code FOR3D with density variations 5.2. Enhancement of Computation Speed 5.2.1. Differential and Difference Operators 5.2.2. The Douglas Scheme 5.2.2.1. The Douglas operator 5.2.2.2. Stability 5.2.2.3. Matrix structure 5.2.3. A Tridiagonal Solver 5.2.4. The Douglas Scheme with Density Variations 5.2.4.1. The Douglas operator with density variations 5.2.4.2. Stability 5.2.4.3. Matrix structure

110 113 114

Chapter 6 Numerical Accuracy Test: An Analytic Solution 6.1. Exact Solutions 6.2. Test Examples 6.3. Numerical Computations

119 123 125 130

Chapter 7 Three-Dimensional Effects 7.1. Concepts of 3D, N x 2D, and 2D 7.1.1. Mathematical Classification

137 138 139

92 93 95 96 97 99 99 102 103 105 110

Contents

7.1.2. Computational Classification 7.1.3. Physical Classification 7.2. Numerical Examples 7.2.1. An Ocean Environment without 6 Coupling 7.2.2. An Ocean Environment with 6 Coupling Chapter 8 The Computer Model — FOR3D

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140 141 144 145 148 155

Chapter 1

Introduction

Due to the weak three-dimensional effects, most existing ocean acoustic propagation models are aimed at providing solutions to problems that are two-dimensional, i.e. range and depth. Thus, even though the ocean is three-dimensional, the development of three-dimensional models to solve realistic three-dimensional problems has received little attention. Moreover, there is inadequate evidence to indicate the significance of three-dimensional effects. The ocean is surrounded by land and is bounded above by air and below by mud, sand, rock, weed, or other materials. Due to these surroundings, the ocean environments are ever changing. For example, the seasonal change of temperature can effect the water density and its sound speed. These ocean environmental changes can influence the intensity level of the acoustic wave field. field. As another example, consider the ocean which involves inhomogeneities over which a strong wind causes formation of internal waves that in turn can cause horizontal refraction of sound in the ocean. A specific illustration is the Gulf stream and its frontal zones. Within these frontal zones, temperature, salinity, density, and sound speed suffer strong variations. Large eddies are observed near such intensive frontal zones. These eddies appear in the rough shape of a ring and are often l

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formed from the separation of meanders from the ocean flow. Two types of eddies, warm and cold, can be distinguished, with the former displaying lower values of sound speed. Environmental effects of the type described above are three-dimensional. Thus, every true three-dimensional model must be able to treat these physical phenomena. Some of them are illustrated by examples in the chapter covering three-dimensional effects. To gain more knowledge of three-dimensional ocean environmental features, readers can consult the available literatures (see Brekhovskikh,1 Urick,2 Officer,3 Shang, 4 Pierce, 5 and Clay 6 ). It is natural to ask the question: How do the physical environmental conditions affect sound propagation? To answer this question satisfactorily, one needs a reliable acoustic model with the required capabilities to conduct investigations. This volume introduces a number of such numerical models and describes one in detail. The numerical model was chosen because it is capable of efficiently handling the threedimensional effects. In 1984, after the return of NASA astronauts who traveled on board the space shuttle Challenger, astronaut Paul D. Scully-Power released a set of historic photographs showing the ocean dynamics. The NASA photo # S1735-094 was used as a cover for our book Computational Ocean Acoustics.7 This photo clearly exhibits the existence of three-dimensional ocean eddies. This pictorial evidence further motivated ocean acousticians to examine true three-dimensional effects. A natural first question is: Where do we look for a realistic description of three-dimensional eddies? This problem is being actively investigated by the community of physical oceanographers, and many results are becoming available. As examples, we deal with sets generated by scientists using the Harvard Open Ocean Model (HOOM). 8 A second question is: Does a general purpose, accurate three-dimensional wave propagation model capable of handling such data correctly exist? These questions stimulated the examination of available three-dimensional acoustic propagation models. One result of this examination was the development of an accurate numerical model by Lee et al.9 using an iterative preconditioning technique. Prior to that work, one contribution to the fully three-dimensional modeling development was by Bear, 10 who applied the split-step algorithm to solve the three-dimensional parabolic equation first introduced the Tappert. 11 Later, Perkins and Baer 12 discussed a simplified approximation to the model in Ref. 10. Their contributions exhibited weak three-dimensional effects, from which they formulated the so-called Nx 2D concept (as we will discuss later). Baer and Perkins' model solves

Introductionon

3

a narrow angle, three-dimensional parabolic equation, but it does not take into consideration the vertical wide angle capability. Their application of the Fast Fourier Transform is efficient for treating constant coefficient parabolic partial differential equations, but an accurate treatment for handling the bottom, particularly a rigid bottom, has been overlooked. In comparison, the accuracy of preconditioning models is highly recognizable, but the computation speed is impractical. Thus, we were motivated to formulate another wave equation and to develop efficient numerical methods to solve this wave equation accurately. During this period, Bayliss, Goldstein, and Turkel 13,14 introduced an iterative method for the solution of the Helmholtz equation and discussed its numerical solution for wave propagation problems in underwater acoustics. In 1986, Goldstein 15 described multigrid preconditioners applied to three-dimensional parabolic equation type models. The multigrid preconditioners were more effective than the one introduced by Schultz et al..9g However, a drawback of Goldstein's solution it that it cannot go out far in range, and, thus, is incapable of handling very long range propagation. Consequently, Lee et a/.16 developed a mathematical model, known as the Lee-Saad-Schultz al. (LSS) wide angle wave equation, and an efficient numerical scheme to accurately solve this wave equation. Botseas et al.17 implemented this solution into a multipurpose computer code (FOR3D) and later applied FOR3D to process the HOOM data and other three-dimensional inputs to obtain various sets of interesting results. To date, there are at least two other three-dimensional wave propagation models. One is based on ray theory, i.e. the notable HARPO (Hamiltonian Acoustic Ray-tracing Program for the Ocean), originally developed by NO A A and later modified by Newhall et al.ls for ocean acoustic applications. The other model, developed by Chiu, 19 is based on a mode-coupling theory and is also suitable for processing ocean data. The HAPRO model is efficient for high frequencies, but is presumably less accurate for lower frequencies. The mode-coupling model does not have frequency restrictions, but its capabilities and computation speed need additional upgrading. A few other fast models for solving three-dimensional problems can be found in Refs. 20 and 21. This book is divided into four main parts: (1) the development of new representative wave equations, (2) the development of new and efficient numerical schemes to solve these representative wave equations, (3) the implementation of these numerical solutions into a workable computer code for both research and application purposes, and (4) the application of

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this computer model to examine true three-dimensional and other related effects. The successful application of finite difference schemes to date and their generality and favorable stability properties are important factors to motivate extensions of these numerical schemes to solve three-dimensional problems. In order to treat the various ocean environments realistically, representative wave equations are developed, each with its own physics and mathematical properties. A separate chapter is devoted to some basic mathematical model developments, in particular, the descriptions of the requirements for developing the model with the desired capabilities. The main representative equation we consider is a one-way outgoing wave equation that can be approximated by a parabolic equation, a pseudopartial differential equation, or a high-order partial differential equation. In the mathematical development, cylindrical coordinates are adopted so as to be useful in realistic environments. The selection of the important physical phenomena to be included in the representative wave equation is part of the art of mathematical modeling. How these representative mathematical models were developed will be discussed in detail in the appropriate section. In Chap. 2, a far-field wave equation is formulated. The formulation of this far-field wave equation for both outgoing and incoming waves is discussed in detail. However, attention in this book is focused on the one-way outgoing waves. This far-field wave equation is the root for developing the outgoing wave equation which is discussed at length in this book. Due to the lack of efficient and capable three-dimensional models, three-dimensional wave propagation problems have typically been solved by available two-dimensional models ignoring the azimuthal azimutha! coupling effects. Results produced by these two-dimensional models are considered acceptable by virtue of weak azimuthal coupling. That is, a three-dimensional sector is divided into N subsectors, each sector at a fixed azimuth angle in which a propagation problem is solved by two-dimensional models, so that the entire problem is solved by N applications of two-dimensional models. This treatment is known as "Nx 2D". Therefore, a distinction must be made among the concepts of "3D", u"iVx Nx 2D" and "2D", each of which is discussed in detail mathematically later in this book. These concepts are also discussed carefully both computationally and physically. These discussions point out the three-dimensional effects in situations where they cannot be ignored, thereby stimulating the need for a model to process three-dimensional wave propagation. These types of considerations motivated us to consider the development of a true three-dimensional

Introductionj n

5

model. Not only did we develop new representative wave equations with required physical capabilities, but we also developed new efficient numerical schemes to solve them. To show the efficiency of our new model, we include one of our early developments, a pseudopartial differential equation which was numerically solved by a preconditioner technique. This early model was, at that time, an accurate development. As it is an iterative scheme, acceleration of its speed of computation required research in the fields of both numerical mathematics and computer science. Our early development is presented in Chap. 3. This chapter gives an in-depth development of a representative pseudopartial differential wave equation that was originally formulated by Siegmann et a/..22 A successful application of an implicit finite difference scheme for two-dimensional problems was extended to solve the new three-dimensional wave equation using the principle that the marching procedure solved a system of equations at every range step. Due to the incorporation of azimuthal coupling effects, the resulting matrix system is not tridiagonal but is instead a sparse seven-diagonal matrix system. This sparse system requires special treatment, one of which is the application of sparse matrix techniques that apply using a preconditioning technique. As our resulting system has no special desirable properties, such as being positive-definite or Hermitian, the iterative preconditioning technique suffers from a slow computation speed, even though the results are accurate. The computation speed could be improved if better preconditioning techniques could be developed. This remains an area that needs increased research in numerical methods in order to improve the efficiency, and hopefully Chap. 3 will stimulate such research. Other than attempting to improve the computation speed of preconditioning techniques, we took a completely different approach, namely, to formulate a new representative wave equation. The details of this development are presented in Chap. 4. This high-order wave equation is derived from the same representative one-way outgoing wave equation discussed in Chap. 3. However, this new equation is second-order in azimuth and fourth-order in depth, which happens to make it much easier to handle. The formulation of the high-order wave equation begins with its dependent variable expressed in terms of matrix exponential form by means of an ordinary differential equation method. The matrix exponentials have differential operators that require an efficient treatment. Our approach is to apply a rational function approximation for these matrix exponentials, which leads to simple first-order rational function expressions that are easy

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Numerical Ocean Acoustic Propagation in Three

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to handle. An important feature in our development is that the rational function operators have the property of being unitary. Thus, for an explicit finite difference marching procedure, the scheme is unconditionally stable. An additional important feature of this finite difference solution is that at most only two tridiagonal systems need to be solved. Another advantage is that the second tridiagonal system controls the azimuthal coupling, in both computational and physical senses. A computer code, FOR3D, was developed to implement the finite difference solution just described. As the development fully incorporates azimuthal coupling, this feature is handled efficiently in the FOR3D code. To date, this model is only one of few in existence from which the user can call for azimuthal coupling. If azimuthal coupling is absent, the code will produce this result automatically whether or not the user asks for it. How this important capability is handled efficiently will be discussed in detail in Chap. 4. The development thus far has created a useful basic model which can process long range, wide angle wave propagation in various three-dimensional ocean environments including shallow or deep water, range dependence, and azimuthal dependence. However, ocean environments are complex, with varying temperature, salinity, and pressure. These changes affect the sound speed and density structure of water masses, effectively creating a layered medium with interfaces. At each interface, the physical properties require that the pressure and normal component of particle velocity are continuous. An elegant numerical treatment of the interface conditions was first introduced by McDaniel and Lee 23 using a Taylor series expansion in conjunction with a finite difference technique. In their treatment, the wave equations in each medium do not involve density; a novel matching technique has to be introduced to satisfy the conditions. This treatment in effect formulates a special representative wave equation with density variations on the interface. This approach handles well our wave equations having the form of pseudopartial differential equations. Extension of this technique to handle high-order wave equations requires more effort, in the cases where a fourth (or higher) derivative in the depth variable in included. For this reason, we chose to handle interface conditions for our higher order equation by a different approach, i.e. by formulating the representative wave equation with density variations. The theoretical development, along with our special numerical treatment, are included in Chap. 5. This particular portion of enhancement is the development of capability to handle density and sound speed changes across the interface boundary and

Introduction)n

7

the theoretical and numerical treatments are justified mathematically. This capability was added to the computer code, thus increasing the FOR3D performance capability. When any code has all the required capabilities, it is obviously desirable that the code computes at a fast speed on any computer. It is a common belief that using newer and faster computers is the way to improve computation speed; however, it is not necessary to rely only on hardware to gain computation speed. In addition to using faster computers, the computation speed can often be improved by (1) formulating a better mathematical model and (2) finding a more efficient numerical solution. An example of (1) is our formulation of a high-order wave equation, as described in Chap. 5. An example of (2) is our approach in which high-order finite difference schemes are developed. Clearly there are a number of other approaches in this category. The new scheme presented in this book is the only one which happens to be easy to implement. In the case where the matrices are symmetric, the recursive formula of the tridiagonal solver is further specialized to gain computation speed; in fact a roughly 40% improvement is achieved. Thus, this high-order numerical scheme offers a distinct advantage, so that long range, large scale problems which require extensive and lengthy computations can sometimes be solved even by conventional computers. If supercomputers are used, a fast computational speed can be expected. The detailed development is also included in Chap. 5. Validity is an extremely important factor in any modeling development. For the class of large-scale and complicated problems, it is usually very difficult to construct an analytic solution for the examination of accuracy as well as capabilities. One of our research results was the construction of an analytic solution that satisfied our representative wide angle wave equation and reasonable prescribed boundary conditions. This analytic solution is suitable to simulate environments of interest so that in some regions the three-dimensional effects are evident, while in other regions they are not. This is ideal for testing not only the computational determination of three-dimensional effects, but also the accuracy of the results. The testing is covered as a vital portion of Chap. 6. It begins with a discussion of the advantages of the analytic solution, and then continues with a check of computational results against the known exact solution for digit-by-digit accuracy. For users of the code with the aim of research or application, it is advised that after implementation the accuracy of the code should be checked. To do so, the analytic solution is recommended. After the accuracy of the FOR3D model is established, the occurrence of

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three-dimensional effects is briefly identified. A special sequence of tests was designed to examine FOR3D's F0R3D's capability to handle three-dimensional effects correctly. We use the test results to demonstrate correct treatment of true three-dimensional effects by comparing results with Nx 2D cases where the differences become quite evident. After the computer code's accuracy and validity were established, we use it to process realistic ocean input data. A set of such realistic data was supplied by scientists at the Center of Earth and Planetary Physics of Harvard University. These data are used to examine the horizontal energy distribution, i.e. the occurrence of three-dimensional effects, the cost of large-scale computations, and practical esitmates for the solution of bigger and more general problems. Since a tremendous amount of effort was required to prepare and process the Harvard data, findings are reported in Chap. 7. This chapter describes the first time that an interface was produced between an ocean prediction model and an acoustic propagation model. The ocean prediction model is an early version of the Harvard Open Ocean Model (HOOM) that employs six vertical levels. The entire ocean region contains information about the Gulf stream, fronts, and eddies, ranging from 33°N to 41°N in latitude and from 60° W to 72° W in longitude. Many source locations and propagation directions were selected to exercise the model's performance and to examine three-dimensional effects. The results present acoustic propagation in different cases involving the Gulf stream, fronts, and warm and cold eddies, including both three-dimensional and two-dimensional model computations. These results clearly indicate the existence and occurrence of three-dimensional effects, thus suggesting the use of efficient three-dimensional models to perform the computations. Of course, this was the main motivation behind our development of the three-dimensional wide angle wave model. The last chapter, Chap. 8, describes the Computer Model — FOR3D. In addition to including the computer code, we discuss what this model can do for users, i.e. its advantages and capabilities. A guide is also provided to assist users of this code and to minimize their difficulties. In short, this book presents a step-by-step development of a new representative wave equation for predicting one-way outgoing acoustic wave propagation in three-dimensional ocean. The development of an efficient numerical scheme effective for solving this particular wave equation is an art. The creation of the research computer code is a contribution available for many applications in the real world. It is believed that high quality, three-dimensional experimental data exist which need a high

Introduction

9

quality, accurate model to verify the measurements. Due to the accuracy of the LSS model, it should be useful in examining high quality measured data. Experimentalists often talk about the GIGO (Garbage In, Garbage Out) principle when examining the actual measurements using available models. No matter which propagation model is applied to examine the high quality experimental input data, the objective is that it should never contribute to the degradation of the accuracy of the results. The LSS model fits into the select category of models capable of handling high quality data, because it was developed to produce accurate results. Of course, disagreements between measured data and numerical results do often exist. In such a case, the user must examine both the model's capabilities as well as the quality of the experimental data. Within the framework of the LSS model's capability, the user can apply it with confidence. Variety of research topics are raised in this book in the areas of numerical analysis and computer applications in order to stimulate research problems in this and related areas. To gain some background for understanding this book, readers are advised to consult a description of a two-dimensional finite difference model, published in the book Ocean Acoustic Propagation by Finite Difference Methods authored by Ding Lee and Suzanne T. McDaniel. 24 References 1. L. Brekhovskikh and Yu. Lysanov, Fundamental of Ocean Acoustic (Springer, Berlin, 1982). 2. R. L. Urick, Sound Propagation in the Sea (Peninsula Pub., California, 1982). 3. C. B. Officer, Introduction to the Theory of Sound Transmission, (McGrawHill, New York, 1958). 4. E. C. Shang and T. S. Wang, Theory of Underwater Sound (Science Pub., China, 1981). 5. A. D. Pierce, Acoustics — An Introduction to Its Physical Principles and Applications (McGraw-Hill, New York, 1981). 6. I. Tolstoy and C. S. Clay, Ocean Acoustics — Theory and Experiment in Underwater Sound American Inst of Physics, New York (1987). 7. M. H. Schultz and D. Lee, Computational Ocean Acoustics (Pergamon, Oxford, 1985). 8. A. R. Robinson and L. J. Walstad, "The Harvard Open Ocean Model: Calibration and application to dynamic process, forecasting, and data assimilation studies," J. Appl. Numer. Math. 3 (1987) 89-132. 9. M. H. Schultz, D. Lee, and K. R. Jackson, "Application of the Yale sparse technique to solve the three-dimensional parabolic wave equation," in Recent Progress in the Development and Application of the Parabolic Equation, eds. P. D. Scully-Power and D. Lee, Naval Underwater Systems Center, TD #7145 (1984).

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Numerical Ocean Acoustic Propagation in Three Dimensions

10. R. N. Baer, "Propagation through a three-dimensional eddy including effects on an array," J. Acoust. Soc. Am. 69 (1981) 70-75. 11. F. D. Tappert, "The parabolic equation method," in Wave Propagation and Underwater Acoustics, Lecture Notes in Physics, Vol. 70, eds. J. B. Keller and J. S. Papadakis (Springer, Berlin, 1977). 12. J. S. Perkins and R. N. Baer, "An approximation to the three-dimensional parabolic equation method for acoustic propagation," J. Acoust. Soc. Am. 72 (1982) 515-522. 13. A. Bayliss, C. I. Godlstein, and E. Turkel, "An interactive method for the Helmholtz equation," J. Comp. Phys. 49 (1983) 443. 14. Ibid, "The numerical solution of the Helmholtz solution for wave propagation problems in underwater acoustics," in Computational Ocean Acoustics, eds. M. H. Schultz and D. Lee (Pergamon, Oxford, 1983). 15. C. I. Goldstein, "Multigrid preconditioners applied to three-dimensional parabolic equation type models," in Computational Acoustics — Wave Propagation, eds. D. Lee, R. L. Sternberg, and M. H. Schultz (North-Holland, Amsterdam, 1986) 57-74. 16. D. Lee, Y. Saad, and M. H. Schultz, "An efficient method for solving the three-dimensional wide angle wave equation," in Computation Acoustics Wave Propagation, eds. D. Lee, R. L. Sternberg, and M. H. Schultz (NorthHolland, Amsterdam, 1986) 75-90. 17. G. Botseas, D. Lee, and D. King, "FOR3D: A computer model for solving the LSS three-dimensional wide angle wave equation," Naval Underwater Systems Center, TR #7943 (1987). 18. A. E. Newhall, J. F. Lynch, C. S. Chiu, and J. R. Daughterty, "Improvements in three-dimensional ray tracing codes underwater acoustics," Computational Acoustics — Ocean Acoustic Models and Supercomputing (North Holland, Amsterdam, 1990) 169-186. 19. C. S. Chiu and L. L. Ehret, "Computing of sound propagation in a three-dimensional varying ocean: A coupled normal mode approach," in Computational Acoustic ^— Ocean Acoustic Models Supercomputing (North Holland, Amsterdam, 1990) 187-202. 20. J. F. Lynch and C. S. Chiu, The 3-D Acoustic Working Group Meeting, Long Beach, MS, July 7-8, 1988, Woods Hole Oceanographic Institution, TR No. WHOI-89-16 (1989). 21. D. Lee, A. Cakmak, and R. Vichnevetsky, Computational Acoustics — Ocean Acoustic Models and Supercomputing (North Holland, Amsterdam, 1990). 22. W. L.Siegmann, G. A. Kriegsmann, and D. Lee, "A wide angle threedimensional parabolic wave equation," J. Acoust. Soc. Am. 78 (1985) 659664. 23. S. T. McDaniel and D. Lee, "A finite-difference finite-difference treatment of interface conditions for the parabolic equation: The horizontal interface," J. Acoust. Soc. Am. 71 (1982) 855-859. 24. D. Lee and S. T. McDaniel, Ocean Acoustic Propagation by Finite Difference Methods (Pergamon, Oxford, 1988).

Chapter 2

Basic Mathematical Model Developments

Successful mathematical modeling leads to the development of a representative mathematical equation that contains as much of the desirable physics as possible. With this in mind, we have focused our development efforts on a three-dimensional, outgoing wave equation. The inclusion of all the desirable physical processes into the mathematical model is conceivable but this approach presents difficulties in finding an efficient solution. In contrast, our basic approach is to develop an accurate mathematical model that is not difficult to solve, even though the representative equation does not contain all the physics we would prefer. The expectation is to develop the mathematical model in such a manner that additional physical mechanisms can be incorporated in subsequent extensions by means of the boundary conditions or the ocean volume. As a starting point, we extend our two-dimensional modeling developments to three-dimensional situations. In many applications, it is appropriate to deal with a harmonic point source; therefore, the hyperbolic wave equation is reduced to the Helmholtz equation. The Helmholtz

11

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Numerical Ocean Acoustic Propagation in Three

Dimensions

equation is an elliptic equation that, with appropriate boundary conditions, constitutes a well-posed problem. Thus, a solution exists and a uniqueness is assured. This Helmholtz equation incorporates both incoming and outgoing waves. One of the major problems remaining in ocean acoustic propagation, still far from being completely solved, is the capability to accurately predict three-dimensional propagation at low frequencies for very long range periods. For this purpose, a representative wave equation to predict one-way wave propagation would be valuable. We develop such a representative wave equation from the Helmholtz equation. As we have done in the two-dimensional case, 1 we develop a three-dimensional representative wave equation which contains the desired physics and which is easy to solve using specifically-designed numerical schemes. To be consistent with, and as an extension of, our two-dimensional wave equation, we use cylindrical coordinates. The three variables are depth z, range r and azimuth variable 6. In the case of harmonic point source with two-way propagation, the following problem is defined. Problem 2.1. The three-dimensional Helmholtz equation in cylindrical coordinates is d22 d2± 2

dr

ld(j) ld(f> \d + r or

d22 4> 2

dz

■+

22

(j) 4> i1 day 2

r r d0 ' 2 2

2



It is now possible to express Eq. (2.17) in two different operator forms. The first is d_ + ik ik000-- ik ik000Vl \/l \/l ++ XX + —+ ik00Vl (— — +ik ikoVl + YY JjJ (( — — + ik ik00 + ik Vl + +X X + YY jJ u dr

dr

d_ d_ /1 ++ XX ++ YY jj= ik - ^ -- ^Vl = (vi Vl -^ -^Vl + +X X+ + YY )\ uu ,, ik000 (((Vl dr

dr

(2.20) (2.20)

and the other is

ikoVl + X + Y )j ([g-+ +X +Y Ju (f jTT -g-- + ih ik0 + ifaVl — +ikoik0 - ik0Vl +X = ik0 (-^\/l (-^Vl+X + X + Y - Vl Vl+X + X + Y ^-^\\ u .

(2.21)

Note that Eq. (2.2) is identical to either Eq. (2.20) or Eq. (2.21). Suppose now that Vl Vl + X + Y ^ = = -?-Vl ^-Vl X + Y Yu. liy/l + X u . or or or or

(2.22)

Basic Mathematical Model Developments 5

17

If Eq. (2.22) holds, it follows that both d

a + ik0-0 - ikoVl ((—+ik — ikoVl+X ik ikoy/l iko j )u ==0 0 +X + YY J) (( — — ++ ik ik00 ++ikoy/1 ik0\/l +XX++YY )u 0Vl + (, (2.23a)

and

(-g-+ik (g-+ik (i

00

d + ikoVl+X X + Y Jj (—+ik ((— — + +0ik -ikoX + Y )u ik00Vl+X VTTX~+Y j u == 00 0 - ik

(2.23b) Thus, the factorization of (2.17) becomes much simpler if (2.22) holds exactly, or holds to good approximation. Equation (2.22) will hold approximately provided some conditions are satisfied. In order to justify these conditions, we will discuss two important properties. We recall that Eq. (2.2) is obtained from Eq. (2.10) by means of a far-field approximation applied to the solution v(r). Specifically, the far-field approximation, k0r > 1, enables the solution v(r) of Eq. (2.9) to be expressed by Eq. (2.12). If we denote the leading term in the far-field approximation of v(r) by v(r), i.e.

fJZ

r v(r) = \l-4- te» ,

^ - ^V rnk0^r " V nkor

'

(2.24) (2.24)

then a measure of the relative error between v(r) and v(r) is E{v) = \v(r) \v{r) - v(r)\ l^^(rr))!!""11 . E(v) =

(2.25)

In a computation employing our approximation equations, suppose we require (2.26) E(v) < 6 , where 6 is a specified positive tolerance. The tolerance condition focuses on the difference in modulus, rather than the difference in phase which is of little interest. It is known 5 that

w r ) i~ HKr)| i c ( r[i - i j g j ^ 2++o+o WDI o ( J J ^ ) ] . kor fkor- — »>. oo . 16(fc r)

W'>i~i*4-ii(5^ ((sb0]~ 4 "i^ ((^r)]' ^°°U M V J ' *»-'»• 0

(22r) (2.27) P.")

The terms inside the square brackets of (2.27) alternate in sign, and they have the property that the remainder after retaining any number of terms is no bigger than the first term neglected. From condition (2.26) and estimate (2.27), it follows that v(r) is regarded as an acceptable approximation of v(r) if

kkor>^=. 0or r> JL* .

* ^47T

(2.28)

18

Numerical Ocean Acoustic Propagation in Three

Dimensions

Property 1. The solution field v(r) of Eq. (2.12) is said to have Property 1 where condition (2.28) is satisfied for an arbitrary assigned 8. In terms of the acoustic frequency / , condition (2.28) requires the range r to satisfy . Co °* 7= _ .. (2.29) rr>r >f=rf = 8nfV6 Inequality is obtained by substituting ko = 2irf/co into condition (2.28), where r? is the minimum range for the far-field approximation to apply. As an example, suppose we give 8 = 0.01, which corresponds to differences between v(r) and v(r) being bounded by 1%. Then, as / increases from, say, 10 to 200 Hz, rt decreases from 60 to 3 m. An alternative mathematical expression for this example is kor > 2.5. By this we mean establishing a lower bound for kor in order that the far-field approximation is valid. This assists in approximately justifying Eqs. (2.23a) or (2.23b). Now let us return to the description of a second property. Since in this chapter we have focused on the basic mathematical development of the simplest parabolic approximation, the narrow angle equation (2.16), we continue to do so here. For simplicity, we write X ++ YY . . Z=X

(2.30)

An expression which represents y/1 + Z consistently with the narrow angle approximation is the linear approximation VT+Z

z

^S 11 + \^ zZ + 0 ( Z 2 ) . Li

(2.31)

As indicated, the right-hand side of (2.31) gives an approximation which is correct to first-order in Z. We recall that (2.22) does not hold exactly in the general case, but it holds approximately, then the factorization approximation Eq. (2.23a) or (2.23b) will be justifiable. That is, we must investigate the terms appearing on the right-hand sides of Eq. (2.20) and (2.21). Using (2.30) and defining an operator E, these terms are

d ^dr

ik0Q (-^VTTZ (Q-VTTZ

-z -

EU. VlTZ-^) = Eu. VTTZ—jg-r)u u = Eu. )'

(2.32)

Basic Mathematical Model Developments

19

Note that the right-hand sides of Eqs. (2.20) and (2.21) differ only by a minus sign. In view of Eq. (2.31), it follow from Eq. (2.32) that

(233

*-*(£*-*!)•• Eu

=

>

(2.33)

-i{d-rZ-Zd-r)U-

Using the definitions of the operators X, Y, and Z in Eqs. (2.18), (2.19), and (2.30), we find the leading term E has the form

v

■, < Q ,dn(r,9,z) dr

i

d2

knr* dd*

(2.34)

By arguments that are analogous to, but lengthier than, those used to derive condition (2.28), Siegmann et a/.6 showed that Eq. (2.34) is negligible compared with other terms retained in Eq. (2.17) provided

and

dn< r r 2 (r,0,2) -- 1\) d r i = (ri,ri)/(pi,pi)Pi) , = Xi + + CliPi CliPi ,, #1+1 =Xi

r%+i =ri-; — adiApi , bbi{ = (r {ri+u ri)l{rUTi) i+uri)/(ri,ri)

Ti)

,

hpi , 6iPi Pi+i = ^.*n-hi A*ri+i 4+ biPi where r0 = f — Ax0, Xo is chosen arbitrarily, and po = A*r 0 . The above loop is repeated starting with i = 0 until convergence. The work per loop requires 5N multiplications plus two matrix vector products. Only 4AT storage locations are required for the vectors a?», r^, r», Pi, pi, and Ap{. In summary, when attempting to solve (3.33), we apply the preconditioning technique to conceptually transform system (3.33) into system (3.35). Then, Craig's method is used to solve system (3.35). It is natural to ask about the need for explicit computation of A* A. However, as expressed by (3.35), A* A need not be carried out explicitly, and this is an advantage of Craig's method. As we have noticed in the literature, 12 there exist two different types of three-dimensional wave equations as a result of the PE approximation. One is the three-dimensional parabolic wave equation (the standard 3DPE) originally used by Tappert 1 to derive the standard two-dimensional PE. Solution to the 3DPE has been developed by Baer and Perkins 2,3 using the split-step algorithm. The second type is the 3D wide angle partial differential equation, developed by Siegmann, Kriegsmann, and Lee in Sec. 1. (We refer to this equation as the 3D wide angle PE.) We chose to concentrate on the solution to the 3D wide angle PE because the standard 3DPE is a special case. We want to remark why we are motivated to solve the 3D wide angle PE instead of the 3DPE; in particular the application of the Yale sparse technique. In this event, the vertical angle of propagation is roughly larger than 15°, due to the irregular nonzero boundary conditions, or other environmental properties where the Fast Fourier transform (FFT) is not easily applicable (this is why a general purpose solution is needed). Now, consider the 3D wide angle PE in a general form, i.e.

d

d_ - u = dr

I

11+Pi +Pl n2(r

(-—iko .*u ++^„,ikoo - - (,

'^-

V

{

\V

1 +

^ ). 1 & i

a2 > ° (fc 0 r) dfl 2 uu ,, r ~ W l & +9a q2+q2

+ p2

i

wl^ 2

1l ++ qigi nirAz) 1 n{rAz) + 1+ {-r A z ) { n 21+9i rfde^j 7 - l + Kr\ ¥ k\&)" 0d^) (k0{k^Wj (3.36) (3.36)

- j^w)

jk^wJ

A Pseudopartial

33

Differential Equation

where n(r,8,z) is the index of refraction andfcois the reference wave number. Note that when p1 = p2 = 1/2 and q\ gi = = q2 = 0, Eq. (3.36) reduces 1 exactly to the 3DPE. Using the split-step algorithm to solve Eq. (3.36) is not easily applicable. One can easily see that alternate general purpose technique is needed to solve Eq. (3.36). One approach that was considered was to multiply both sides of Eq. (3.36) by the operator in the denominator of the right-hand side of Eq. (3.36). The following was obtained:

I",

1 d\ l.JL\ 2

( 2, a ^ -,

2 h(n (r,0,z)1 + 01

+

kldz>)

92

2 1 a d2 1] dd_ u

ary 89* dr

2 2 2 = iko[(Pi-qi)([n (rAz)-l} + ^^ ^ )j = t*b [(Pi ~ ft) ([* (r,M) - I] 2 +

,

d2 d]

1

+ {p

*-q2)(k^Wd-r\U-

(3.37) (3 37)

u .

'

Equation (3.37) is not a PE, but a third-order partial differential equation known as the pseudopartial differential equation. (A reminder to the reader here is that Eq. (3.36) is called the 3D wide angle PE because the 3DPE is a special case and the terminology is a very familiar term.) In solving Eq. (3.37), St. Mary and Lee 13 attempted to seek a finite difference solution. Since their analyses indicated a too restrictive stability condition, we attempted a similar implicit finite difference scheme as was used for the 2D wide angle PE because of its favorable unconditional stability. The solution by means of an iterative technique is the main topic of this section, and its efficient solution by means of Craig's technique will be the main result. To deal with the solution of Eq. (3.36), we must first discuss how to convert this evaluation into finite difference form in order to apply Craig's method. Before this formulation, we have a few definitions to state. Let m indicate the discretization index in the z-direction and Az = h indicate the z-increment. Similarly, £ is used to indicate the index in the ^-direction; A0 is the ^-increment; k is used to indicate the range step Ar. When the integer n is used as a superscript, it indicates the range level. Also, for brevity, use the definitions X = n2(r,e,z)-l ) - l - + ^ - ^

kldz*

and

Y Y Y

d2

l

= T^&2

~ d62 ' = (k (&?&■ 0r)

(2.18)

19) (2.19) (2(219) --

34

Numerical Ocean Acoustic Propagation in Three

Dimensions

Then (3.36) can be expressed in a short expression using the above definitions, i.e. d_ ( U dr -{ V dr Write

1+PiX 1+PlX+P2Y + paY^ it.lit

lko lklk lk0 +

)

(3.38)

U —T + °l°l \ ++qiqiXX-— q2Y) ++q,Y)

(/ .. , . Cu Cu== tko+lko iko+

., l+ l+PlPlXX + pp2Y\ 2Y\ X + ^l ++ qiqi X + q2q2Y)Y)UU--



so that Eq. (3.38) (which is (3.36) using (2.18) and (2.19)) can be written in a short operator form, i.e. d . —uu = = £u Cu . — or or

(3.40)

Numerical solution to Eq. (3.40) can be expressed as Numerical solution to Eq. (3.40) can be expressed as n+1 un+l =ekC un , kC n u

=

e

u

(3.41) )

(3

?

41

where k = Ar. Using a half-half splitting of exponential and setting up the solution to Eq. (3.41) by the implicit finite difference Crank-Nicolson scheme, we find an implicit finite difference discretization to Eq. (3.36) as

n+l (i hc\: uun n.. (l - hcj hzC\: u u n + 1 == ((1 l ++\kc\ )

(3.42)

)

Using the definition of £, X and y , Eq. (3.42) becomes

^ [l [l ++ + ft ft(„>(r,*,*)-l (nHr,6,z)-l (f{rAz) - 1++ ^ ^ ))) + +92J^J^^\ 9 2 ^ ^ ^ ] «> In *+n+11 ft 2

- ^ikokUp! -ifcofcUpx ^ikokUpi. -qi) -qi (n)2(r,0,z) (n (r,0,z) - 1 + -pg-^j 1+ p^j) ^ikokUpi-qi)

+iP2 + q2) q2))q + qp2

1

r) 1

r) 1 un+1 un+1 un+1 un+1

^^-(- *-wrTwm WrTwm wrwm * WTWw\ )

2,

= [i +ft(„»M,*)-i+ li ^a ) + f c ^ ^ ] « 22 ik k k n 2(r,0,z)-l + k\(pi-qi) + -ik (Pi-«i) (r,0,z)-l+ + 22ikoQ° (Pi ~ 9 i ) (\n (r'6''z)~1

++{P2 +

un

^-^wM q2) q2) iP2

un ~ ~kj^w\ ^d¥\- un-

+ ^2^2 ^ ^ tf-Qj,J 2 "g^-^j 2 )

+

(3(3.43) 43)

-

A Pseudopartial

Differential Equation

Using central differences for both operators -^ 68 = A0, we simplify to obtain l"i ii. l"i I"i

// 22

1 + 9l(n

l

+

qi) +

ipi 2

%h jfco/i

11

11

-

*«« 77 77/ 7 //

0

1

* * /

(ft ft)

11

2

2

N \

1 \ -n+l n+1

» fc (, _ , 1 11\\ {P2 q2 2 {P2 '(r (r + 22 ifc '(r ++ k) ifcofc0 ^0 **' k? 8* P ))

11

nn+ +i Um Ume+1 e+1 Um e+1

'''

N

i k ,,

2

U ( r + *) 11 11 \\ ,

(P2 ?2 2 U +1 u U[¥ ^ 22^0{P2 ^ U ff0^P ^ ++ 2T ^~q2)""W) « W^ jJUm -.^ 'e+1

(Q2 / (t foe 1ll l 1

i fc k ,

x

l1 1 \\

_

+ f e ( ,P 2 9)2 ) + ((|^^ ^ + 2 +^2^ - -* H^^j P + qi(n u

2qi X

2g2 1 *^ -1)-^---^—^-)

k$ h*

-il-k k(p! -qi)(n -qy 2 2 - 1 ) -il-hokip! 02*o*(Pi - Mil-kokip! -tfi)(rc - M 2*o*(Pi -tfiX™ 2 - 1 )

fc{Pl1 qi)

{P2iP2q2) q2) -hh0h{pi--qi)-V -hvh? - ~)' )' -V J7TW 0 n0 = 9i 1 .1 p*i , 9 i ), 1 n ^-W 9i 1 *2k,vh?> 1 fcg^¥.12fco" -h?2k- -- - ^' h?> ¥ -h?0^ ^=_9 (7TW P+ ~ kj L w « 2 ,x 1\ ^ = ( l + 9i(n - l ) - ^ - - - - ^2 - - _ j

'

+ + ii\ikf --knk(v-i A ;( 0pf ci ( p—-ig-ai)(n gi )i )( (nn22-- l ) W

_ _ D+

R

*fc11

^^

( p( pi i_ 99ll))

ib:^

gga2 1 1

= ¥

0

^

-

fell i

l l

A

( (pP22 gg22 )) A

- f c : ^ ^ -- J'' -^^^

.11 * , 1 1 + 2k l iP2 q2) 0 2T0 - w-

(3.45)

Wee can can see see Eq. H/q. (3.45) ^o.^o; in in aa simpler simpler form, iorm, i.e. i.e. RP < / P> +! !TP+J > + ! oo oo oo ff ll + + o o o o o o ff ll + + o o o o o o ll [[ uu ii ,, !! ]] 77 11 p+,_ oo P + ! oo oo oo p fl+ , + oo oo oo ff ll + +o o oo ii ll U21 P+! op,+ U2fl U2fl 0 0

P+ 2 0 0 P+j 0

fl+ fl+ o0 + R+ 0 R 0

0

0 P+x

0

0O fi li ++ 00

u3tl

0

+

0

0

0

fl+

n44, i U

0o

0o

o ii ll

uu11)2 )2

0o

o i I l I tt u22,2 ,2 O

0

0

R+ R

+

0o R R+ + °0

/*+ 0o R+

R+

0

0

0

0

0

0

fl+

0

0

0

O f l + 0 0 00 0 f fl+ O l + 00

0

0 f fl+ l + 00

P ++22 0° 0° o0 P 0o P p+ 0 °0 2+2

o0 0

fl+

L 0

0

0

P+2 0 P+2

0

0

0 P+2

+ R+

0

0

+ 0O f Rl + 0 0 00 00 R+ R+

O Jf ll + 0

0

0

R+ 0 0

I

Q*«0,2

_i_ I

°

u 3 ti 2

//"} o 47^ A7\

0

u4)2

Q*«5,2 Q*«0,3

0 P 22 + 33 0 0 00 00 P+3 P+3 00

"2,3 t*2,3 "3,3 w3,3

0

fl+

o Q*t*5,l

ui,3

0 R+ 0

0 0

o i l

I

0

0 P+3 0 00

0

n ("Q*«o,il ("Q*«o,iln °°

II

0 P P+^J ^ J LU4'3 J

° 00

II

L^* U 5 ' 3 J

We can now extend the illustration above to the case of arbitrary M and L. In general, the large sparse system to be solved is in the form Aun+1

= Bun + v%+1 + + uS ,

(3.48)

where UQ+1 contains surface and bottom boundary information at the advanced range level and UQ contains surface and bottom boundary information at the present range level. The A and B matrices both possess the format [Ti ii nn [Ti D D 00 .... .. oo oo D T D ... 0 0 0 2 D T2 D ... 0 0 0 00 D 00 00 00 D T T33 .. .. .. (3.49) • . 0 0 Li?

0 0 0

0 0 0

... ... ...

Tn_2 D 0

D T n _i D

0 D Tn\

All the block matrices (T, D and R) are of the same order M x M. Each T matrix is tridiagonal, whereas both off-diagonal block matrices D and R are diagonal matrices. The entire A and B matrices are 7-diagonal with the symmetry property A = AT and B — BT. The right-hand side of Eq. (3.48) can be carried out by one matrix vector operation and two vector additions. Equation (3.48) is a large, sparse system, which we want to solve by taking advantage of the Yale sparse technique, Craig's method. Note that when we consider wave propagation around a complete 360°, then we are dealing with a system where A and B are of the form (3.49), i.e. a 7-diagonal matrix. If we consider that the wave propagates only in an

A Pseudopartial

Differential Equation

39

azimuthal sector, then the periodic boundary condition for the azimuthal plane is absent, and we then solve system (3.48) where A and B are in the following simpler form: rTj D 0 ... [Ti J) \ D T \D T22 D ... ... ... 0 D T3 ... 0 0 L0

0 0 0

0 0 0

0 0 0 0

0 0 0 0

] 01 0 0 0

: • . . . Tn_2 D 0 . . . D T n _ i! D D ... 0 D Tn\

(3.50) (3.50)

These now have the form of 5-diagonal matrices. Further, if we consider wave propagation only in a vertical plane, a two-dimensional case results. Then we deal with the system (3.48) where A and B are tridiagonal matrices, i.e. of the form ["Ti TTi 0 0

o0 o0

LO Lo

0 0 ... T2 0 . . . 0 T33 .... .. :

0 0 0

0 0 0 0 ... ...

0 0

0 0 0

Rl 0 0

0 .. T 0 0 o 0 o .... r„_ o o n _22 0 0 .. 0 _i o 0 o o .... o T r nn_i

.

(3.51) (3.51)

0 0 T Tnn\

It is important to note that when p\ px = p2 = 1/2 and q1 = q2 = 0, the system (3.46) reduces to the 3DPE. The Yale sparse technique (Craig's method) was implemented on a VAX 11/780 computer to solve Eq. (3.38) using the system of equations expressed by Eq. (3.46). We used a known exact solution as a check for accuracy of the implementation. To describe the exact solution, we use the formulation of Eq. (3.37). We seek a solution to Eq. (3.37) in the form im6 u(r, 6, z) = sm{Slz)eirne <j>{0) (j)(6) ..

(3.52)

Substituting Eq. (3.52) into Eq. (3.37), we find

[[ll++ g i ( [ n ftV([„» A zM) -,*)-l]-^)-^]«r l]-J)-^]Wr

*L*L- ak^)-i]4)-^ik. k^)-i]4)-^il.

([nV,)-l]4)-^^. = =* L - ^ g ft) L

V

«o /

*o

r j

(3 53)

40

Numerical Ocean Acoustic Propagation in Three

Dimensions

We select n2{r,0, z) — 1 — p - = 0 for computational simplicity. Since k = kon(r, 0, z) = u/c, then it follows that rr

k03=—

(-)

9 2

-ft

" -1I/2 I1/2

2

(3.54)

.

Equation (3.53) can be simplified using the ko defined by Eq. (3.54) to give 2 2 2 d _fikfiko(p d fik /(k0r) /(k\ 0r)2\ x x _ 0(p 22 - q2)m 0(p 2-q2)m 2 2 2 /(r) Tr -~ ) *' ' ^* '' Tr = { ll-q - ?22mm/(k V0(r)V ) * = "~~llf{r)

(3.55) (3 55)

* ^

which is a first-order ordinary differential equation. The solution to Eq. (3.55) can readily be expressed in the form i f< T)dr f == Ae Ae^ The effort needed to find <j)(r) u(r u(ro,0,z) ,0,z) o

is the evaluation of Jf(r)dr. iTO iTn0m V(ro) = sin(ftz)e sm(nz)e {r)= =AeAet^ ^ . . (3.60)

A Pseudopartial

Differential

41

Equation

Table 3.1 describes the results. Each of the first rows indicates the completed values, and each of the second rows indicates the exact solution. The results are taken at the boundary between the third and fourth sectors at 108° and at a range of 50.4 m. Table 3.1. Results of narrow angle propagation.

/

Z (m)

LOSS (dB)

u(I)

3 3

30.00 30.00

12.636 12.636

(0.18834E+00 (0.18886E+00

-0.13793E+00) -0.13722E-I-00)

6 6

60.00 60.00

6.859 6.859

(0.36627E+00 (0.36729EH-00

-0.26824E+00) -0.26685E+00)

9 9

90.00 90.00

3.749 3.749

(0.52397E+00 (0.52541E+00

-0.38372E+00) -0.38174E+00)

12 12

120.00 120.00

1.841 1.841

(0.65270E+00 (0.65451E+00

-0.47800E4-00) -0.47553E+00)

15 15

150.00 150.00

0.688 0.688

(0.74537E+00 (0.74743E+00

-0.54587E+00) -0.54304E+00)

18 18

180.00 180.00

0.108 0.108

(0.79685E+00 (0.79906E+00

-0.58357E+00) -0.58055E-f00)

Imaginary

Real

Case 2: Wide Angle Propagation (pi = p2 = 3/4, ^i = q2 = 1/4) An evaluation of / f(r)dr gives the solution • rn(po-qo)

<j)(r) =Ae Ae~l (j>(r) =

%

2
61 )

Numerical results are presented in Table 3.2 in the same manner as in Case 1.

42

Numerical Ocean Acoustic Propagation in Three

Dimensions

Table; 3.2. Results of wide angle propagation.

J

Z (m)

LOSS (dB)

3 3

30.00 30.00

12.645 12.636

(0.22578E+00 (0.22613E-HK)

-0.58389E+00) -0.57994E+00)

6 6

60.00 60.00

6.868 6.859

(0.43904E+00 (0.43976E+00

-0.11362E+00) -0.11268E+00)

9 9

90.00 90.00

3.757 3.749

(0.62819E+00 (0.62909E+00

-0.16245E+00) -0.16134E+00)

12 12

120.00 120.00

1.852 1.841

(0.78225E-I-00 (0.78365E+00

-0.20236E+00) -0.20098E+00)

15 15

150.00 150.00

0.694

(0.89377E+00 (0.89492E+00

-0.23150E+00) -0.22952E+00)

18 18

180.00 180.00

0.117

(0.95544E+00 (0.95673E+00

-0.24602E+00)

0.688 0.108

u(I) Real

Imaginary

-0.24537E+00)

Before leaving this exact solution accuracy comparison, we briefly mention the relationship between the narrow angle and wide angle solutions, Eqs. (3.60) and (3.61). Specifically, we examine the behavior of the solution of Case 2 for large for. First, we consider the real part of the solution, i.e. X

_ m(p2 -Q2)p -Q2)* ffoT-rriy/q^ /fcor-m^/gi" \ in 2v^2 \k0r + my/(h \for my/q2~ )J

-^--{^[(-^-iw--) cos(rr) - coz(m{P2~q2)\( ™ ^ - - ^ ^ - ^ cos(z)-cosj ^ ^ ...J 2 ^ _ 2(fcor)2 (my/to 1 m2q2 \ 11 V KQTfor 2(for)2^'")\j *

•)]}•

(3.62)

Then, ignoring terms of O(l/(fcor) 3 ) in the argument of (3.62),

/ x

cos(x) = = COS cos COS(x)

frn(p2-q2)

—\ \ _ _ —

V ?

(4.2)

(4 2)

where u+(r,0,z) z). and u~(r,0,z) are initial conditions at the range r. This first term in the right side of Eq. (4.2) is the outgoing wave, and the second term is the incoming wave. As the standard parabolic approximation suppresses backscattering, the desired one-way solution for the outgoing wave equation is obtained from (4.2) by dropping the incoming wave term. This yields the local solution svl+x+Y Y u(r + Ar, 6, 0, z) = ee~6see6V1 +x+u(r, u(r, u(r, 9, 0, z) ,

(4.3)

where in this chapter the parameter 6 is ik0Ar . 6 = ikoAr

(4.4) (4.4)

50

Numerical Ocean Acoustic Propagation in Three

Dimensions

Note that solution (4.3) is, as expected a local solution of the one-way outgoing wave equation from Chap. 2, i.e. urr = ik u iko(-l 0(-l

+ + y/l Vl + +X X + +Y Y )u )u ..

(2.37)

The new approach taken by Lee-Saad-Schultz consists of approximating the term e 5 v / 1 + x + y in a convenient and accurate manner. The simplest way in which this can be done is to use a linear polynomial approximation, as described in Eq. (2.31), i.e. Vl + + iIpT +X + + Y £Sil1 + ( X + yY), ),

(4.5)

which yields the standard three-dimensional narrow angle PE (2.15). However, the resulting Eq. (2.15) accurately represents only narrow angle propagation. To accommodate wide angle propagation, LSS apply the high-order approximation + \Y■Y . Vl + X + + Y S 1 + lx \ ;x* - \x2X" + (4.6) 22 8o 22 Using Eq. (4.6) in the outgoing wave equation, Eq. (2.37), a different form is obtained: ^ 22 + + (4.7) (4.7) Ur = = ik^o[-l+(l \x i x- -^X i y Ml«. ) 1u . 0 [-1 + ( l ++ Wr This is the new Lee-Saad-Schultz (LSS) three-dimensional wide angle wave equation in operator form. Then the formula (4.3) for the local solution of Eq. (4.8) becomes s s 1+ x x2+ Y Ar,0,z) = = ee^e^^^-i^^^u^O.z) u(r + Ar,6,z) e ( i -s i MrAz)

Y

K(rAz) .

(4.8)

Assuming that the index of refraction n(r, 0, z) varies slowly with respect to 0, the operators X and Y are nearly commutative, and Eq. (4.7) itself yields the local solution 6 1+ x x s u(r + + Ar,9,z) Ar,0,s) = = ee - e^ V < i1 +-i*^e * -^u(r,e,z) * * V * t i ( r , M ) .•

(4.9) (4.9)

Note that Eqs. (4.8) and (4.9) are identical if and only if X and Y commute. In any application, the commutativity of X and Y can always be wellapproximated numerically by taking a small Ar. An important step in the formulation is how to treat the two matrix 2 exponentials es(1+2x~&xi* )) and e^/ 2 ^ y efficiently. Not only should they be easy and economical to compute, but also their approximations should yield stability in the marchingo solution. For convenience we write G(6,X)^e6^ix-ix2K X -

i*2)

(4.10)

A High-Order Wave Equation

51

Here the function G(8,X) and its approximations should be regarded as functions of the real variable X and 6 is an independent parameter. Applying a Taylor series expansion to (4.10) about X = 0 gives 22 2 G(6,X) -X++III ((£(££---000 X 0(X3)3).. G(6, X) == e* es [l[l++66-X XX ]]]+++0(X*) 0(X

(4.11) (4.11) (4.11)

At the same time, we seek a rational function approximation to G(£, G( \jW> IT&c> ^**

where p is a complex number to be determined and p is the complex conjugate of p. The purpose of Eq. (4.12) is to address the question of stability, as will be seen. A question is whether there exists a p such that the approximation Eq. (4.12) can be made. The resolution of this question becomes clear as we try to determine the p. We impose the condition that the right side of (4.11) is equal to the right side of (4.12), giving 6

2

*!_*W

1 2—l U

1 +f - X +\-

47

1 + pX

"TTW

(4-13)

From (4.13) it can easily be determined that p is p = yP=l Hence, we obtain

« 1 - ++ -\ . 4 4

(4.14)

1 G(S - + ii/*±+! ji )±XM G(6,X)=X) .

l + (J-f)-X"

A similar development for the term in Eq. (4.9) A similar development for the term in Eq. (4.9) H(6,Y) = e$Y H(6,Y)=e%Y leads to the approximation leads to the approximation

w> =TTfr> }±f, mY)

=

(4.15) (4.15)

(4.16) (4.16)

((4.17) 417)

with q = | . Hence, we obtain 1 + -Y H(6tY) = \ ± ^ . 1

4

r

(4.18) (4.18)

52

Numerical Ocean Acoustic Propagation in Three

Dimensions

Therefore, the final form of the local solution (4.9) to the LSS equation (4.7) takes the expression

*+*••■«> = {TT^W))x. (0) «'•»■ *> • )*'

Equation (4.19) can be rewritten as u(r + Ar, 0, z) = Lu(r, 0, z) ,

(4.20)

L = L^LxLy^Ly ,

(4.21)

where in which L = 1+

*

'1

(i

+

L y = 1 + 8-Y . 4'

6s

l)*'

(422) (4.22)

(4.23)

In Eqs. (4.21), (4.22), and (4.23), L~l* stands for the inverse of L* and L* is the adjoint of L r , where r is either X or Y. Equation (4.19) can formally be regarded as an explicit marching scheme. It can be easily seen that the two operators in the denominator of (4.19) are nonsingular, because 6 is purely imaginary and X and Y are both self-adjoint. We next analyze the accuracy and stability of this scheme and show how to discretize it and use it numerically. 4.1.1.1. Stability The operators X and Y defined earlier are self-adjoint and therefore their corresponding eigenvalues are real. Since the numerator and the denominator of each term between brackets in Eq. (4.19) are conjugates of each other, the norms of the bracketed term are both unity, i.e. these two rational function operators are unitary. This property ensures the unconditional stability of the marching scheme (4.19). After discretization, the eigenvalues of X and Y will remain real, provided boundary conditions are properly handled and the discretizations in both numerators and denominators are evaluated at the same range. Under these conditions, the scheme is stable. Note that the second part of scheme (4.20) corresponds to the usual Crank-Nicolson approximation applied here to the term e ^ / 2 ) y .

53

A High-Order Wave Equation

4.1.1.2. Accuracy To analyze the local error of the marching scheme (4.19), one must attempt to find an estimate of the difference between the operator X+Y e-6e6VT+X+Y

( 4 2 4 ) (4.24)

and the operator on the right side of (4.19). In the following, consider X and Y as two independent real variables. After calculation one finds that the second-order Taylor expansion of the operator (4.24) is e-6e6Vi+xW

+Y

=1: 1

+ t(X X 2

+ Y)

+ ^(S - 1)(X2 + 2XY + Y2) 8

+ 0(||(X,r)||3).

(4.25)

On the other hand, the second-order Taylor expansion of the operator of the right side of (4.19) is given by y

11 (i -f) X 11■\y [l +Hit-f)f)* (l-l)x)\l-±Y) U= [l + ^ + ^ - l■)i :)X^ 22 + ...] [i + | y + £y» +...] = l+6(X (X- + Y) + ^XY+6-(6-l)X* -1)X2 2

4

8

+ ^Y> + 8

.... (4.26)

The larger terms in the difference between (4.25) and (4.26) are e"
Y(Ly is also unitary, as is readily seen by forming LL which is found to be the identity operator. The question as to which of the various schemes is preferred is certainly worth further investigation but will not be pursued in this book. Let us denote by A the finite difference approximation of the operator Lx = I + (\ - {)X and by B that of LY = I + {Y. Both matrices are tridiagonal with the structure indicated below. Upper diagonal of A : Main diagonal of A :

1 £ 2 2^ (\ ( 44 - '4*\ ) A k$h V

'

(I

^ 2 (I 1 + Q - - J [n2(r,6,z) - 1] - ^ 2 [\ ~ 4 ) ' jfcg/i \ ,4 ' 4 4,

f

1

U

a.

*> ^---J , 4 .4

Lower diagonal of A :

_

Upper diagonal of B:

- ^4jfcg - 2L -

klh?

8_ 1

r

fi 1 1 6_ Main diagonal of B : 1 ++ -

1 , (A0)2 11

A g ) 22

' l\fc0V ((A0)

Lower diagonal of 2?:

,

5 1 1 4&§r (A0)

- | ^ - 2- ^ _ 2 .

When solving tridiagonal systems with the matrices A and 5 , it is of interest to know whether or not these matrices are diagonally dominant. While the matrix B can be shown to be always diagonally dominant, the situation is more complicated for A. In the simple case where n(r,0, z) = 1, the matrix is conditionally diagonally dominant (i.e. when h > 1/ko) in the sense that the modulus of the diagonal term is greater than or equal to the sum of the moduli of the off-diagonal terms in the same row. The more general case, when n is arbitrary, is not easy to analyze. The scheme (4.28) becomes j +1 uj+1 = A-*AB-*BuBv? . u'

(4.30)

Note that we evaluate the matrices A and B at mid-distance between u^1 and u\ i.e. at range r + Ar/2. This is in order to ensure that the operators A* and A, as well as B* and 5 , form two pairs of operators that are

56

Numerical Ocean Acoustic Propagation in Three

Dimensions

conjugates of each other. The choice will guarantee stability, as was seen in the previous section. To perform one step of (4.30), we must start bycomputing wj = BvP and solve the tridiagonal system £ V + 1 = wj . Then we compute v J + 1 = A"1A*u^+1

(4.31)

and solve the tridiagonal system

A*uj+X = Av^1

.

(4.32)

Thus, there are two multiplications of a tridiagonal system by a vector and two tridiagonal systems to solve at every step. Proposed Problem 4.1. A variation of the LSS wave equation can be achieved by considering different approximations of y/1 + X + Y . Show that if another approximation of y/1 + X + Y is made to include the XY term, then the resulting system is NOT tridiagonal. Construct the resulting system matrix. To find an economical solution is a research problem. 4.3. Model Appraisal It was seen in Chap. 3 that preconditioning technique used to solve the SKL pseudopartial differential equation is an iterative marching scheme. Using this model to predict long range propagation is time consuming and thus, impractical. Chan-Shen-Lee's introduction of the Du Ford-Prankel scheme improved the speed. However, this improvement is not considered to be satisfactory because this scheme is only applicable to the narrow angle standard PE and is not capable of handling the wide angle propagation. To improve the computation speed and to maintain the wide angle capability, the LSS model has the advantage. In this section we selected a simple test problem on which all three of these models are applicable. For the same accuracy requirement, we examine the computation speed and give an appraisal of the computer code of LSS implementation (FOR3D). Computations are made on the VAX-11/780 computer using single precision complex arithmetic. This test problem deals with low frequency propagation whose exact solution is known, t*(r, 0, z) = c - ° V

m

V ^ .

(4.33)

A High-Order Wave Equation

57

Expression (4.33) is used to generate the initial field. The surface condition is taken to be 2 u(r,0,z) = eiTneei"%F (4.34) and the bottom condition is taken to be ijoL

t*(r, 0, ^max) = e-nz™*eiTneei7&,'5137 .

(4.35)

The scalar Q in Eq. (4.33) is chosen to be 2k0 and the angular model number M is assumed to be 3. To obtain an accuracy of 10~ 2 , as in Ref. 2, the methods need to take a range step size of 0.001 m. The input parameters are listed in Table 4.1 for convenience. Table 4.1. Input parameters. Input parameters Source Initial range Source frequency Bottom depth Sound speed Reference sound speed Receiver depth Propagation sector Maximum range Depth increment Range step size Angular increment Surface condition Bottom condition Size of matrices A and B

Test problem 10 m 10 m 20 Hz 20 m 1500 m 1500m/s 5m

- 5 ° , 5° 10.5 m 0.2m 0.001m, 0.25 m

1° Dirichlet Dirichlet 1000

We tested two different step sizes, i.e. 0.001 and 0.25 m. The experiment with the first range step size is only done in order to permit comparison with Ref. 2. We should point out that such a small step size is necessary for the Du Ford-Prankel method of Chan-Shen-Lee because the scheme is explicit. In this test computation, we found that the FOR3D method was approximately 1.6 times faster than the Du Ford-Frankel explicit method in Ref. 2, and 17 times faster than the preconditioning solution, described in Sec. 2. Note that as discussed therein, the preconditioning technique

58

Numerical Ocean Acoustic Propagation in Three

Dimensions

uses a stable version of the conjugate gradient method applied to the normal equations. Using the second step size of 0.5 m, we found that the same accuracy could be achieved by FOR3D as with Ar = 0.001, but the execution was much faster. Here our method is approximately 160 times faster than the explicit five-point method and 1600 times faster than the preconditioner. The results are displayed in Table 4.2. Table 4.2. Results of comparison.

Method

Ar

SKL (precond.) Explicit LSS (FOR3D) LSS (FOR3D)

0.001 0.001 0.001 0.25

Relative error (0.18E-01, (0.10E-01, (0.26E-02, (0.22E-02,

-0.12E-01) -0.11E-01) -0.11E-02) -0.11E-01)

CPU time h-m-s 03-47-10 00-21-35 00-13-12 00-00-09

Since obtaining solutions to ocean acoustic propagation in three dimensions can be complicated as well as computationally expensive, fast computation speed is important. Moreover, it is now clear that twodimensional models are no longer sufficiently representative. Efficient methods and clever implementation for dealing with three-dimensional wave propagation are therefore very important. The new LSS method and its implementation introduced in this chapter is not only a fast and accurate method, but is also just as representative as a model as other well-known existing 3D models. The numerical results have demonstrated that the method is efficient and have confirmed the theory that it is also stable. The LSS approach uses a form of the 3D wave equation as an ordinary differential equation with respect to range. Then a formal expression of the solution is written in terms of the exponential of the square root of some operator. The artifice used in this paper is to approximate this exponential in a clever way by the product of two rational functions of the type (1.1). As a consequence, the resulting ODE integration process requires only two successive tridiagonal system solutions. The theory shows that our method is unconditionally stable. Moreover, it is so accurate that larger step sizes can be afforded resulting in substantial savings in computational times. This has been widely confirmed by a set of comprehensive numerical accuracy tests. Moreover, angles of propagation as wide as 31 degrees have been accurately handled.

A High-Order Wave Equation ion

59

4.4. Computer Code — FOR3D A computer code 3 was developed to implement the marching implicit scheme (4.29). The complete development from the mathematical model, Eq. (4.7), to the numerical solution of Eq. (4.30) consists of a Finite difference solution, an Ordinary differential equation solution, and Rational function approximations; this combination is specifically and efficiently developed for solving three-dimensional (3D) problems. That is why the computer code is named FOR3D. Since a complete computer listing is included in this book, the development of FOR3D will be described. Descriptive materials used to design FOR3D should be useful information to help the user to understand the program structure. Materials presented in this section are the basic developments which contain no density variations and will be discussed further in a separate chapter. However, the entire code contains the capability enhancement and the density variation. In addition to a version available for VAX 11/780 computer, an identical version is also available on Cray X-MP also at the Naval Underwater Systems Center, New London, Connecticut, U.S.A. 4.4.1. Computer

Algorithm: \hm. Two-Step>p P\Procedure

In previous sections, it was discussed how Lee, Saad, and Schultz applied finite difference techniques to develop a method for numerically solving the LSS three-dimensional, wide angle wave equation. In this section, the algorithm used to implement the LSS method into computer code will be discussed. Equation (4.29) may be solved in two steps as follows: Writing Eq. (4.29) in the matrix form A * B V + 1 = ABuuj and substituting v J + 1 for B*uj+1

j

(4.36)

and vJ for Bu^, we have

A*vj+1

[vjj .. = Av

(4.37)

Now, solve as follows: Step 1 Compute Ayi and, using a tridiagonal solver, solve the system j

4 V ' + 1 = Avvj J +1

for the vector v ' ,

(4.38)

60

Numerical Ocean Acoustic Propagation in Three

Dimensions

Step 2 Now that yi+1 is known, use a tridiagonal solver to solve the system fl V

+ 1

= vj+1

(4.39)

for the vector u*+1. As previously stated, vP+1 is the acoustic field at all receivers in the cylindrical plane at the advanced range r + Ar. Throughout the test of this section, a capital U is used to denote the wave field w(r, 0, z) because of the computer program designation.

4.4.2. Computation for Step 1

0 = zero

AL =

(; - J) wt? * '°""r J ' < * - i f BMlJ' i r,6,z) P = 0 (5.6) (r, - p ] pdz) [dr2 r dr r2>\P Tz

'ae u>0*/

(IL + e

First, expressing k(r,6,z) = fcon(r,0, z) where k0 is the reference wave number, 7i(r,0, z) is the three-dimensional index of refraction. Then, following the development in Chap. 3 by applying the transformation P(r,0)Z) (kor) and substituting it into Eq. (5.6) and .*) = U(V^0,Z)HQ simplifying, a counterpart wave equation of Eq. (2.37) can be obtained in an identical form, viz ur = (-iko + ik0y/l+X++ +

+y +

) u ,

(5.7)

*♦-.■(„•..)-l + £p£(i£)

(5.8)

where

82

Numerical Ocean Acoustic Propagation in Three

and Y+

Dimensions

1

d (\ d" 2 IP fcgr 'M ;PdO)

(5.9)

when p is constant in both z and 0, and X+ and Y+ reduce to X and F , respectively. Applying the same rational function approximation as described in Eq. (4.7) for the square root operator in Eq. (5.7), the following is obtained:

Vi + x+ + r+ s i +1 \x+ -81,V\{x+)2 + iy+ . 2 2

8

2

(5.10) (5.10)

Substituting (5.10) into (5.7) gives uT = iko ( - 1 4- ( l + \X+

- i(X+)2 + i r + ) ) u .

(5.11)

Substituting the definitions of X+ and y + into Eq. (5.11), a partial differential equation is obtained in the form

«, = *,[-!+(l + i {.'(r,»,») -1 + 4 H G I ) ] } 1 f 22 n■ (r,e) , z ) - 1 8 I

+

1 \d d k0

2^^(p^JJU'

'i a\

V%\pdzj

w

(512)

which is the three-dimensional wide angle wave equation with density variations in operator form. Equation (5.12) accommodates the interface conditions in the ocean. Due to the discontinuity of p(0, z) on the interface boundary, the handling of p j j \\lk)dz ) a n ( ^ P~§o 1 1 must satisfy the interface conditions on the interface boundary. To fulfill this requirement, a finite difference technique, suggested by Varga,2 will be applied to treat the interface boundary. Consistent with the theory in developing the LSS model, the index of refraction, n(r, 0, z), satisfies the same assumption that it varies slowly with respect to 0, and the operators X + and Y+ are nearly commutative, thus Eq. (5.11) yields the solution u(r + Ar,0,z)

= e-ses(1+ix+-*(x+f)es*u(rAz)H wi(r,r,0,z) .

(5.13)

Enhancement of the High- Order Wave Equation

83

Again, following the rational function approximations for the exponentials, one can find es(i+ix*-i(x*)') s

and

,u

1+

(i + f ) X + i + (i-f)*+

(5.14)

£y+

«•* - 1 ^

•



How to obtain formulas (5.14) and (5.15) is left for the reader as an exercise. Therefore, the final solution of Eq. (5.13) can be expressed in the form u(r

+

X+ (14 Hf) ' 4/

'1ArAz)=(^l^+_f)Xx+^

r ■a 4

4/

Jt+

/ i . £y+ ( l ± iA £ ) « M , z ) . 1- 4 2 1

(5.16)

There are three issues here which need to be discussed: (i) the nonsingularity of both denominators in Eq. (5.16), (ii) the stability of the marching scheme (5.16), and (iii) the accuracy of the theoretical development. Note that X+ and Y+ are different from X and F , respectively, because X+ and F + both have a discontinuity at the interface boundary. This is due to the special numerical treatment for the interface plus the fact that this discontinuity is denumerable so that the continuity conditions are satisfied. As a consequence, X + and y + are both self-adjoining after special discretization and, with pure imaginary £, the two operators in the denominator of (5.16) can be expected to be nonsingular. Then it is seen in Eq. (5.16) that between two brackets the numerator and its associated denominator are conjugate of each other. Thus, their norms are unitary, i.e. these two rational function operators are unitary. Therefore, the marching scheme (5.16) is unconditionally stable. The accuracy analysis can be done identically to Sec. 4.1.1.2 by replacing X and Y with X+ and y + , respectively. 5.1.2. A NumericalcaZ ScSolution A numerical solution to the three-dimensional wide angle wave equation with density variations, Eq. (5.12), can be developed by using the same scheme as expressed by either (4.29) or (4.30), where X is replaced by X + and Y replaced by y + , which are defined in formulas (5.8) and (5.9). Let the symbol r represent either the depth variable z or the azimuth variable 6. Since X+ and Y + both involve expressions in the form p-jfr ( p ^ ) ) which is discontinuous on the interface boundary, a numerical

84

Numerical Ocean Acoustic Propagation in Three

Dimensions

technique suggested by Varga2 is used to handle the density discontinuity on the interface boundary such that the interface conditions are satisfied on the interface boundary. This is the KEY solution to Eq. (5.12) . Write counterpart expressions of (4.28) and (4.29) as follows: EXPLICIT form:

u' +1

■1-1

[» G- 1)M

x (l - ^y+) (l +

6

[14

'1 v4

K

6* X+ 4,

-Y^ u> ;

(5.17)

IMPLICIT form:

1

e -9

)*

6, ) ^ f> > 4°-r

+

= 1+

[ (3 + iH(1

+ y+

+1

J )"-

(5is)

Exactly the same theory with regard to stability used to prove scheme (4.29) for solving (4.7) can be applied for scheme (5.18) in solving (5.12). If p(0, z) is continuous in both 6 and z, no discontinuity can occur and no interface conditions will exist. What is needed is an efficient numerical technique to handle the discontinuity of p(0, z) on the interface boundary. As a direct generalization of the finite difference, described in Chap. 4, Sec. 4.2, for solving Eq. (4.29), a similar numerical solution to solve Eq. (5.18) can be developed in the form PQuj+1

= P*Q*uj ,

(5.19)

where P and Q are to be determined. To determine the operators P and Q, a technique to discretize the operators A"+ and Y+ needs to be introduced to handle the density function p(0, z) which is discontinuous on the interface boundary. Writing Quj+1 = w J + 1 , then solving system (5.19) can be done in two steps, i.e. (i) (ii)

PWJ+1 = P*Q*ui +1

Qui

1

= WJ+ .

and

(5.20) (5.21)

To obtain elements of P and Q, substitute the definitions of X + and F+ into Eq. (5.18), which gives

Enhancement of the High-Order Wave Equation ition

85

{'♦G-9[»'M->-K(£)]} 8 . ds 8 2 p Um e 4k 4k 0r2 de\pde)\ >

l

x

[

= {1+(l + £)[„>, s , 2 ) - 1 + i4(l|)]} a (Lia\

• 1

X 1+

[ J5F"IG^)]

k% dz \pdz)

P

d (i d\\

,

,gM.

+

[ p iIFw (,*)]*

Pm1-i,e

}

Pm--\*)

2't

1klh2 \4 ^ 1) \p 1 m+i,e

Lower diagonal:

-1

mPm-

Pm-\,t)

Pm

1 f 6\ 1 2>* Lower diagonal: -r^—r 7 + 7 . k$h2 V4 4 / p m + i ) £ The elements of matrix Q can be determined in the same way. System (5.21) can be rewritten below,

ri e+i -I + Pm,e-$ ) %mi+$

1

, 2

_ L

I

PmPrn,l-\) y

t^_L 1_ / 1 + 4fc 2 r 2 (A0) 2 \pmMi+pmMi) Lower diagonal:

-

6 1

89

Wave Equation ion

1

1

TTHTTMO

1

\

Pm,£-\

'

2

•

4fc2r2(A0)2pm^+i

The matrix Q* should contain the following elements: Upper diagonal: Main diagonal:

1 - -j-^ —— , 4fcgr 2 (Ar) 2 p m ) £ _i 1 - ( 1 2 \Pmt£+i 2i _6_1 2 2

)

PmPm,i-iJ

1

4fc r (A0) Lower diagonal:

1 / 2

1

1

+

\

\pm,t+l Pmj+l) X-

1 "2

'

1 1 -yrr, >2 . 4 f c 2 r 2 ( A 0 ) 2 p m ^ + "2 i1

Prom the information of upper and lower diagonals of matrices P and Q, it is not easily seen that both matrices are self-adjoint. As an example, an examination of matrix P on the interface boundary (indexed by m) reveals that

Pm

-^ = W?(\-i)^rr

p +

™ ^ = w (i" 9 ^

•

(5 30) (5.30)

-

{5 31)

-

Note that the element above the interface boundary m is

P

— = ^ U - 4J ^

'

(5 32)

-

90

Numerical Ocean Acoustic Propagation in Three

Dimensions

and the element below the interface boundary is Pm,m+l

like

1

= ^klh? ( I - ! )1 _ i _ . Pm-\-h*

(5.33)

It is seen within that small block that a portion of the matrix P looks

/

\

•

Pm,m-1

•

Pm-i,m

•

Pm+i,m

•

Pm,m+1

•

\

0 *]•""■"""■