Computional Acoustics: Theory & Implementation

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 Paraxial Approximation 199 9.3 Operator Factoring 201 9.4 Pauli Spin Matrices 204 9.5 Reduction of Order 205 9.5.1 The Pade 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

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