Theoretical Acoustics of Underwater Structures

THEORETICAL ACOUSTICS OF UNDERWATER STRUCTURES

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THEORETICAL ACOUSTICS OF UNDERWATER STRUCTURES

E A Skelton Imperial College, UK

J H James

Imperial College Press

Published by Imperial College Press 516 Sherfield Building Imperial College London SW7 2AZ Distributed by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 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.

THEORETICAL ACOUSTICS OF UNDERWATER STRUCTURES Copyright © 1997 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 1-86094-085-4

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

V

CONTENTS

PREFACE ACKNOWLEDGEMENTS

xi xiii

INTRODUCTION

1

1 MATHEMATICAL M E T H O D S

7

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

Fourier Series Fourier Transforms Delta Functions Vector and Scalar Identities Integral Theorems Bessel Functions Spherical Bessel Functions Legendre Functions Stationary Phase Matrices Numerical Integration

7 9 11 14 18 19 23 26 28 30 34

2 R E S P O N S E OF D Y N A M I C A L SYSTEMS 2.1 Time-Harmonic Response 2.2 Transient Response 2.3 Time-Harmonic Power Flow

43 43 47 55

3 ACOUSTIC EQUATIONS 3.1 Differential Equation 3.2 Plane Wave Solutions 3.3 Green's Function in Cartesian Coordinates

61 61 66 67

VI

3.4 3.5 3.6 3.7 3.8

Green's Function in Cylindrical Coordinates Green's Function in Spherical Coordinates Arbitrary Source Distribution Arbitrary Boundaries Helmholtz Integral Equation

70 73 75 77 77

4 SCATTERING FROM H A R D A N D SOFT S T R U C T U R E S 4.1 Planar Surface 4.2 Spherical Surface 4.3 Cylindrical Surface 4.4 Axisymmetric Slender Body 4.5 Monopole in Axisymmetric Channel

83 83 88 92 97 106

5 ACOUSTIC FINITE ELEMENTS 5.1 Variational Principle 5.2 Finite Element Discretization 5.3 Acoustic Finite Elements 5.4 Radiation Boundary Condition 5.5 Sources of Sound 5.6 Axisymmetric Elements

115 115 119 121 123 125 126

6 ELASTIC EQUATIONS A N D CONSTITUTIVE RELATIONS 133 6.1 Strain and Stress 133 6.2 Isotropic Displacement Equations 136 6.3 Finite Element Variational Principle 140 6.4 Fluid Constants 144 6.5 Isotropic Elastic Constants 144 6.6 Anisotropic Solid -. 148 6.7 Unidirectional Fibre Reinforcement 150 6.8 Thin Layered Composite 151 6.9 Local to Global Transformation 155 6.10 Effective Constants of Symmetric Stack 159 7 ACOUSTICS OF SPHERICAL SHELL 7.1 Equations of Motion 7.2 Mechanical Excitation

163 163 165

vii 7.3

Acoustical Excitation

171

8 ACOUSTICS OF THIN PLATE 8.1 Equations of Motion 8.2 Mechanical Excitation 8.3 Monopole Excitation 8.4 Plane Wave Excitation 8.5 Transient Sound Radiation 8.6 Frame and Bulkhead Stiffened Plate 8.7 Orthogonally Stiffened Plate 8.8 Plate with Point Attachments 8.9 Parallel Plates with Attachments 8.10 Plate with Periodic Resonators

179 179 180 189 191 193 198 207 213 221 228

9 ACOUSTICS OF CYLINDRICAL SHELL 9.1 Equations of Isotropic Shell 9.2 Equations of Anisotropic Shell 9.3 Mechanical Excitation 9.4 Monopole Excitation 9.5 Plane Wave Excitation 9.6 Periodic Rib Stiffening 9.7 Axisymmetric Attachments 9.8 Fluid Coating

241 241 244 246 253 259 263 273 278

10 SPHERICALLY LAYERED MEDIA 10.1 Introduction 10.2 Acoustic Fluid Layer 10.3 Isotropic Elastic Layer 10.4 Viscous Fluid Layer 10.5 Assembly of Elements 10.6 Mechanical Excitation 10.7 Acoustical Excitation 10.8 Numerical Examples

283 283 284 287 291 293 294 295 297

vm

11 P L A N A R L A Y E R E D M E D I A

301

11.1 Introduction

301

11.2 Acoustic Fluid Layer

303

11.3 Isotropic Elastic Layer

305

11.4 Orthotropic Elastic Layer

311

11.5 Viscous Fluid Layer

317

11.6 Assembly of Elements

319

11.7 Mechanical Excitation

320

11.8 Monopole Excitation

325

11.9 Plane Wave Excitation

329

l l . l O L o w Frequency Sound Propagation

333

12 C Y L I N D R I C A L L Y L A Y E R E D M E D I A

343

12.1 Introduction

343

12.2 Acoustic Fluid Layer

345

12.3 Isotropic Elastic Layer

347

12.4 Orthotropic Elastic Layer

354

12.5 Viscous Fluid Layer

359

12.6 Assembly of Elements

361

12.7 Mechanical Excitation

362

12.8 Monopole Excitation

366

12.9 Plane Wave Excitation

369

13 S I M P L Y S U P P O R T E D C Y L I N D E R

373

13.1 Isotropic Thin Cylinder

373

13.2 Point Force Excitation

376

13.3 Plane Wave Excitation

381

13.4 Anisotropic Layered Cylinder

385

14 F I N I T E A X I S Y M M E T R I C S T R U C T U R E

395

14.1 Introduction

395

14.2 Fourier Series

397

14.3 Helmholtz Integral Equation

397

14.4 Far Field Acoustics

403

14.5 Fluid-Structure Equations

406

14.6 Conical Shell Finite Element

408

ix 14.7 Axisymmetric Solid Finite Element 14.8 System Dynamic Stiffness 14.9 Numerical Examples

414 420 423

LIST OF COMMONLY O C C U R R I N G SYMBOLS

429

INDEX

433

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XI

PREFACE

T h e m a t h e m a t i c s of sound radiation from, and sound scattering by, elastic structures has received much attention in the latter half of this century due to its relevance to the problem of detecting and classifying submarines and other submersibles. T h e theoretical work described herein is substantially a compilation of our research interests for the Navy on the acoustics of basic elastic struc­ tures excited by mechanical forces and acoustic sources. It is not intended as a comprehensive t r e a t m e n t of the subject for which even a list of textbooks and research publications is far too lengthy to record here. The standard reference work on theoretical structural acoustics is the textbook "Sound, Structures, and Their Interaction", published by the M I T Press, and authored by Junger & Feit. T h e Journal of Sound and Vibration and the Journal of the Acoustical Society of America are the main repositories of refereed papers. While the textbook has been written primarily as a reference work at a postgraduate level, experience has shown t h a t undergraduates are well able to cope with most aspects provided they are supervised accordingly. The theo­ retical methods, developed originally for Naval noise control problems, should also find civil application in the acoustic modelling of novel structures fabricated from b o t h fibre reinforced and isotropic materials, such effort being increasingly desirable and necessary in this noisy world. The textbook should be a useful source of information for those scientists who need to develop skills in theoretical a n d / o r numerical modelling in dynamics and acoustics, and it is hoped t h a t it will also be a profitable source of material for scientists whose research interests lie elsewhere in continuum mechanics. T h e first edition of any sizeable mathematical publication is likely to contain algebraic and typographical errors.

T h e responsibility for those here is ours

alone. When formulae are extracted from published literature it is generally advisable to check t h e m against originals and also, if possible, to re-work the mathematics.

xii

E.A. Skelton Mathematics Department Imperial College of Science, Technology and Medicine London, SW7 2BZ

J.H. James Brynteg House, High Street Mountain Ash Mid Glamorgan, CF45 3PB

May 1997

xm

ACKNOWLEDGEMENTS

Much of the work described here is contained in Ministry of Defence tech­ nical reports and m e m o r a n d a written by the authors and colleagues over the years 1974-1994. Our thanks to the Procurement Executive, Ministry of De­ fence for providing funding for t h a t research as part of an underwater acoustics research p r o g r a m m e . T h e Defence Research Agency provided access to reports and software, for which we are grateful to C.J. Jenkins, D. Ealey and librarian J. Sawyer. Particular thanks are due to E.J. Clement who ably has assisted us over the years. Work done with ex-colleagues S.J. Ansley, G. Jones and V. Burke is also included here. Students J o h n Melotte, Julie Pestell, William Spicer, Sherrie Lester, Nicola W h i t e and J a m e s Mason worked gainfully during their sum­ mer vacations. Navy scientists A.L. Kendrick, G. Barber, J. Cook, A.N. Hicks, D. Emery, G. Alker, A. Baird, G. Jones, A. James, and C. Jenkins have sup­ ported a background of theoretical modelling as a necessary and valuable input to applied research programmes. T h e research output of University scientists has also provided a worthy contribution: B.R. Mace, M. Petyt, C.R. Fuller and F.J. Fahy, working at the Institute of Sound and Vibration Research, have pro­ vided quality research on wave propagation in fluid-loaded plates and shells; I.C. Mathews and research students at Imperial College have provided up to date expertise in finite element analysis of

fluid-structure

interaction problems.

A . G . P W a r h a m , G . F . Miller, R. Borland and A.J. Burton, at the National Physi­ cal Laboratory, have always been happy to take on difficult theoretical problems. We are grateful to the publisher Academic Press Limited London for permission to use material from certain papers published in The Journal of Sound and Vi­ bration. Finally, we apologize to those who perceive t h a t their work has been omitted or not properly acknowledged; it sometimes can be difficult to attribute credit a n d / o r priority because similar work has sometimes been carried out in m a n y research organizations, and not always published in refereed journals or publicly released technical reports.

INTRODUCTION

The acoustic characteristics of a ship, building or machine are likely to be but one of a number of sometimes conflicting requirements of an engineering or building project. Nonetheless, because it is likely to retain some importance, there will be some acoustic objective to be achieved subject to various constraints such as cost and operational requirements. Generally, theoretical modelling as well as experimental testing should be built into an acoustic design procedure, the former being particularly valuable because it increases the range of options which can be assessed while at the same time helping to reduce the chances of introducing unwanted acoustic features, particularly in structures of novel design and/or new materials. It is the role of an acoustician to provide theoretical modelling support, either for paper assessment studies or for experimentally based structural and materials research programmes. To effectively provide this support it is desirable for an analyst to have access to a wide range of theoretical tools, in both textbook and software form, and to be experienced in their use. These tools can then be used as basic building blocks for solving more complex acoustical problems, and can also be used as examples of how new building blocks can be developed. Thus, herein are provided many of the basic building blocks for modelling the dynamics and acoustics of underwater elastic structures, and it will be found that many of the models described can also find a wide range of applicability in the field of architectural acoustics. A single theoretical model for covering very low to very high frequency structural acoustics, 0.1 Hz to 100 kHz, say, is entirely impractical due to the restrictions imposed by the limited speed of current computers. Even if such a model were possible, the computer output would not help greatly with physical understanding of the wave processes involved. However, when the solution of a problem appears to be somewhat intractable, the way forward is to solve simple substitute problems from which, it is hoped, qualitative predictions and physical insight will emerge. These substitute problems can turn out to be basic building 1

Acoustics of Underwater Structures

2

blocks. T h e art of noise and vibration modelling is to select and enhance those models which yield the required information in a cost effective way. Thus, a layered approach to modelling is desirable. At the concept stage of an engineer­ ing project the acoustic modelling should be based on uncomplicated models with a m a x i m u m of physical understanding and a minimum of computational detail; these are called fundamental models and they are used for making qual­ itative predictions. It is essential t h a t the concept stage is properly researched and documented because a bad concept can result in acoustical problems in the prototype. At the design stage more elaborate models are required and certain quantitative predictions will be found desirable; these are called phenomenological models. At the prototype stage finite element models may be needed, even if only to avoid lengthy experimentation; these are called one-to-one models. Con­ tained herein are a wide range of acoustic models covering fundamental models, through phenomenological models, to a simple one-to-one model based on the finite element m e t h o d and the Helmholtz integral equation. T h e m a t h e m a t i c a l utilities necessary for solving problems in theoretical acoustics are s t a n d a r d to courses in classical mathematical physics. As a re­ fresher, Chapter 1 gives certain formulae obtained from the theory of Fourier integral transforms and their stationary phase asymptotic approximation, delta functions, vector and scalar identities, integral theorems, Bessel and Legendre functions, matrices and numerical integration. This is followed in Chapter 2 by a brief description of the time-harmonic response, transient response and mechan­ ical power flow in dynamical systems. T h e foundations for the acoustic theory used herein are set out in Chapter 3 in which the acoustic wave equation is de­ rived in a simple way. There follows separated solutions of the wave equation, plane wave expansions and free-space Green's functions in cartesian, cylindrical and spherical coordinate systems. In later chapters the plane wave is the basic excitation used to described the transmission and reflection properties of elas­ tic structures, and the Green's function, approximating to a localized source of sound, can be used to construct more complex sound sources, either directly or via the Helmholtz integral equation which is central to numerical acoustics. A m o n g the canonical problems of acoustics are those for the scattering of sound by h a r d or soft spherical, planar and cylindrical boundaries which are ensonified by plane waves or pressure fields from acoustic monopoles located near to the boundaries.

Closed-form solutions, developed in Chapter 4, are

Introduction

3

used in later chapters as part of the solution when elastic effects are included. Also considered in Chapter 4 is an approximate treatment of sound scattering from a finite axisymmetric body and an exact solution for the sound field in an axisymmetric channel, these two models being examples of how the free-space Green's function can be used as a building-block. For awkward geometries, for which closed-form expressions are impractical, acoustic finite elements can be used to solve low frequency problems, as considered in Chapter 5. Here, after some discussion of variational principles for solving linear field problems, simple axisymmetric acoustic finite elements are developed, which can, for example, be used to investigate the sound radiation characteristics of sound sources in circular ducts. For most practical problems the elastic response of a structure must be ad­ dressed. This is particularly i m p o r t a n t in underwater acoustics as the mechani­ cal loading terms stemming from the surrounding fluid are generally comparable to a structure's mass and stiffness. In Chapter 6 are obtained the displacement differential equations of linear elasticity for isotropic materials. These are de­ rived from strain, stress equilibrium and stress-strain relations which are given in cartesian, cylindrical and spherical coordinates. The equations in the various coordinate systems are reduced to wave equations by means of simple substitu­ tions, so t h a t well-established solution methods for wave equations are equally applicable to elastic equations. There follows a general variational principle on which the finite element method, used in later chapters, is based.

A section

on the elastic constants of an isotropic elastic solid is accompanied by a ta­ ble of Young's modulus, Poisson's ratio and density for a variety of materials. T h e constitutive m a t r i x is a 6 x 6 m a t r i x which relates the stress vector to the strain vector. For an isotropic material its m a t r i x elements are formed from two constants, but for an anisotropic material there are up to twenty one different constants. It is shown how a certain mixture theory can be used to find the effective orthotropic elastic constants required for fibre reinforced material, how effective elastic constants can be obtained for a stack of fibre reinforced layers comprising a thin shell or a thick three-dimensional solid, and how a constitutive m a t r i x can be transformed from a local axes set, in which one or more axis is parallel to a fibre direction, to a global axes set. T h e spherical shell is the only finite geometry for which simple closed-form solutions are available for its vibration, sound radiation and sound scattering.

Acoustics of Underwater Structures

4

This is the baseline problem of fluid-structure interaction. In Chapter 7, closedform expressions are obtained for the sound radiation from a spherical shell excited by a point force, and the sound scattered by the shell when it is excited by a plane wave or monopole acoustic source. Plots of intensity vectors demon­ strate the considerable distortion in an acoustic field when a structure is excited at a resonant frequency. The most extensively studied fluid-structure system, driven by mechanical forces and acoustic sources, is perhaps a uniform infinite plate with attached nonuniformities such as periodically spaced ribs or arbitrary arrangements of point attachments.

This system is considered in Chapter 8.

It contains the essential physics of wave propagation in fluid loaded structures, and this can be extracted with minimal computation. The non-uniformities on the otherwise uniform plate act as secondary sources of acoustic radiation and their presence considerably can enhance or reduce sound radiation. In certain circumstances resonators attached to plates can be used as sound absorbing de­ vices, and it is shown how simplified "lumped parameter" methods can be used to obtain simple formulae for studying the effect of uniform distributions of res­ onators embedded in different types of plate cavity. In Chapter 9 the complicated differential equations of motion of thin isotropic and anisotropic cylinders are given. These are solved, for an infinite cylinder with both interior and exterior fluid loading, by using standard Fourier series and integral transform methods. T h e effects of periodically spaced ribs and finite numbers of axisymmetric con­ straints are also considered. Because the ribs on a shell are usually deep, such as those T-ribs found inside a pressure hull, it is generally necessary to account for deformation of their cross-sections, which can be achieved by using the finite element m e t h o d to model the ribs. Finally, it is shown how a coating, modelled as a fluid, can be attached to the outer surface of an infinite cylinder. The coat­ ing simulates an anechoic coating for absorbing sound waves from an exterior noise source, or a decoupling coating for isolating mechanical vibrations from a surrounding fluid. In Chapters 10-12 is considered the acoustics of spherically layered media, infinite planar layered media and cylindrically layered media. Each layer can be an acoustic fluid satisfying the acoustic wave equation, a viscous fluid satisfying the Navier-Stokes equations, or an isotropic/anisotropic elastic solid satisfying the equations of elasticity. T h e layers are represented by spectral dynamic stiff­ ness matrices which relate surface displacements and stresses. The layers are

Introduction

5

combined to form the layered system m a t r i x by the method of dynamic stiffness coupling, which is a m a t r i x method for combining finite element matrices to form a structure m a t r i x . Numerical solution of the system matrix leads to the solution of the problem of sound transmission through layered systems when the excitation is a plane wave, and the far field sound radiation when the excitation is a mechanical force or a monopole acoustic source. For the case of planar layered media, the problem of low frequency sound propagation in the ocean, with a lay­ ered seabed, is extracted as a special case. Sound transmission through layered media is a canonical problem in both underwater and architectural acoustics; the addition of a capability to include fibre reinforced composites provides a major enhancement of applicability because high-strength, lightweight, fibre re­ inforced materials are increasing being used as components in the construction of buildings and underwater submersibles. In Chapter 13 the acoustic behaviour of a simply supported finite cylinder is considered. This is a relevant geometry because it approximates to a compart­ ment in an underwater submersible. Two cases are considered: the first is for which the cylinder is modelled by using the thin shell theory of Chapter 9; the second is for which the cylinder is modelled as a layered composite by using the layered media theory of Chapter 12. Numerical results show the considerable enhancement of sound t h a t occurs when the excitation frequency coincides with a resonant frequency, both for the separate cases of point force and plane wave excitation. Finally, in Chapter 14, the acoustics of a finite axisymmetric structure sub­ ject to plane wave or point force excitation is considered. T h e finite element m e t h o d is used to model the structure by using conical shell finite elements or by using axisymmetric solid finite elements, both of which can be fibre re­ inforced layered composites. T h e exterior fluid is modelled by the Helmholtz integral equation which is a relation between the acoustic pressure in a fluid and a surface vibration which is the cause of the acoustic disturbance. This type of modelling increasingly will be used for practical structures; it is the direction of much leading-edge research aimed at software packages for solving large scale fluid-structure interaction problems. T h e model described herein is useful for certain low frequency sound radiation studies, where, for example, the propul­ sion system and associated machinery in an underwater submersible can cause large structural vibrations.

6

Acoustics of Underwater Structures

Fortran computer programs have been written for obtaining numerical pre­ dictions of the acoustics of the dynamical systems considered here, but program­ ming and numerical details are not considered in the text. However, numerous numerical examples are given of both time-harmonic and transient pressure fields stemming from mechanical and acoustical excitation of structures. These are an aid to physical insight and provide benchmark computations of sound radiation and sound scattering which can be of particular value to those scientists, who, by being users of software packages, provide numerical modelling support and consultancy.

CHAPTER 1. MATHEMATICAL METHODS

1.1

Fourier Series

If f(x) is defined in the interval (a, 6) and is also periodic with period 6— a, viz. f(x + 6 — a) — / ( # ) , then a Fourier series may be defined as an cos o— a n=0

where an

=

h V^ bn sin , *-^ o—a

(1-1.1)

n=l

—^— / f(x)cosdx, o- a Ja b- a

— / /(x)sin^dx, (1.1.2) » - a Ja b- a with e n = 1 for n = 0 and e n = 2 otherwise. Because the functions are periodic the limits of integration may be changed from (a, 6) to (ao,ao + 6 — a). The Dirichlet conditions, viz. f(x) is single valued except at a finite number of points in (a, 6) and f(x) and f'(x) are continuous except at the discontinuities, are a sufficient but not necessary condition for the Fourier series to exist and to converge to f(x) if x is a point of continuity, and to converge to [/(#+)+/(#_ )]/2 if x is a point of discontinuity. The notation x+ and x~ refers to values of x on the right and left, respectively, of the discontinuity. For a half range Fourier sine series of the odd function f(x) — —/(—#), set a = —L and b — L and note that an — 0 and the integral for 6n is twice the value in Eq. (1.1.2) with limits (0, L). For a half range Fourier cosine series of the even function f(x) = f(—x), set a = —L and b — L and note that bn = 0 and the integral for a n is twice the value in Eq. (1.1.2) with limits (0, L). bn

=

7

8

Acoustics of Underwater Structures

A complex notation is usually desirable when solving certain equations of m a t h e m a t i c a l physics and engineering. Thus,

J2 c " e x P y r 7 '

f(x)=

(L1-3)

n = —oo

where cn = b-

1

fb 0, cn - 0.5(a n - i6 n ), c _ n = 0.5(a n + i6 n ), an - cn + c _ n , 6 n = i(c„ - c _ n ) . A Fourier series of the function f{<j>) defined over the range (0, 27r) is frequently required, in which case the formulae are valid provided x = , a = 0 and 6 = 2ir. Another case of interest is the time function f(t)

defined over the range ( 0 , T ) ;

here set x — t, a = 0 and b = T. Parseval's relation for the functions f(x)

and g(x), represented by the com­

plex Fourier series of Eq. (1.1.3) with coefficients cn and d n , respectively, is oo

b

f*(x)g(x)dx

/

= (b-a)

T

c*ndn,

(1.1.5)

n = — oo

where the asterisk superscript denotes complex conjugate. A special form of the Fourier series is appropriate to "simply supported", f(-L)

= f(L)

= f"(-L)

= f"(L) f(x)

= 0, boundaries at ( - L , L), viz.

= J2bnsm

\

r

21

',

(1.1.6)

n= l

where n =

1 [ Lj

r, / W

. . Sm

rnr(x + L) , 2L

, ( L L 7 )

Generalization of Fourier series to more than a single dimension is straight­ forward. For example, for the two-dimensional periodic function f(x,y)

defined

in the range (a, b) in the ^-direction and the range (c,d) in the ^-direction, CO

/(*>!/) =

V

CO

V

m = —oo n = —oo

cmnexp-

exp-—-,

(1.1.8)

Mathematical Methods

9

where 1

1

fd fb ti

^

-ilvmx

-i2irnj/J

°™ = Jjb^JdZj Je Ja f^y^P-^r^^ZT^^dy. 1.2

3

„ , „x

(1.1.9)

Fourier Transforms

If f(x) is defined and is absolutely integrable in the interval (-oo,+oo), then its Fourier transform or spectral form, F(a), is defined by the equations /+°°

1 f(x) = —

F(a)exp(iaz)da,

(1.2.1)

and /-pOO

f(x)exp(-iax)dx.

(1-2.2)

■CO

The Dirichlet conditions, viz. f(x) is single valued except at a finite number of points in (a, 6) and f(x) and f'(x) are continuous except at the discontinuities, are a sufficient but not necessary condition for the Fourier integral in Eq. (1.2.1) to exist and to converge to f(x) if a; is a point of continuity; and to converge to [/(#+) + / ( * _ ) ] / 2 if * ^ a point of discontinuity. An alternative representation of the Fourier transform pair is also in common use, viz. /+oo •oo

and

/+oo -co

ra(a)exp(2xiax)da,

(1.2.3)

ja{X) exp(—zwuyx)Q.x.

yi.2.^ij

Interchanging between the two versions is accomplished by noting that /(*) = /„(*) and F(a) = Fa(a/2*) are the transform pair m the first system. ParsevaPs relation for the functions f{x) and g?x) whose transforms are F(a) and G(a) is /+oo

-j

f*(x)g(x)dx=■CO

r+OO

F*(a)G(a)da.

(1.2.5)

Acoustics of Underwater Structures

10

The convolution theorem states that the Fourier transform of the convolu­ tion, -foo

f(u)g(x - n)dii,

/

(1.2.6)

-oo

is simply C(a) = F(a)G(a). The convolution obeys the commutative, associa­ tive and distributive rules of algebra. Two other useful relations comprise the generalized Poisson summation for­ mula, oo

1

J2

f(x-nd)=-

oo

d ^

F(2Trn/d)exp(2mnx/d),

(1.2.7)

n = — oo

and

oo

^

^2

f(nd)exp(-iand)

oo

= - ^ d

F(a + 2irn/d).

(1.2.8)

n = — oo

These formulae are of prime importance when analyzing periodic systems, and they can also be useful for converting a slowly convergent series to a rapidly converging series. For example, the second formula, with a = 0, can be used to show that oo

i—

exp(-n2d2)

J2

= ^

d

oo

J2

exp(-ir2n2/d2)

(1.2.9)

n = — oo

when the transform pair f(x) = exp(—x2) and F(a) — yJH. exp(—ct2/4) is used. The series on the left converges slowly for small c?, while the series on the right converges rapidly. Generalization of the Fourier transform to more than a single dimension is straightforward. For example, for the two-dimensional Fourier transform, the function f(x,y) and its transform F(a,f3) are defined as follows, i

r+oo

y+oo

f(x>y) = T~5 ^T

/ J — oo

F(a,/3)exp(iax

+ ipy)dad(3,

(1.2.10)

J — oo

and +oo

/

/»+oo

/ -oo

f{x,y)exp{-iax

ParsevaPs relation for the functions f(x,y) +oo

/

/*+oo

/ -oo

J — oo

-i(ly)dxdy.

(1.2.11)

J — oo

-j

r(x,y)g(x,y)dxdy=-1 ^^"

and g(x,y) is /♦-f-oo

/*-foo

/

/

J — oo

J — oo

F*(a,p)G(a,

P)dadp, (1.2.12)

Mathematical Methods

11

and the Poisson summation formula is CO

CO

XI X f(x ~mdx^y-ndy) = m = —oo n = —oo .j

CO

CO

—jdxdv

2^

22

FC^nm/dx, 27rn/dy) exp{2iri(mx/dx

+

ny/dy)}.

* m = —co n — — oo

(1.2.13) In a cylindrical coordinate system (r, yz) an integral transform in the z direction and a series transform in the <j> direction is the most appropriate de­ composition. Thus, for a field variable /(r, ,z), f(r,, z) is +oo

/

p2ic

/ -co

°°

f*(r,(l>,z)g(r,(f))z)d<j)dz=

JO

^ n =

y*+co

/

F*(r, n, a)G(r, n , a ) d a .

_00J-oo

(1.2.16) 1.3

Delta Functions

The Dirac delta function has a preeminent place in mathematics due to its simplicity and its usefulness for facilitating integral transform solutions of the equations of mathematical physics. It is a representation of a point source forcing function which leads to a Green's function, viz. the response of a system at field point i due to unit excitation at field point j . Some Green's functions of the reduced acoustic wave equation are derived in Chapter 3. Here, basic properties and some useful formulae involving delta functions are given.

Acoustics of Underwater Structures

12

Table 1.1: Some s t a n d a r d Fourier transform pairs. defined in §1.3.

/(*) d" xnf(x) S(x - x0) exp(iao#) f(ax)exp(ibx) { li \x\ < a \ 0, \x\ > a cos(ao^)

The delta function S is

F(a) (ia)nF(a) dn dan v J exp(—ixoa) 27r6(a — ao) -F((a-b)/a) a 2 sm(aa) a 7r{S(a + a 0 ) + S(a - a 0 )}

exp(-a0|z|)

i7r{ 0 l x -f cr

— exp(—aasgna)

sin(aotf)

— (i7rsgna) exp(—aasgna)

1

Mathematical Methods

13

The delta function 6(x), which is a generalized function, exists by virtue of its sampling property. If f(x) is continuous in the neighbourhood of the point x — #o, then +00

f(x)6(x-x0)dx

/

= f(x0),

(1.3.1)

which can immediately be extended to cover derivatives and finite limits of integration, viz. fb

(

n

/ f(x)S < \x Ja

[ (-l)n/(n)(z0), - x0)dx = ' dz

5^ 1 W

1

0»\

dR' RdO ' RsmO d<j)J di dl vA

= V.A =

(1-4.1)

1 ^ + ^ + ^

5x dy dz dAr Ar 1 dAA dAz + — +—?rr + dr r r d dz (dAR 2AR \_d

. „, ,

1

dA*\ (1.4.2)

curlA = V x A

=

dAz _ dAy dAx _ dAz 9Ay _ dAx dy dz ' dz dx dx dy 1 dAz dAA dAr dAz 1 d r d dz ' dz dr ' r dr (rAj) 1 d . 1 dA, Rsin0d0K (Af9 sin 0)' - Rsm9 d ' 1 dAR l1 «0 . n i . 1l «d , n i . i£sin# a

18Ar r d

IdAi 1< 9^A (1.4.3)

_,. ,

JT ,

„2T

/ a 2 * a2* s 2 * \ d2^ dr2 fd2V \9R2 + +

199 l_d^ d2H>\ 2 2 r dr r d<j> dz2) ' 2 99 1 d ( . n99 2 . „ -^r sin 0 T : TTP: + R,-,-, RdR sin0 90 \ d0

*WW>

(L44)

Acoustics of Underwater Structures

16

Figure 1.1: Coordinate systems used: (r, , z) a n d (c) spherical (R,9,<j>).

(a) cartesian (x,y,z),

Table 1.2: Coordinate systems. | Cartesian

1

x

1

z

y

Cylindrical Spherical | x — r cos <j> x — Rsin9 cos<j> 1 y — r sin y — R sin 9 sin z —z z = R cos 9 1

(b) cylindrical

Mathematical Methods

17

A number of relations between the various quantities find frequent applica­ tion when solving physical problems. If ^ and $ are scalars and A and B are vectors, div(curlA)

0,

curl(grad^)

0,

div(tfA)

#divA + A.grad^,

curl(^A)

grad^ x A 4- ^curlA,

grad(A.B)

(A.grad)B + (B.grad)A + A x curlB + B x curlA,

div(A x B)

B.curlA — A.curlB,

curl(A x B)

(B.grad)A - B(divA) - (A.grad)B + A(divB),

2

tf V 2 $ + $V 2 tf + 2gradtf .grad$,

V (tf#) V2A

grad(divA) — curl(curlA).

(1.4.5)

The final relation in Eq. (1.4.5) is important because while V 2 operating on a scalar is straightforward, V 2 operating on a vector is not simply V 2 operating on the vector's separate components, except in cartesian coordinates. Thus, in cartesian coordinates V2A = ( V 2 ^ , V % , V 2 ^ ) .

(1.4.6)

In cylindrical coordinates V 2 A = ( V'A,

2 8A4 ,V%r2 d<j>

2 dAr 2 ,V AZ d(j>

+ r2

(1.4.7)

In spherical coordinates the expressions are much more complicated, VaA

■

(

dA, 2AR d (sin0Ae)R2sin0 d<j> R2 B?sm0d0 2 cos $ dA))

(1.6.1)

(1.6.2)

of the reduced wave equation, V 2 \£ -\- k2^f = 0, in cylindrical coordinates, where kr — +\J{k2 — a 2 ) . The two independent solutions of Eq. (1.6.1) are the Bessel function of the first kind Zn(z) and the Bessel function of the second kind Yn(z). Bessel functions of the third kind, viz., H ^ ( * ) = J n (z) + iY n (z) and E(n2\z) = Jn(^) — iY n (^) are also independent solutions; these are the Hankel functions of the first and second kinds. The superscripts on the Hankel function denote the type of Hankel function, not its derivative. In later chapters, the symbol H n is used as an abbreviated form for Hn . For the purposes of formulae given here, n is an integer and z any complex number with Im(z) > 0; and it is to be noted that these are not necessary conditions, for example, n can be an arbitrary complex number. The function J n (z) is an analytic function everywhere in the complex z plane. The functions Yn(z), H n (z) and H n (z) are analytic functions everywhere in the complex z plane cut along the negative real axis. For analytical continuation in the complex

Acoustics of Underwater Structures

20

plane, values on the upper edge of the cut along the negative real axis are related to values on the positive real axis by the formulae J„(-ar)

=

(-l)nJ„(z),

Yn(-x)

=

( - l ) n Y n ( * ) + 2i(-l) n J„(;r),

!#>(-*)

=

(-ly^HWOr),

=

(-l)nE™(x)

4

2)

(-^)

+ 2(-l)nJn(x).

(1.6.3)

The Bessel functions satisfy the recurrence relations C„+i(z)

=

—Cn(z)-C„_i(z), Z

C'n(z)

=

nCn(z)

=

nCn(z)

=

i(C„-i(2r)-Cn+i(z)), zCn-xW-zCfa), zC'n(z) + zCn^(z),

with CQ(Z) — —C\{z), and C-n(z) both for numerical calculations and The small and large argument cal quantities in their near and far expansions are limJ„(z)

(1.6.4)

n

— (—l) C n (z). These formulae are useful simplifying algebraic expressions. expansions are required to evaluate physi­ fields, respectively. The first terms in the

=

zn/2nnl

=

(2/JT) (In z-

=

-2n(n

z-+0

limYo(z)

0.1159),

Z—»-0

lim Yn(z) z—vQ

- l)\/wzn,

\

lim J n (z)

=

lim Yn(z)

— \/2/7rzsm(z

/

n > 0,

i

\f2/-KZQ,os(z — nir/2 — 7r/4), — n,7r/2 — 7r/4),

2—»-00

limHW^)

=

^2/wzexp

(i(z - nw/2 -

lim HW(z)

=

y/2/7rzexp (-i(z -

UTT/2

TT/4))

-

,

TT/4))

.

(1.6.5)

The function J„(z) is finite at z = 0, at which point the functions Y n (z), H^ '(z) and Hn (2) are singular.

Mathematical Methods

21

Some of the Bessel function Wronskian relations are 3n(z)Y'n(z)-Yn(z)3'Jz)

=

2/TTZ,

Jn(z)EW'(z)-E(n1\z)3'n(z)

=

21/TTZ,

=

-2/TTZ2,

=

-2/irz,

Jn(z)YZ(z)-Yn(zmz) 3n(z)Yn+1(z)-Yn(z)3n+1(z) RVKz^lM-EWWEZliz)

= -4i/™.

(1.6.6)

These relations can sometimes help to simplify algebraic expressions that occur when solving acoustic problems. A special form of Graf's "addition" theorem is given in reference [1], as C

< « (

sinj

= ^

a

+

nMJ

n

(^)|

'

(1.6.7)

where z is any complex number, d = A/( Q 2 -f 62 — 2a6 cos 0), a — b cos 0 = d cos z/>, 6sin0 = dsin^>, |6exp(±i#)| < \a\. The final constraint is not required when C = J and z/ is any integer including zero. Variants of this addition theorem are required for developing particular forms of Green's functions. A number of relations involving integrals have found useful application when solving physical problems by transform methods. A commonly required integral is J^

{ sTnnt « * « « » ( * - * » d * = 2 " " J - W { s T n t

W

If a, 6 and k are positive, although with certain limitations they may be complex, then it is shown in reference [2] that exp{iJby/(q2 + 62)} _ f°° e x p { - q y V - k2)} V V + b2 )

~ Jo

^(s2 - P)

Jo(bs)sds.

(1.6.9)

This integral has a branch point at s — &, and the path of integration must be indented to avoid it. An e-prescription is used in which the real valued k is given a small imaginary part, viz. k = k(\ +ie). This lifts the the branch point above the real s axis, or equivalently indents the contour below the real axis. When s > k the positive value of y/(s2 — k2) is implied. A related integral is given in

Acoustics of Underwater Structures

22

reference [3] as expii^y^TP)} \/(a2 + b2)

i p ^ J-oo

=

H

^(avW^))exp(1S6)dS,

(1.6.10)

where 0 < arg y/{k2 — s2) < 7r, 0 < arg(fc) < 7r and a and 6 are real. A particular case of the Hankel integrals discussed in reference [2] is

Jo

Jn(sa)Jn(sb) s2 ~ 7 2

=

^ ^

f (m/2)Jn(ya)Eln\jb), for b > a, 1 (7ri/2)J n (76)Hi 1) ( 7 a), for a > 6,

with Im(7) > 0 and Re(7) > 0. This formula is valid for both positive and negative values of n. The integration is facilitated by extending the range of integration to the entire real axis by using the identity Jn(x) = i { I # > ( * ) - (-iru£X-x)},

(1.6.12)

which is obtained from the relation J n (ic) = (l/2){Hn (x) + Wn (x)} and the third of Eq. (1.6.3). First, for b > a, take x = bs and after some re-arrangement of terms the integral r + °° ,

z

J-oo

Jnisa^isb) * —y

results. The integration path is initially along the upper edge of the branch cut extending from s = — oo to 5 = 0, and then along the positive real axis extending from s — 0 to s — oo. The e-prescription, viz. 7 = 7(1 + e), lifts the pole at s = + 7 above the real axis, when Im(7) = 0; the pole at s = —7 does not contribute as it is moved below the real axis. This is equivalent to indenting the s integration path, on the positive real axis, below the real pole. An indentation at the origin is necessary because of the singularity of the Hankel function. When this function is replaced by the first term of its small argument expansion, viz. (2i/7r) ln(s6) for n = 0 and ((n — l)!/i7r)(2/s&)n for n > 0, the contribution from the indentation at the origin can be shown to vanish. The residue theorem then allows evaluation of the integral as the first of Eq. (1.6.11). The procedure is much the same when a > 6; take x — as etc., to end up with the second of Eq. (1.6.11).

Mathematical Methods

23

T h e Hankel transform stems from an extension of Neumann's integral by Hankel, as discussed in reference [2]. For field problems in cylindrical coordi­ nates, the Hankel transform pair, f(x)

-

/ Jo

F(a)

=

/ Jo

F(a)Jn(ax)ada,

/»oo

f(x)Jn(ax)xdx,

(1.6.14)

can sometimes be used to facilitate solving the reduced Helmholtz wave equa­ tion. In other versions of this transform, certain factors of 2ir are present. The transform stems from the more general transform pair, /»oo

g{x)

=

/ Jo

G(a)

=

/

G(a)^(ax)(xa)^2day

/•OO

g(x)Ju(ax)(xa)1/2dx.

(1.6.15)

Jo

1.7

Spherical Bessel Functions T h e spherical Bessel functions cn(z)

are solutions of the homogeneous dif­

ferential equation

z 2

^

L

+ 2

^

+

{ z 2 _ n ( n +

1)}Cn =

0

'

(L7

-1}

which arises from independent separable solutions, V(R, 0, ) = A(k, n)cn(kR)Sn(cos of the reduced wave equation, V 2 \£ + k2^

0),

(1.7.2)

= 0, in spherical coordinates. The

two independent solutions of Eq. (1.7.1) are the spherical Bessel function of the first kind }n{z)

and the spherical Bessel function of the second kind y n ( z ) . For

the purposes herein, n is an integer and z any complex number with lm(z)

> 0;

but it must be noted t h a t these are not, in general, necessary conditions. Spher­ ical Bessel functions of the third kind, defined as h^ \z) z

hn (z) — jn(^) — iy n ( )>

are a

^

so

— }n(z) + lYn{z)

anc

*

independent solutions; these are the spherical

Acoustics of Underwater Structures

24

Hankel functions of the first a n d second kinds.

In later chapters, t h e sym­

bol h n is used as an abbreviated form for hn . The spherical Bessel func­ tions are related t o cylindrical Bessel functions in a straightforward way, viz. cn(z)

= >/(7r/2z)C f n + i/2(^). For analytical continuation in the complex plane,

values on the upper edge of the cut along the negative real axis are related to values on the positive real axis by the formulae jn(-x)

=

(-l)nj„(*),

yn{-x)

=

(-l)"+1y„(*),

b£\-x)

= (-l) n h«(*),

tfH-x)

= (-l)"^ 1 ^)-

(1-7-3)

T h e functions have particularly simple forms for small values of integer n, viz. . sin z Jo(z) = , z , x cos z z , yo(*) =

b£\z) = - i ^ ^ l ,

sin z cos z Ji(z) = —-2 , z1 z cos z sin z z yi(*) = - - z*

h[%) = -i(-i + l / z ) ^ M .

(L7 . 4)

T h e spherical Bessel functions satisfy the recurrence relations cn+i(z) c

=

2n+ 1 cn(2:)-cn_i(z), z

=

-Cn(z)-Cn

=

n c n _ 1 ( 2 ) - ( n + l)cn+1(2),

=

cn-1(z)-^——cn(z),

71

'n(z)

(2n + l)c'n(z) c'n(z)

+

i(z),

(1.7.5)

with c'0(z) = -Cl(z); h ( _ ^ _ i W = i(-l)nh£\z) and h(_2;>_i(s) = -i(-l)"h