Allan D. Pierce

Acoustics

An Introduction to Its Physical Principlesand Applications

Third Edition

^prIss ^ Springer

Contents

1 The Wave Theory of Sound 1

1.1 A Little History 3

1.2 The Conservation of Mass 6

1.3 Euler's Equation for a Fluid 8

1.4 Pressure-Density Relations 11

1.4.1 Laplace's Hypothesis 12

1.4.2 Interpretation in Terms of Entropy 13

1.4.3 Incorporation of Heat Conduction into Fluid

Dynamics 14

1.5 Equations of Linear Acoustics 15

1.6 The Wave Equation 18

1.6.1 The Velocity Potential 20

1.7 Plane Traveling Waves 21

1.7.1 Processes Occurring During Passage of a Sound

Wave 24

1.8 Waves of Constant Frequency 25

1.8.1 Time Average of a Product 27

1.8.2 Spatially Dependent Complex Amplitudes 28

1.8.3 Plane Waves of Constant Frequency 29

1.9 Speed of Sound and Ambient Density 30

1.9.1 Speed of Sound in Gases 30

1.9.2 Acoustic Properties of Liquids 32

1.10 Adiabatic Versus Isothermal Speeds 37

1.11 Energy, Intensity, and Source Power 39

1.11.1 Acoustic-Energy Corollary 39

1.11.2 Energy Conservation in Fluids 40

1.11.3 Acoustic Power of Sources 42

1.12 Spherical Waves 44

1.12.1 Spherical Spreading of Acoustic Energy 44

1.12.2 Spherically Symmetric Solution of the Wave

Equation 46

xix

xx Contents

1.12.3 Fluid Velocity in a Spherically Symmetric Wave 47

1.12.4 Intensity and Energy Density 48

1.12.5 Field at Large Distances from Source of Finite

Extent 49

1.13 Problems 51

2 Quantitative Measures of Sound 61

2.1 Frequency Content of Sounds 61

2.1.1 Frequency Bands 61

2.1.2 Frequency Partitioning of Mean Squared Pressure 62

2.1.3 Frequency Partitioning of Intensity, Acoustic

Power, and Energy Density 63

2.2 Proportional Frequency Bands 64

2.2.1 Standard Frequencies and Bands 65

2.2.2 Equally Tempered Musical Scales 65

2.3 Levels and the Decibel 68

2.3.1 Sound-Pressure Levels 68

2.3.2 Levels and Sound-Pressure Ratios 69

2.3.3 Logarithms and Antilogarithms 69

2.3.4 History of the Decibel 71

2.3.5 Intensity and Power Levels 73

2.4 Frequency Weighting and Filters 74

2.4.1 Frequency Weighting Functions 74

2.4.2 Linear Filters 75

2.5 Combining of Levels 77

2.6 Mutually Incoherent Sound Sources 80

2.7 Fourier Series and Long-Duration Sounds 83

2.7.1 Spectral Density 85

2.7.2 Levels and Spectral Density 85

2.7.3 White and Pink Noise 87

2.8 Transient Waveforms 87

2.8.1 Dirac Delta Function 89

2.8.2 Sound-Exposure Spectral Density 91

2.9 Transfer Functions 92

2.10 Stationary Ergodic Processes 95

2.10.1 Wiener-Khintchine Theorem 97

2.11 Bias and Variance 100

2.12 Problems 106

3 Reflection, Transmission, and Excitation of Plane Waves 115

3.1 Boundary Conditions at Impenetrable Surfaces 115

3.1.1 Stationary Surfaces 117

3.1.2 Vibrating Surfaces 117

3.1.3 Continuity of Normal Component of Displacement ...119

3.2 Plane-Wave Reflection at a Flat Rigid Surface 120

Contents XX1

3.3 Specific Acoustic Impedance 122

3.3.1 Plane Traveling Waves and Specific Acoustic

Impedance 124

3.3.2 Plane-Wave Reflection at a Surface with Finite

Specific Impedance 124

3.3.3 Locally Reacting Surfaces 126

3.3.4 Theory of the Impedance Tube 127

3.4 Radiation of Sound by a Vibrating Piston Within a Tube 130

3.4.1 Causality 131

3.4.2 Tube with Rigid End: Resonance 132

3.4.3 Constant-Frequency Oscillations 135

3.4.4 Tube with Impedance Boundary Condition at End 137

3.4.5 The Q of a Resonance 138

3.5 Sound Radiation by Traveling Flexural Waves 140

3.5.1 Sound Generated by Supersonic Flexural Waves 141

3.5.2 Trace-Velocity Matching Principle 141

3.5.3 Outgoing Versus Incoming Waves 143

3.5.4 Acoustic Disturbances Created by Subsonic

Flexural Waves 144

3.5.5 The Coincidence Frequency 145

3.5.6 Specific Radiation Impedance 147

3.6 Reflection and Transmission at an Interface Between

Two Fluids 148

3.6.1 Water-Air Interfaces 153

3.6.2 Transient Reflection 154

3.7 Multilayer Transmission and Reflection 156

3.8 Transmission Through Thin Solid Slabs, Plates, and Blankets...

160

3.8.1 Transmission Loss 160

3.8.2 Slab Specific Impedance 160

3.8.3 Oblique-Incidence Mass Law 163

3.8.4 Transmission Through Euler-Bernoulli Plates 164

3.8.5 Transmission Through Porous Blankets 167

3.9 Problems 168

4 Radiation from Vibrating Bodies 177

4.1 Radially Oscillating Sphere 177

4.1.1 Low-Frequency Approximation 179

4.2 Transversely Oscillating Rigid Sphere 180

4.2.1 Force Exerted by Transversely Oscillating Sphere 183

4.2.2 Small-/ca Approximation 183

4.3 Monopoles and Green's Functions 184

4.3.1 Concept of a Point Source 184

4.3.2 Point Mass Source 187

4.3.3 Green's Functions 188

xxii Contents

4.4 Dipoles and Quadrupoles 191

4.4.1 Dipoles 191

4.4.2 Point Force in a Fluid 192

4.4.3 Quadrupoles 193

4.4.4 Multipole Expansions 196

4.5 Uniqueness of Solutions of Acoustic Boundary-ValueProblems 198

4.5.1 Poisson's Theorem and Its Implications 198

4.5.2 Closed Regions 201

4.5.3 Uniqueness and Open Regions 204

4.5.4 Sommerfeld Radiation Condition 204

4.5.5 Uniqueness of Constant-Frequency Fields 206

4.6 The Kirchhoff-Helmholtz Integral Theorem 208

4.6.1 Multipole Expansions of the

Kirchhoff-Helmholtz Integral 210

4.7 Sound Radiation from Small Vibrating Bodies 211

4.8 Radiation from a Circular Disk 219

4.8.1 Oblate-Spheroidal Coordinates 220

4.8.2 Solutions of Laplace's Equation 221

4.8.3 Determination of the Outer Solution 223

4.9 Reciprocity in Acoustics 224

4.9.1 Reciprocity in Vibrating Systems 224

4.9.2 Reciprocity and the Linear Acoustic Equations 226

4.9.3 Interchange of Source and Listener 227

4.9.4 Reciprocity and Green's Functions 228

4.10 Transducers and Reciprocity 229

4.11 Problems 233

5 Radiation from Sources Near and on Solid Surfaces 241

5.1 Sources Near Plane Rigid Boundaries 241

5.1.1 Image Sources 241

5.1.2 Remarks Concerning Acoustic Power and

Spherical Spreading 242

5.1.3 Cases When More Than One Wall Is Present 2435.1.4 Dependence of Acoustic Far Field and Net

Acoustic Power Output on Distance from a Wall 244

5.2 Sources Mounted on Walls: The Rayleigh Integral;Fresnel-Kirchhoff Theory of Diffraction by an Aperture 246

5.2.1 Green's-Function Derivation of Rayleigh Integral 248

5.2.2 Fresnel-Kirchhoff Theory of Diffraction 249

5.3 Low-Frequency Radiation from Sources Mounted on Walls 251

5.3.1 Pressure on Vibrating Circular Piston at Low

Frequencies 252

5.3.2 Force Exerted by the Slowly Oscillating Baffled

Piston 254

Contents xxiii

5.4 Radiation Impedance of Baffled-Piston Radiators 255

5.4.1 Electroacoustic Significance of Radiation

Impedance 255

5.4.2 Evaluation of Radiation Impedance for a Baffled

Circular Piston 256

5.5 Far-Field Radiation from Localized Wall Vibrations 261

5.6 Transient Solution for Baffled Circular Piston 264

5.7 Field on and Near the Symmetry Axis 268

5.7.1 Field on Symmetry Axis 268

5.7.2 Field Near Symmetry Axis 270

5.8 Transition to the Far Field 271

5.8.1 Properties of the Diffraction Integral 275

5.8.2 Field Near Edge of Main Beam 276

5.8.3 Characteristic Single-Edge Diffraction Pattern 278

5.8.4 Field Far Outside the Central Beam 282

5.9 Problems 283

6 Room Acoustics 291

6.1 The Sabine-Franklin-Jaeger Theory of Reverberant Rooms 292

6.1.1 Energy-Conservation Equation for Rooms 293

6.1.2 Spatial Uniformity 293

6.1.3 Reverberation Time 295

6.1.4 Sabine's Equation 296

6.1.5 Diffuse Sound Fields 298

6.1.6 Equivalent Area of Open Windows 301

6.1.7 Absorbing Power of Objects and Persons 301

6.2 Some Modifications 303

6.2.1 Mean Free Path 303

6.2.2 Limitations of Sabine's Equation 305

6.2.3 Norris-Eyring Reverberation Time 306

6.2.4 Rooms with Asymmetric Absorption 308

6.2.5 The Room Constant 310

6.3 Applications of the Sabine-Franklin-Jaeger Theory 313

6.3.1 Design and Correction of Rooms 313

6.3.2 Measurement of Absorption Coefficients

and Reverberation Times 316

6.3.3 Measurement of Source Power 318

6.3.4 Simultaneous Conversations in a Reverberant

Room 319

6.4 Coupled Rooms and Large Enclosures 321

6.4.1 Transmission of Reverberant Sound Througha Panel 321

6.4.2 Transmission Out Through an Open Window 324

6.4.3 Theory of Large Enclosures 324

6.4.4 Coupled Rooms 326

6.4.5 Reverberant Decay in Coupled Rooms 327

xxiv Contents

6.5 The Modal Theory of Room Acoustics 328

6.5.1 The Eigenvalue Problem 329

6.5.2 Modes for a Rectangular Room 329

6.5.3 Orthogonality of Modal Eigenfunctions 331

6.5.4 Modal Expansion of Functions 332

6.5.5 Field of a Point Source in a Room with Walls

of Large Impedance 333

6.5.6 Acoustic Energy in a Room 335

6.5.7 Modal Description of Power Injection 336

6.6 High-Frequency Approximations 337

6.6.1 The Modal Density 337

6.6.2 The Schroeder CutoffFrequency 339

6.6.3 Approximation of Modal Sums by Integrals 340

6.6.4 Modal Averages of Squares of Eigenfunctions 341

6.6.5 Evaluation of Modal Integrals 342

6.7 Statistical Aspects of Room Acoustics 343

6.7.1 Frequency Correlation 344

6.7.2 The Poisson Distribution 347

6.7.3 Effect of Finite-Frequency Bandwidth 349

6.8 Spatial Correlations in Diffuse Sound Fields 351

6.8.1 The Spatial Autocorrelation Function

for Acoustic Pressure 352

6.8.2 Spatial Averaging 355

6.8.3 Frequency Averaging Versus Spatial Averaging 356

6.9 Problems 357

7 Low-Frequency Models of Sound Transmission 361

7.1 Guided Waves 361

7.1.1 Duct Cross-Sectional Eigenfunctions 361

7.1.2 Duct with Rectangular Cross Section 363

7.1.3 Duct with Circular Cross Section 363

7.1.4 Cutoff Frequencies and Evanescent Modes 364

7.1.5 Point Source in a Duct 365

7.2 Lumped-Parameter Models 367

7.2.1 Volume Velocity and Average Pressure 368

7.2.2 Acoustic Impedance 368

7.2.3 Acoustical Two-Ports 369

7.2.4 Continuous-Volume-Velocity Two-Port 371

7.2.5 Continuous-Pressure Two-Port 373

7.3 Guidelines for Selecting Lumped-Parameter Models 373

7.3.1 Continuity of Pressure 373

7.3.2 Continuity of Volume Velocity 375

7.3.3 Reflection and Transmission at a Junction 379

Contents xxv

7.4 Helmholtz Resonators and Other Examples 380

7.4.1 The Helmholtz Resonator 380

7.4.2 Helmholtz Resonator as a Side Branch 382

7.4.3 Composite Example 384

7.5 Orifices 385

7.5.1 Matched-Asymptotic-Expansion Solution

for Orifice Transmission 386

7.5.2 Acoustic-Radiation Resistance 388

7.5.3 Helmholtz Resonator with Baffled Opening 388

7.5.4 Acoustic Inertance of a Circular Orifice

in a Thin Plate 389

7.5.5 Diffraction of Plane Wave by a Circular Orifice 391

7.6 Estimation of Acoustic Inertances and End Corrections 392

7.6.1 Principle of Minimum Kinetic Energy 392

7.6.2 Principle of Minimum Acoustic Inertance 394

7.6.3 Effect of Relaxing of Constraints 394

7.6.4 Lower Bound for Acoustic Inertance 395

7.6.5 Flanged Opening in a Duct 397

7.6.6 Circular Orifice in a Plate of Finite Thickness 399

7.6.7 End Corrections 399

7.6.8 Effective Neck Lengths of Helmholtz Resonators 400

7.6.9 Boundary Conditions at Open Ends of Ducts 401

7.7 Mufflers and Acoustic Filters 403

7.7.1 The Transmission Matrix and Its Consequences 403

7.7.2 Insertion Loss 404

7.7.3 Reactive and Dissipative Mufflers 405

7.7.4 Helmholtz Resonators as Filters 406

7.7.5 Expansion-Chamber Muffler 407

7.7.6 Commercial Muffler Designs 408

7.8 Horns 410

7.8.1 The Webster Horn Equation 413

7.8.2 Salmon's Family of Horns 414

7.8.3 Concept of a Semi-Infinite Horn 415

7.8.4 The Cutoff Frequency 416

7.8.5 Other Considerations in Horn Design 417

7.9 Problems 419

8 Ray Acoustics 427

8.1 Wavefronts, Rays, and Fermat's Principle 427

8.1.1 Ray Paths in Moving Media 427

8.1.2 Fermat's Principle 432

8.2 Rectilinear Sound Propagation 435

8.2.1 Parametric Description of Wavefronts 436

8.2.2 Variation of Principal Radii of Curvature

Along a Ray 437

8.2.3 Caustics 438

xxvjContents

8.3 Refraction in Inhomogeneous Media 4418.3.1 Refraction by Sound-Speed Gradients 441

8.3.2 Rays in a Medium with Constant-Sound-SpeedGradient 443

8.3.3 Refraction by Wind Gradients 444

8.4 Rays in Stratified Media 446

8.4.1 The Ray Integrals 447

8.4.2 Channeling of Ray Paths 447

8.4.3 Rays in Fluids Without Ambient Flow 449

8.4.4 Abnormal Sound 4518.5 Amplitude Variation Along Rays 455

8.5.1 Wave Amplitudes in Homogeneous Media 455

8.5.2 Energy Conservation Along Rays 4588.6 Wave Amplitudes in Moving Media 459

8.6.1 Linear Acoustics Equations for Moving Media 459

8.6.2 Conservation of Wave Action 461

8.6.3 The Blokhintzev Invariant 4658.7 Source Above an Interface 468

8.7.1 Sound Field Above the Interface 4698.7.2 Field Below the Interface 471

8.8 Reflection from Curved Surfaces 4738.8.1 General Geometrical Considerations 4738.8.2 Ray-Tube Area After Reflection 4778.8.3 Echoes from Curved Surfaces 4788.8.4 Sound Beam Incident on a Sphere 478

8.9 Problems 480

9 Scattering and Diffraction 4879.1 Basic Scattering Concepts 488

9.1.1 Scattering by a Rigid Object 488

9.1.2 Scattering Cross Section 4919.1.3 Higher-Frequency Scattering 4939.1.4 Scattering by Inhomogeneities 4949.1.5 Spherical Inhomogeneity 4979.1.6 Inertia Effect for Freely Suspended Particle 4989.1.7 Resonant Scattering 499

9.2 Monostatic and Bistatic Scattering: Measurement

Configurations 5039.2.1 Monostatic Pulse-Echo Sounding 5049.2.2 Inhomogeneities and the Born Approximation 5069.2.3 Scattering Volumes Delimited by Electroacoustic

Transducers 5089.2.4 Acoustic Radar Equation 5119.2.5 Incoherent Scattering 5129.2.6 The Echosonde Equation 513

Contents xxvii

9.3 The Doppler Effect 517

9.3.1 Doppler Shift for a Moving Source 518

9.3.2 Galilean Transformations 520

9.3.3 Echoes from Moving Targets 521

9.3.4 Doppler-Shift Velocimeters 523

9.4 Acoustic Fields Near Caustics 527

9.4.1 The Airy Function 530

9.4.2 Generalization to Inhomogeneous Media 532

9.4.3 Field Near a Turning Point 534

9.4.4 Phase Shift at a Caustic 535

9.5 Shadow Zones and Creeping Waves 537

9.5.1 Point Source Above a Locally Reacting Surface

in a Stratified Medium 538

9.5.2 Residues Series for the Shadow Zone 541

9.5.3 Creeping Waves 544

9.5.4 Ray Shedding by a Creeping Wave 546

9.6 Source or Listener on the Edge of a Wedge 547

9.6.1 Source on Edge 548

9.6.2 Listener on the Edge 549

9.7 Contour-Integral Solution for Diffraction by a Wedge 550

9.7.1 Method of Images for Integer Wedge Index 552

9.7.2 Generalization to Noninteger Wedge Indices 554

9.8 Geometrical-Acoustic and Diffracted-Wave Contributions

for the Wedge Problem 556

9.8.1 The Geometrical Acoustics Portion of the Field 556

9.8.2 The Diffracted Wave 559

9.8.3 Asymptotic Expression for the Diffracted Wave 560

9.8.4 Physical Interpretation of the Diffracted Wave 562

9.9 Applications of Wedge-Diffraction Theory 566

9.9.1 Insertion Loss of Single-Edged Barriers 566

9.9.2 Far Field of a Source on the Side of a Building 569

9.9.3 Backscattering from an Edge 571

9.10 Problems 573

10 Effects of Viscosity and Other Dissipative Processes 583

10.1 The Navier-Stokes-Fourier Model 583

10.1.1 The Stress Tensor 583

10.1.2 The Energy Equation 585

10.1.3 Constitutive Relations for a Fluid 586

10.1.4 Values of Viscosity and Thermal Conductivity 589

10.1.5 Transport Properties of Water 589

10.2 Linear Acoustic Equations and Energy Dissipation 590

10.2.1 Linear Acoustic Equations 590

10.2.2 The Energy Conservation-Dissipation Corollary 591

10.2.3 Attenuation of Plane Sound Waves 593

xxviii Contents

10.3 Vorticity, Entropy, and Acoustic Modes 595

10.3.1 Dispersion Relations for the Component Modes 595

10.3.2 A Generalization Based on the SuperpositionPrinciple 598

10.3.3 Vorticity Mode 598

10.3.4 Acoustic Mode 599

10.3.5 The Entropy Mode 599

10.4 Acoustic Boundary-Layer Theory 600

10.4.1 Boundary Conditions at a Solid Surface 601

10.4.2 Vorticity- and Entropy-Mode Fields Near a Solid

Surface 602

10.4.3 Boundary Condition on the Acoustic-Mode Field 604

10.4.4 Energy Loss from the Acoustic Mode at a Boundary.... 605

10.4.5 Plane-Wave Reflection at a Solid Surface 606

10.5 Attenuation and Dispersion in Ducts and Thin Tubes 609

10.5.1 Propagation in Wide Ducts 609

10.5.2 Propagation in Narrow Tubes 612

10.5.3 Slab with Circular Pores 615

10.6 Viscosity Effects on Sound Radiation 616

10.6.1 Revision of the Kirchhoff-Helmholtz Theorem 617

10.6.2 Transversely Oscillating Rigid Bodies 618

10.6.3 Stokes Flow Limit 619

10.6.4 Thin-Boundary-Layer Approximation 620

10.6.5 Gutin's Principle 622

10.6.6 Helicopter Rotor Noise 623

10.7 Relaxation Processes 627

10.7.1 Partitioning of Internal Energy 627

10.7.2 The Bulk Viscosity 629

10.7.3 Instantaneous Entropy Function 630

10.7.4 Fluid-Dynamic Equations with Relaxation Included... 631

10.7.5 The Relaxation Equations 633

10.7.6 Numerical Values for the Constants of the Model 634

10.8 Absorption of Sound 636

10.8.1 Linear Acoustic Equations for Air 636

10.8.2 Energy Corollary 637

10.8.3 Dispersion Relation for Plane Traveling Waves 638

10.8.4 Absorption by Relaxation Processes 639

10.8.5 Phase-Velocity Changes Due to Relaxation

Processes 643

10.9 Problems 644

11 Nonlinear Effects in Sound Propagation 649

11.1 Nonlinear Steepening 649

11.1.1 Plane Waves in Homogeneous Media 650

11.1.2 Steepening of Waveforms 653

Contents xxix

11.2 Generation of Harmonics 654

11.2.1 Fourier-Series Representation 655

11.2.2 Conservation ofEnergy 657

11.3 Weak-Shock Theory 658

11.3.1 The Rankine-Hugoniot Relations 658

11.3.2 Weak Shocks 661

11.3.3 The Equal-Area Rule 662

11.4 N Waves and Anomalous Energy Dissipation 664

11.4.1 Plane-Wave Propagation of an N Wave 664

11.4.2 Dissipation of Acoustic Energy 666

11.5 Evolution of Sawtooth Waveforms 667

11.6 Nonlinear Dissipative Waves 672

11.6.1 Approximate Equations for Transient Plane Waves.... 673

11.6.2 Modification to Include Nonlinear Effects 674

11.6.3 The Burgers Equation 674

11.6.4 Rise Times and Thicknesses of Weak Shocks 676

11.6.5 Relaxation Effects on Shock Structure 677

11.7 Transition to Old Age 680

11.7.1 Reduction to the Linear Diffusion Equation 681

11.7.2 Solution of the Boundary-Value Problem 681

11.7.3 Relative Importances of Nonlinear and

Dissipative Effects 683

11.7.4 Transition from Sawtooth to Old Age 683

11.8 Nonlinear Effects in Converging and Diverging Waves 685

11.8.1 Corrected Travel Time Along Ray Path 685

11.8.2 Weak Shocks 687

11.8.3 Asymptotic Form of a Transient Pulse 690

11.9 N Waves in Inhomogeneous Media: Spherical Waves 691

11.9.1 N-Wave Propagation 691

11.9.2 Waves with Spherical Spreading 692

11.10 Ballistic Shocks: Sonic Booms 694

11.10.1 Linear Acoustic Model for Sound Generation

by a Moving Body 695

11.10.2 Geometrical-Acoustics Interpretation 698

11.10.3 Nonlinear Modifications 700

11.11 Problems 704

Appendix: Answers and Hints to Problems 711

Name Index 743

Subject Index 753