Computational Acoustics

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.

Autorentext

David R. Bergman, PhD is Owner and Chief Scientist, Exact Solution Scientific Consulting LLC. He has
a PhD in physics with a specialization in General Relativity and High Energy Theory. Among other things, he has developed simulations for testing algorithms used in acoustics, modeled electromagnetic remote sensing devices, and modeled underwater and aero-acoustic propagation, acoustic propagation in transducer layers, and performed mechanical vibrational analysis in bio mechanical systems.

Inhalt

Series Preface ix

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 Paraxia...