Fundamentals of Acoustics

Preface 13

Chapter 1. Equations of Motion in Non-dissipative Fluid 15

1.1. Introduction 15

1.1.1. Basic elements 15

1.1.2. Mechanisms of transmission 16

1.1.3. Acoustic motion and driving motion 17

1.1.4. Notion of frequency 17

1.1.5. Acoustic amplitude and intensity 18

1.1.6. Viscous and thermal phenomena 19

1.2. Fundamental laws of propagation in non-dissipative fluids 20

1.2.1. Basis of thermodynamics 20

1.2.2. Lagrangian and Eulerian descriptions of fluid motion 25

1.2.3. Expression of the fluid compressibility: mass conservation law 27

1.2.4. Expression of the fundamental law of dynamics: Euler’s equation 29

1.2.5. Law of fluid behavior: law of conservation of thermomechanic energy 30

1.2.6. Summary of the fundamental laws 31

1.2.7. Equation of equilibrium of moments 32

1.3. Equation of acoustic propagation 33

1.3.1. Equation of propagation 33

1.3.2. Linear acoustic approximation 34

1.3.3. Velocity potential 38

1.3.4. Problems at the boundaries 40

1.4. Density of energy and energy flow, energy conservation law 42

1.4.1. Complex representation in the Fourier domain 42

1.4.2. Energy density in an “ideal” fluid 43

1.4.3. Energy flow and acoustic intensity 45

1.4.4. Energy conservation law 48

Chapter 1: Appendix. Some General Comments on Thermodynamics 50

A.1. Thermodynamic equilibrium and equation of state 50

A.2. Digression on functions of multiple variables (study case of two variables) 51

A.2.1. Implicit functions 51

A.2.2. Total exact differential form 53

Chapter 2. Equations of Motion in Dissipative Fluid 55

2.1. Introduction 55

2.2. Propagation in viscous fluid: Navier-Stokes equation 56

2.2.1. Deformation and strain tensor 57

2.2.2. Stress tensor 62

2.2.3. Expression of the fundamental law of dynamics 64

2.3. Heat propagation: Fourier equation 70

2.4. Molecular thermal relaxation 72

2.4.1. Nature of the phenomenon 72

2.4.2. Internal energy, energy of translation, of rotation and of vibration of molecules 74

2.4.3. Molecular relaxation: delay of molecular vibrations 75

2.5. Problems of linear acoustics in dissipative fluid at rest 77

2.5.1. Propagation equations in linear acoustics 77

2.5.2. Approach to determine the solutions 81

2.5.3. Approach of the solutions in presence of acoustic sources 84

2.5.4. Boundary conditions 85

Chapter 2: Appendix. Equations of continuity and equations at the thermomechanic discontinuities in continuous media 93

A.1. Introduction 93

A.1.1. Material derivative of volume integrals 93

A.1.2. Generalization 96

A.2. Equations of continuity 97

A.2.1. Mass conservation equation 97

A.2.2. Equation of impulse continuity 98

A.2.3. Equation of entropy continuity 99

A.2.4. Equation of energy continuity 99

A.3. Equations at discontinuities in mechanics 102

A.3.1. Introduction 102

A.3.2. Application to the equation of impulse conservation 103

A.3.3. Other conditions at discontinuities 106

A.4. Examples of application of the equations at discontinuities in mechanics: interface conditions 106

A.4.1. Interface solid – viscous fluid 107

A.4.2. Interface between perfect fluids 108

A.4.3 Interface between two non-miscible fluids in motion 109

Chapter 3. Problems of Acoustics in Dissipative Fluids 111

3.1. Introduction 111

3.2. Reflection of a harmonic wave from a rigid plane 111

3.2.1. Reflection of an incident harmonic plane wave 111

3.2.2. Reflection of a harmonic acoustic wave 115

3.3. Spherical wave in infinite space: Green’s function 118

3.3.1. Impulse spherical source 118

3.3.2. Green’s function in three-dimensional space 121

3.4. Digression on two- and one-dimensional Green’s functions in non-dissipative fluids 125

3.4.1. Two-dimensional Green’s function 125

3.4.2. One-dimensional Green’s function 128

3.5. Acoustic field in “small cavities” in harmonic regime 131

3.6. Harmonic motion of a fluid layer between a vibrating membrane and a rigid plate, application to the capillary slit 136

3.7. Harmonic plane wave propagation in cylindrical tubes: propagation constants in “large” and “capillary” tubes 141

3.8. Guided plane wave in dissipative fluid 148

3.9. Cylindrical waveguide, system of distributed constants 151

3.10. Introduction to the thermoacoustic engines (on the use of phenomena occurring in thermal boundary layers) 154

3.11. Introduction to acoustic gyrometry (on the use of the phenomena occurring in viscous boundary layers) 162

Chapter 4. Basic Solutions to the Equations of Linear Propagation in Cartesian Coordinates 169

4.1. Introduction 169

4.2. General solutions to the wave equation 173

4.2.1. Solutions for propagative waves 173

4.2.2. Solutions with separable variables 176

4.3. Reflection of acoustic waves on a locally reacting surface 178

4.3.1. Reflection of a harmonic plane wave 178

4.3.2. Reflection from a locally reacting surface in random incidence 183

4.3.3. Reflection of a harmonic spherical wave from a locally reacting plane surface 184

4.3.4. Acoustic field before a plane surface of impedance Z under the load of a harmonic plane wave in normal incidence 185

4.4. Reflection and transmission at the interface between two different fluids 187

4.4.1. Governing equations 187

4.4.2. The solutions 189

4.4.3. Solutions in harmonic regime 190

4.4.4. The energy flux 192

4.5. Harmonic waves propagation in an infinite waveguide with rectangular cross-section 193

4.5.1. The governing equations 193

4.5.2. The solutions 195

4.5.3. Propagating and evanescent waves 197

4.5.4. Guided propagation in non-dissipative fluid 200

4.6. Problems of discontinuity in waveguides 206

4.6.1. Modal theory 206

4.6.2. Plane wave fields in waveguide with section discontinuities 207

4.7. Propagation in horns in non-dissipative fluids 210

4.7.1. Equation of horns 210

4.7.2. Solutions for infinite exponential horns 214

Chapter 4: Appendix. Eigenvalue Problems, Hilbert Space 217

A.1. Eigenvalue problems 217

A.1.1. Properties of eigenfunctions and associated eigenvalues 217

A.1.2. Eigenvalue problems in acoustics 220

A.1.3. Degeneracy 220

A.2. Hilbert space 221

A.2.1. Hilbert functions and L 2 space 221

A.2.2. Properties of Hilbert functions and complete discrete ortho-normal basis 222

A.2.3. Continuous complete ortho-normal basis  223

Chapter 5. Basic Solutions to the Equations of Linear Propagation in Cylindrical and Spherical Coordinates 227

5.1. Basic solutions to the equations of linear propagation in cylindrical coordinates 227

5.1.1. General solution to the wave equation 227

5.1.2. Progressive cylindrical waves: radiation from an infinitely long cylinder in harmonic regime 231

5.1.3. Diffraction of a plane wave by a cylinder characterized by a surface impedance 236

5.1.4. Propagation of harmonic waves in cylindrical waveguides 238

5.2. Basic solutions to the equations of linear propagation in spherical coordinates 245

5.2.1. General solution of the wave equation 245

5.2.2. Progressive spherical waves 250

5.2.3. Diffraction of a plane wave by a rigid sphere 258

5.2.4. The spherical cavity 262

5.2.5. Digression on monopolar, dipolar and 2n-polar acoustic fields 266

Chapter 6. Integral Formalism in Linear Acoustics 277

6.1. Considered problems 277

6.1.1. Problems 277

6.1.2. Associated eigenvalues problem 278

6.1.3. Elementary problem: Green’s function in infinite space 279

6.1.4. Green’s function in finite space 280

6.1.5. Reciprocity of the Green’s function 294

6.2. Integral formalism of boundary problems in linear acoustics 296

6.2.1. Introduction 296

6.2.2. Integral formalism 297

6.2.3. On solving integral equations 300

6.3. Examples of application 309

6.3.1. Examples of application in the time domain 309

6.3.2. Examples of application in the frequency domain 318

Chapter 7. Diffusion, Diffraction and Geometrical Approximation 357

7.1. Acoustic diffusion: examples 357

7.1.1. Propagation in non-homogeneous media 357

7.1.2. Diffusion on surface irregularities 360

7.2. Acoustic diffraction by a screen 362

7.2.1. Kirchhoff-Fresnel diffraction theory 362

7.2.2. Fraunhofer’s approximation 364

7.2.3. Fresnel’s approximation 366

7.2.4. Fresnel’s diffraction by a straight edge 369

7.2.5. Diffraction of a plane wave by a semi-infinite rigid plane: introduction to Sommerfeld’s theory 371

7.2.6. Integral formalism for the problem of diffraction by a semi-infinite plane screen with a straight edge 376

7.2.7. Geometric Theory of Diffraction of Keller (GTD) 379

7.3. Acoustic propagation in non-homogeneous and non-dissipative media in motion, varying “slowly” in time and space: geometric approximation 385

7.3.1. Introduction 385

7.3.2. Fundamental equations 386

7.3.3. Modes of perturbation 388

7.3.4. Equations of rays 392

7.3.5. Applications to simple cases 397

7.3.6. Fermat’s principle 403

7.3.7. Equation of parabolic waves 405

Chapter 8. Introduction to Sound Radiation and Transparency of Walls 409

8.1. Waves in membranes and plates 409

8.1.1. Longitudinal and quasi-longitudinal waves. 410

8.1.2. Transverse shear waves 412

8.1.3. Flexural waves 413

8.2. Governing equation for thin, plane, homogeneous and isotropic plate in transverse motion 419

8.2.1. Equation of motion of membranes 419

8.2.2. Thin, homogeneous and isotropic plates in pure bending 420

8.2.3. Governing equations of thin plane walls 424

8.3. Transparency of infinite thin, homogeneous and isotropic walls 426

8.3.1. Transparency to an incident plane wave 426

8.3.2. Digressions on the influence and nature of the acoustic field on both sides of the wall 431

8.3.3. Transparency of a multilayered system: the double leaf system 434

8.4. Transparency of finite thin, plane and homogeneous walls: modal theory 438

8.4.1. Generally 438

8.4.2. Modal theory of the transparency of finite plane walls 439

8.4.3. Applications: rectangular plate and circular membrane 444

8.5. Transparency of infinite thick, homogeneous and isotropic plates 450

8.5.1. Introduction 450

8.5.2. Reflection and transmission of waves at the interface fluid-solid 450

8.5.3. Transparency of an infinite thick plate 457

8.6. Complements in vibro-acoustics: the Statistical Energy Analysis (SEA) method 461

8.6.1. Introduction 461

8.6.2. The method 461

8.6.3. Justifying approach 463

Chapter 9. Acoustics in Closed Spaces 465

9.1. Introduction 465

9.2. Physics of acoustics in closed spaces: modal theory 466

9.2.1. Introduction 466

9.2.2. The problem of acoustics in closed spaces 468

9.2.3. Expression of the acoustic pressure field in closed spaces 471

9.2.4. Examples of problems and solutions 477

9.3. Problems with high modal density: statistically quasi-uniform acoustic fields 483

9.3.1. Distribution of the resonance frequencies of a rectangular cavity with perfectly rigid walls 483

9.3.2. Steady state sound field at “high” frequencies 487

9.3.3. Acoustic field in transient regime at high frequencies 494

9.4. Statistical analysis of diffused fields 497

9.4.1. Characteristics of a diffused field 497

9.4.2. Energy conservation law in rooms 498

9.4.3. Steady-state radiation from a punctual source 500

9.4.4. Other expressions of the reverberation time 502

9.4.5. Diffused sound fields 504

9.5. Brief history of room acoustics 508

Chapter 10. Introduction to Non-linear Acoustics, Acoustics in Uniform Flow, and Aero-acoustics 511

10.1. Introduction to non-linear acoustics in fluids initially at rest 511

10.1.1. Introduction 511

10.1.2. Equations of non-linear acoustics: linearization method 513

10.1.3. Equations of propagation in non-dissipative fluids in one dimension, Fubini’s solution of the implicit equations 529

10.1.4. Bürger’s equation for plane waves in dissipative (visco-thermal) media 536

10.2. Introduction to acoustics in fluids in subsonic uniform flows 547

10.2.1. Doppler effect 547

10.2.2. Equations of motion 549

10.2.3. Integral equations of motion and Green’s function in a uniform and constant flow 551

10.2.4. Phase velocity and group velocity, energy transfer – case of the rigid-walled guides with constant cross-section in uniform flow 556

10.2.5. Equation of dispersion and propagation modes: case of the rigid-walled guides with constant cross-section in uniform flow 560

10.2.6. Reflection and refraction at the interface between two media in relative motion (at subsonic velocity) 562

10.3. Introduction to aero-acoustics 566

10.3.1. Introduction 566

10.3.2. Reminder about linear equations of motion and fundamental sources 566

10.3.3. Lighthill’s equation 568

10.3.4. Solutions to Lighthill’s equation in media limited by rigid obstacles: Curle’s solution 570

10.3.5. Estimation of the acoustic power of quadrupolar turbulences 574

10.3.6. Conclusion 574

Chapter 11. Methods in Electro-acoustics 577

11.1. Introduction 577

11.2. The different types of conversion 578

11.2.1. Electromagnetic conversion 578

11.2.2. Piezoelectric conversion (example) 583

11.2.3. Electrodynamic conversion 588

11.2.4. Electrostatic conversion 589

11.2.5. Other conversion techniques 591

11.3. The linear mechanical systems with localized constants 592

11.3.1. Fundamental elements and systems 592

11.3.2. Electromechanical analogies 596

11.3.3. Digression on the one-dimensional mechanical systems with distributed constants: longitudinal motion of a beam 601

11.4. Linear acoustic systems with localized and distributed constants 604

11.4.1. Linear acoustic systems with localized constants 604

11.4.2. Linear acoustic systems with distributed constants: the cylindrical waveguide 611

11.5. Examples of application to electro-acoustic transducers 613

11.5.1. Electrodynamic transducer 613

11.5.2. The electrostatic microphone 619

11.5.3. Example of piezoelectric transducer 624

Chapter 11: Appendix 626

A.1 Reminder about linear electrical circuits with localized constants 626

A.2 Generalization of the coupling equations 628

Bibliography 631

Index 633