# Computational Acoustics

## Theory and Implementation

Wiley Series in Acoustics Noise and Vibration

1. Edition February 2018

296 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-27728-6**

Covers the theory and practice of innovative new approaches to modelling acoustic propagation

There are as many types of acoustic phenomena as there are media, from longitudinal pressure waves in a fluid to S and P waves in seismology. This text focuses on the application of computational methods to the fields of linear acoustics. Techniques for solving the linear wave equation in homogeneous medium are explored in depth, as are techniques for modelling wave propagation in inhomogeneous and anisotropic fluid medium from a source and scattering from objects.

Written for both students and working engineers, this book features a unique pedagogical approach to acquainting readers with innovative numerical methods for developing computational procedures for solving problems in acoustics and for understanding linear acoustic propagation and scattering. Chapters follow a consistent format, beginning with a presentation of modelling paradigms, followed by descriptions of numerical methods appropriate to each paradigm. Along the way important implementation issues are discussed and examples are provided, as are exercises and references to suggested readings. Classic methods and approaches are explored throughout, along with comments on modern advances and novel modeling approaches.

* Bridges the gap between theory and implementation, and features examples illustrating the use of the methods described

* Provides complete derivations and explanations of recent research trends in order to provide readers with a deep understanding of novel techniques and methods

* Features a systematic presentation appropriate for advanced students as well as working professionals

* References, suggested reading and fully worked problems are provided throughout

An indispensable learning tool/reference that readers will find useful throughout their academic and professional careers, this book is both a supplemental text for graduate students in physics and engineering interested in acoustics and a valuable working resource for engineers in an array of industries, including defense, medicine, architecture, civil engineering, aerospace, biotech, and more.

1 Introduction 1

2 Computation and Related Topics 5

2.1 Floating-Point Numbers 5

2.1.1 Representations of Numbers 5

2.1.2 Floating-Point Numbers 7

2.2 Computational Cost 9

2.3 Fidelity 11

2.4 Code Development 12

2.5 List of Open-Source Tools 16

2.6 Exercises 17

References 17

3 Derivation of the Wave Equation 19

3.1 Introduction 19

3.2 General Properties of Waves 20

3.3 One-Dimensional Waves on a String 23

3.4 Waves in Elastic Solids 26

3.5 Waves in Ideal Fluids 29

3.5.1 Setting Up the Derivation 29

3.5.2 A Simple Example 30

3.5.3 Linearized Equations 31

3.5.4 A Second-Order Equation from Differentiation 33

3.5.5 A Second-Order Equation from a Velocity Potential 34

3.5.6 Second-Order Equation without Perturbations 36

3.5.7 Special Form of the Operator 36

3.5.8 Discussion Regarding Fluid Acoustics 40

3.6 Thin Rods and Plates 41

3.7 Phonons 42

3.8 Tensors Lite 42

3.9 Exercises 48

References 48

4 Methods for Solving the Wave Equation 49

4.1 Introduction 49

4.2 Method of Characteristics 49

4.3 Separation of Variables 56

4.4 Homogeneous Solution in Separable Coordinates 57

4.4.1 Cartesian Coordinates 58

4.4.2 Cylindrical Coordinates 59

4.4.3 Spherical Coordinates 61

4.5 Boundary Conditions 63

4.6 Representing Functions with the Homogeneous Solutions 67

4.7 Green's Function 70

4.7.1 Green's Function in Free Space 70

4.7.2 Mode Expansion of Green's Functions 72

4.8 Method of Images 76

4.9 Comparison of Modes to Images 81

4.10 Exercises 82

References 82

5 Wave Propagation 85

5.1 Introduction 85

5.2 Fourier Decomposition and Synthesis 85

5.3 Dispersion 88

5.4 Transmission and Reflection 90

5.5 Attenuation 96

5.6 Exercises 97

References 97

6 Normal Modes 99

6.1 Introduction 99

6.2 Mode Theory 100

6.3 Profile Models 101

6.4 Analytic Examples 105

6.4.1 Example 1: Harmonic Oscillator 105

6.4.2 Example 2: Linear 108

6.5 Perturbation Theory 110

6.6 Multidimensional Problems and Degeneracy 118

6.7 Numerical Approach to Modes 120

6.7.1 Derivation of the Relaxation Equation 120

6.7.2 Boundary Conditions in the Relaxation Method 125

6.7.3 Initializing the Relaxation 127

6.7.4 Stopping the Relaxation 128

6.8 Coupled Modes and the Pekeris Waveguide 129

6.8.1 Pekeris Waveguide 129

6.8.2 Coupled Modes 131

6.9 Exercises 135

References 135

7 Ray Theory 137

7.1 Introduction 137

7.2 High Frequency Expansion of the Wave Equation 138

7.2.1 Eikonal Equation and Ray Paths 139

7.2.2 Paraxial Rays 140

7.3 Amplitude 144

7.4 Ray Path Integrals 145

7.5 Building a Field from Rays 160

7.6 Numerical Approach to Ray Tracing 162

7.7 Complete Paraxial Ray Trace 168

7.8 Implementation Notes 170

7.9 Gaussian Beam Tracing 171

7.10 Exercises 173

References 174

8 Finite Difference and Finite Difference Time Domain 177

8.1 Introduction 177

8.2 Finite Difference 178

8.3 Time Domain 188

8.4 FDTD Representation of the Linear Wave Equation 193

8.5 Exercises 197

References 197

9 Parabolic Equation 199

9.1 Introduction 199

9.2 The Paraxial Approximation 199

9.3 Operator Factoring 201

9.4 Pauli Spin Matrices 204

9.5 Reduction of Order 205

9.5.1 The Padé Approximation 207

9.5.2 Phase Space Representation 208

9.5.3 Diagonalizing the Hamiltonian 209

9.6 Numerical Approach 210

9.7 Exercises 212

References 212

10 Finite Element Method 215

10.1 Introduction 215

10.2 The Finite Element Technique 216

10.3 Discretization of the Domain 218

10.3.1 One-Dimensional Domains 218

10.3.2 Two-Dimensional Domains 219

10.3.3 Three-Dimensional Domains 222

10.3.4 Using Gmsh 223

10.4 Defining Basis Elements 225

10.4.1 One-Dimensional Basis Elements 226

10.4.2 Two-Dimensional Basis Elements 227

10.4.3 Three-Dimensional Basis Elements 229

10.5 Expressing the Helmholtz Equation in the FEM Basis 232

10.6 Numerical Integration over Triangular and Tetrahedral Domains 234

10.6.1 Gaussian Quadrature 234

10.6.2 Integration over Triangular Domains 235

10.6.3 Integration over Tetrahedral Domains 239

10.7 Implementation Notes 240

10.8 Exercises 240

References 241

11 Boundary Element Method 243

11.1 Introduction 243

11.2 The Boundary Integral Equations 244

11.3 Discretization of the BIE 249

11.4 Basis Elements and Test Functions 253

11.5 Coupling Integrals 254

11.5.1 Derivation of Coupling Terms 254

11.5.2 Singularity Extraction 256

11.5.3 Evaluation of the Singular Part 260

11.5.3.1 Closed-Form Expression for the Singular Part of K 260

11.5.3.2 Method for Partial Analytic Evaluation 261

11.5.3.3 The Hypersingular Integral 266

11.6 Scattering from Closed Surfaces 267

11.7 Implementation Notes 269

11.8 Comments on Additional Techniques 271

11.8.1 Higher-Order Methods 271

11.8.2 Body of Revolution 272

11.9 Exercises 273

References 273

Index 275