FUNDAMENTALS OF PHYSICAL ACOUSTICS

DAVID T. BLACKSTOCK

University of Texas Austin, Texas

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

New York • Chichester • Weinheim • Brisbane • Singapore • Toronto

CONTENTS

Preface xix

Chapter 1 Introduction 1

A. What Is a Wave? 1

B. Plane Waves: Some Basic Solutions 3

1. General Solution of the Wave Equation 4

2. Free Waves 10

3. Forced Waves 14

4. Relation between Derivatives for a Progressive Wave 16

C. Derivation of Wave Equations. Impedance 18

1. Electrical Transmission Line 18

2. Waves on a String 22

3. Sound Waves 27

D. Spherical and Cylindrical Sound Waves of One Dimension 39

1. Three-Dimensional Wave Equation 40

2. Solutions for One-Dimensional Waves 41

3. Sound from a Pulsating Sphere 42

vii

viii CONTENTS

E. Signals, Impedance, Intensity and Power, and Levels 44

1. Time and Frequency Domains 44

2. Impedance 46

3. Intensity and Sound Power 48

4. Sound Pressure Level and Other Levels 51

References 55

Problems 55

Chapter 2 Detaiied Development of the Acoustical Wave Equation 65

A. Conservation Equations and Constitutive Relation 65

1. Equation of Continuity 65

2. Momentum Equation 69

3. Energy Equation 77

4. Equation of State (Constitutive Relation) 80

5. Entropy Equation 82

6. Summary and Discussion 83

B. Nonlinear Wave Equation 84

1. Introduction 84

2. Plane Progressive Waves of Finite Amplitude 86

3. Second-Harmonic Distortion 89

4. Sum- and Difference-Frequency (Intermodulation) Distortion 91

C. Small-Signal Wave Equation 91

1. Lossless Medium at Rest 91

2. Lossless Medium Moving with Constant Velocity 93

3. Lossless Medium in a Gravitational Field 95

4. Viscous Fluid 96

5. Viscous, Thermally Conducting Fluid 97

6. Relaxing Fluid 98

References - 98

Problems 99

CONTENTS ix

Chapter 3 Reflection and Transmission of Normally Incident Plane Waves of Arbitrary Waveform 108

A. Reflection and Transmission Coefficients for an Interface between Two Ideal Fluids 108

1. Pressure Signals • 109

2. Sound Power 111

3. Transmission Loss 112

B. Special Cases 112

1. Rigid Wall 112

2. Pressure Release Surface 113

3. Matched Impedance Interface 114

C. Change in Cross-Sectional Area 114

D. Examples 116

1. Rectangular Pulse in an Air-Filled Tube of Finite Length 117

2. ShockTube 119

3. Bursting Balloon 121

References 125

Problems 125

Chapter 4 Normal Incidence Continued: Steady-State Analysis 130

A. Introduction 130

B. Single Impedance Termination 134

1. Pressure Release Termination (Z„ = 0) 134

2. Rigid Termination (Z„ = oo) 138

3. General Resistive Termination 139

4. General Impedance Termination 140

5. Change in Cross-Sectional Area 144

C. Lumped-Element Approximation 144

1. Electrical Analogs 144

X CONTENTS

2. Short Closed Cavity 145

3. End Correction for an Open Tube 151

4. Short Open Cavity 153

5. Helmholtz Resonator 153

6. Orifice 156

D. Examples 156

1. Side Branches, Filters 156

2. Probe Tube Microphone 160

E. Three-Medium Problems 163

1. Three Different Media, Constant Cross Section 163

2. Cross Sections Different for the Three Media 168

3. Sound Power Reflection and Transmission Coefficients 170

F. Wall Transmission Loss: Lumped-Element Approach 171

References 173

Problems 174

Chapter 5 Transmission Phenomena: Oblique Incidence 186

A. Simple Derivation of Snell's Law and Specular Reflection 186

B. Plane Interface Separating Two Fluids 189

1. Alternative Derivation of Snell's Law; R, T, and r Coefficients 190

2. Special Cases 193

C. Transmission through Panels at Oblique Incidence 198

1. Transmission Dominated by Panel Mass: The Mass Law 200

2. Panel Stiffness: The Coincidence Effect 203

D. Composite Walls 208

References 211

Problems 211

CONTENTS XI

Chapter 6 Normal Modes in Cartesian Coordinates: Strings, Membranes, Rooms, and Rectangular Waveguides 218

A. Vibrating String (and Other One-Dimensional Problems)

1. String with Fixed Ends

2. Other Boundary Conditions

3. The Struck String

B. Vibrating Membrane

C. Sound in a Rectangular Enclosure

D. Rectangular Waveguide

1. Membrane Waveguide

2. Forward Traveling Waves, Phase Velocity, and Cutoff

3. Physical Interpretation

4. Source Conditions

References

Problems

218

219

226

227

229

233

236

236

238

240

242

243

244

Chapter 7 Horns 250

A. Webster Hörn Equation 251

1. Continuity Equation 251

2. Momentum Equation 252

3. Webster Hörn Equation 254

B. Example: Exponential Hörn 254

1. Exponential Hörn Equation and Solution 254

2. Amplitude Decay and Phase Velocity 255

C. Impedance, Power Transmitted, and Transmission Factor 257

1. Impedance and Power 258

2. Conical Hörn 259

3. Transmission Factor 260

xii CONTENTS

D. More General Approach: WKB Method 260

1. Application to the Hörn Equation: Direct Approach 262

2. Modified Approach 263

3. Impedance and Transmission Factor 264

4. Examples 265

E. Hörn Duals 266

References 267

Problems 268

Chapter 8 Propagation in Stratified Media 273

A. Static Properties of the Atmosphere and the Ocean 274

1. Atmosphere 274

2. Ocean 276

B. Vertical Propagation of Plane Waves 278

1. One-Dimensional Wave Equation 278

2. Vertical Propagation through an Isothermal Atmosphere 280

3. General Solution by Means of the WKB Method 281

C. Ray Theory 284

1. Ray Paths 284

2. Rays in a Fluid Having a Linear Sound Speed Profile 288

3. Time of Travel along a Ray Path 292

References 294

Problems 294

Chapter 9 Propagation in Dissipative Fluids: Absorption and Dispersion 298

A. Introduction 298

B. Viscosity and Heat Conduction 303

1. Viscous Fluids 304

2. Thermally Conducting Fluids 306

CONTENTS xui

3. Thermoviscous Fluids 313

C. Relaxation 315

1. Introduction 315

2. Equation of State 317

3. Wave Equation 318

4. Dispersion Relation 318

D. Boundary-Layer Absorption (and Dispersion) 322

1. Physical Phenomenon: Viscous Boundary Layer 322

2. Thermal Boundary Layer 323

3. Effect of the Two Boundary Layers 324

E. Summary of Sound Absorption in Fluids 325

1. Viscous Fluids 325

2. Thermally Conducting Fluids 326

3. Thermoviscous Fluids 326

4. Relaxing Fluids 326

5. Boundary-Layer Absorption: Thermoviscous Fluids 327

References 327

Problems 328

Chapter 10 Spherical Waves 335

A. Introduction 335

B. Solution by Separation of Variables 337

1. Legendre Polynomials 338

2. Spherical Bessel Functions 341

3. Spherical Hankel Functions 344

4. Summary of Solutions for Axially Symmetrie Wave Motion 345

5. Most General Spherical Waves; Spherical Harmonics 346

6. Example: Bipolar Pulsating Sphere 349

C. Standing Spherical Waves: Enclosure Problems 352

xiv CONTENTS

1. Pressure Release Sphere 353

2. Hollow Sphere 355

D. Radiation Problems 356

1. Introduction: Multipole Expansion 356

2. Monopoles 358

3. Dipoles 367

References 375

Problems 376

Chapter 11 Cylindrical Waves 386

A. Solution of the Wave Equation in Cylindrical Coordinates 386

1. Solution by Separation of Variables 387

2. Properties of Bessel Functions 389

B. Circular Membrane 398

1. Introduction 398

2. Example: Membrane with Uniform Initial Displacement 400

3. Variations 401

C. Three-Dimensional Cylindrical Coordinates

1. Enclosure Problems

2. Radiation Problems

References

Problems

Waveguides

A. Introduction

B. Rectangular Waveguide

1. General Solution

2. Source Conditions and Mode Excitation

3. Example

4. Pressure Release Walls

404

404

407

413

413

420

420

421

421

423

424

427

CONTENTS xv

5. Phase and Group Velocity 427

C. Cylindrical Waves in Waveguides 428

1. Cylindrical Tube 429

2. Parallel Planes 430

References 432

Problems 433

Chapter 13 Radiation from a Baffled Piston 440

A. General Solution: The Rayleigh Integral 441

1. Time-Harmonic Piston Vibration 442

2. Example: Ring Piston 442

3. Circular Piston (Disk) 445

B. Farfleld Radiation 446

1. Rayleigh Distance 447

2. Size of ka > 449

3. First Null, Minor-Lobe Suppression, Beamwidth, and Phase 450

4. Intensity, Power, and Source Level 451

C. Pressure Field on the Axis 452

1. Transition to the Farfleld 454

2. Nearfield Structure 454

3. Intensity 455

D. Pressure on the Face of the Piston 457

E. Transient Radiation from a Piston 460

1. Signal on the Axis 460

2. Farfleld 461

F. Nonuniform Piston 463

References 465

Problems 465

xvi CONTENTS

Chapter 14 Diffraction 472

A. Introduction 472

B. Helmholtz-Kirchhoff Integral Theorem 473

1. Derivation 473

2. Time Domain Version 475

C. Circular Aperture 476

1. Plane Wave Normally Incident on a Circular Aperture 477

2. Spherical Wave Incident on a Circular Aperture 480

D. Reflection by a Rigid Disk 481

E. Babinet's Principle 486

References 489

Problems 489

Chapter 15 Arrays 495

A. Directivity: Nomenclature and Definitioiis 495

B. Array of Two Point Sources 499

C. Array of N Point Sources 502

D. Continuous Line Array 504

E. Array of Directional Sources: Product Theorem 506

References 506

Problems 506

Appendix A Elastic Constants, Velocity of Sound, and Characteristic

Impedance 510

Appendix B Absorption Formulas for the Atmosphere and Ocean 513

1. Atmosphere 513

2. Ocean 516

References 518

Appendix C Absorption due to Tube Wall Boundary-Layer Effects 519

1. Viscous Boundary Layer 520

2. Quasi-Plane-Wave Equation 524

CONTENTS xvu

3. Effect of the Thermal Boundary Layer 525

References 525

Appendix D Solution of Legendre's Equation by Power Series 526

Appendix E Directivity and Impedance Functions for a Circular Piston 528

Index 531