Fundamentals of Acoustics

Michel Bruneau is Emeritus Professor at the University of Maine in France. He is the founder of the Laboratoire d’Acoustique from University of Maine (LAUM) affiliated to the Centre National de la Recherche Scientifique (CNRS) where he was the director of the postgraduate studies in acoustics. This book results mainly from his lectures and publications.

• Preface 13Chapter 1. Equations of Motion in Non-dissipative Fluid 151.1. Introduction 151.1.1. Basic elements 151.1.2. Mechanisms of transmission 161.1.3. Acoustic motion and driving motion 171.1.4. Notion of frequency 171.1.5. Acoustic amplitude and intensity 181.1.6. Viscous and thermal phenomena 191.2. Fundamental laws of propagation in non-dissipative fluids 201.2.1. Basis of thermodynamics 201.2.2. Lagrangian and Eulerian descriptions of fluid motion 251.2.3. Expression of the fluid compressibility: mass conservation law 271.2.4. Expression of the fundamental law of dynamics: Euler’s equation 291.2.5. Law of fluid behavior: law of conservation of thermomechanic energy 301.2.6. Summary of the fundamental laws 311.2.7. Equation of equilibrium of moments 321.3. Equation of acoustic propagation 331.3.1. Equation of propagation 331.3.2. Linear acoustic approximation 341.3.3. Velocity potential 381.3.4. Problems at the boundaries 401.4. Density of energy and energy flow, energy conservation law 421.4.1. Complex representation in the Fourier domain 421.4.2. Energy density in an “ideal” fluid 431.4.3. Energy flow and acoustic intensity 451.4.4. Energy conservation law 48Chapter 1: Appendix. Some General Comments on Thermodynamics 50A.1. Thermodynamic equilibrium and equation of state 50A.2. Digression on functions of multiple variables (study case of two variables) 51A.2.1. Implicit functions 51A.2.2. Total exact differential form 53Chapter 2. Equations of Motion in Dissipative Fluid 552.1. Introduction 552.2. Propagation in viscous fluid: Navier-Stokes equation 562.2.1. Deformation and strain tensor 572.2.2. Stress tensor 622.2.3. Expression of the fundamental law of dynamics 642.3. Heat propagation: Fourier equation 702.4. Molecular thermal relaxation 722.4.1. Nature of the phenomenon 722.4.2. Internal energy, energy of translation, of rotation and of vibration of molecules 742.4.3. Molecular relaxation: delay of molecular vibrations 752.5. Problems of linear acoustics in dissipative fluid at rest 772.5.1. Propagation equations in linear acoustics 772.5.2. Approach to determine the solutions 812.5.3. Approach of the solutions in presence of acoustic sources 842.5.4. Boundary conditions 85Chapter 2: Appendix. Equations of continuity and equations at the thermomechanic discontinuities in continuous media 93A.1. Introduction 93A.1.1. Material derivative of volume integrals 93A.1.2. Generalization 96A.2. Equations of continuity 97A.2.1. Mass conservation equation 97A.2.2. Equation of impulse continuity 98A.2.3. Equation of entropy continuity 99A.2.4. Equation of energy continuity 99A.3. Equations at discontinuities in mechanics 102A.3.1. Introduction 102A.3.2. Application to the equation of impulse conservation 103A.3.3. Other conditions at discontinuities 106A.4. Examples of application of the equations at discontinuities in mechanics: interface conditions 106A.4.1. Interface solid – viscous fluid 107A.4.2. Interface between perfect fluids 108A.4.3 Interface between two non-miscible fluids in motion 109Chapter 3. Problems of Acoustics in Dissipative Fluids 1113.1. Introduction 1113.2. Reflection of a harmonic wave from a rigid plane 1113.2.1. Reflection of an incident harmonic plane wave 1113.2.2. Reflection of a harmonic acoustic wave 1153.3. Spherical wave in infinite space: Green’s function 1183.3.1. Impulse spherical source 1183.3.2. Green’s function in three-dimensional space 1213.4. Digression on two- and one-dimensional Green’s functions in non-dissipative fluids 1253.4.1. Two-dimensional Green’s function 1253.4.2. One-dimensional Green’s function 1283.5. Acoustic field in “small cavities” in harmonic regime 1313.6. Harmonic motion of a fluid layer between a vibrating membrane and a rigid plate, application to the capillary slit 1363.7. Harmonic plane wave propagation in cylindrical tubes: propagation constants in “large” and “capillary” tubes 1413.8. Guided plane wave in dissipative fluid 1483.9. Cylindrical waveguide, system of distributed constants 1513.10. Introduction to the thermoacoustic engines (on the use of phenomena occurring in thermal boundary layers) 1543.11. Introduction to acoustic gyrometry (on the use of the phenomena occurring in viscous boundary layers) 162Chapter 4. Basic Solutions to the Equations of Linear Propagation in Cartesian Coordinates 1694.1. Introduction 1694.2. General solutions to the wave equation 1734.2.1. Solutions for propagative waves 1734.2.2. Solutions with separable variables 1764.3. Reflection of acoustic waves on a locally reacting surface 1784.3.1. Reflection of a harmonic plane wave 1784.3.2. Reflection from a locally reacting surface in random incidence 1834.3.3. Reflection of a harmonic spherical wave from a locally reacting plane surface 1844.3.4. Acoustic field before a plane surface of impedance Z under the load of a harmonic plane wave in normal incidence 1854.4. Reflection and transmission at the interface between two different fluids 1874.4.1. Governing equations 1874.4.2. The solutions 1894.4.3. Solutions in harmonic regime 1904.4.4. The energy flux 1924.5. Harmonic waves propagation in an infinite waveguide with rectangular cross-section 1934.5.1. The governing equations 1934.5.2. The solutions 1954.5.3. Propagating and evanescent waves 1974.5.4. Guided propagation in non-dissipative fluid 2004.6. Problems of discontinuity in waveguides 2064.6.1. Modal theory 2064.6.2. Plane wave fields in waveguide with section discontinuities 2074.7. Propagation in horns in non-dissipative fluids 2104.7.1. Equation of horns 2104.7.2. Solutions for infinite exponential horns 214Chapter 4: Appendix. Eigenvalue Problems, Hilbert Space 217A.1. Eigenvalue problems 217A.1.1. Properties of eigenfunctions and associated eigenvalues 217A.1.2. Eigenvalue problems in acoustics 220A.1.3. Degeneracy 220A.2. Hilbert space 221A.2.1. Hilbert functions and L 2 space 221A.2.2. Properties of Hilbert functions and complete discrete ortho-normal basis 222A.2.3. Continuous complete ortho-normal basis  223Chapter 5. Basic Solutions to the Equations of Linear Propagation in Cylindrical and Spherical Coordinates 2275.1. Basic solutions to the equations of linear propagation in cylindrical coordinates 2275.1.1. General solution to the wave equation 2275.1.2. Progressive cylindrical waves: radiation from an infinitely long cylinder in harmonic regime 2315.1.3. Diffraction of a plane wave by a cylinder characterized by a surface impedance 2365.1.4. Propagation of harmonic waves in cylindrical waveguides 2385.2. Basic solutions to the equations of linear propagation in spherical coordinates 2455.2.1. General solution of the wave equation 2455.2.2. Progressive spherical waves 2505.2.3. Diffraction of a plane wave by a rigid sphere 2585.2.4. The spherical cavity 2625.2.5. Digression on monopolar, dipolar and 2n-polar acoustic fields 266Chapter 6. Integral Formalism in Linear Acoustics 2776.1. Considered problems 2776.1.1. Problems 2776.1.2. Associated eigenvalues problem 2786.1.3. Elementary problem: Green’s function in infinite space 2796.1.4. Green’s function in finite space 2806.1.5. Reciprocity of the Green’s function 2946.2. Integral formalism of boundary problems in linear acoustics 2966.2.1. Introduction 2966.2.2. Integral formalism 2976.2.3. On solving integral equations 3006.3. Examples of application 3096.3.1. Examples of application in the time domain 3096.3.2. Examples of application in the frequency domain 318Chapter 7. Diffusion, Diffraction and Geometrical Approximation 3577.1. Acoustic diffusion: examples 3577.1.1. Propagation in non-homogeneous media 3577.1.2. Diffusion on surface irregularities 3607.2. Acoustic diffraction by a screen 3627.2.1. Kirchhoff-Fresnel diffraction theory 3627.2.2. Fraunhofer’s approximation 3647.2.3. Fresnel’s approximation 3667.2.4. Fresnel’s diffraction by a straight edge 3697.2.5. Diffraction of a plane wave by a semi-infinite rigid plane: introduction to Sommerfeld’s theory 3717.2.6. Integral formalism for the problem of diffraction by a semi-infinite plane screen with a straight edge 3767.2.7. Geometric Theory of Diffraction of Keller (GTD) 3797.3. Acoustic propagation in non-homogeneous and non-dissipative media in motion, varying “slowly” in time and space: geometric approximation 3857.3.1. Introduction 3857.3.2. Fundamental equations 3867.3.3. Modes of perturbation 3887.3.4. Equations of rays 3927.3.5. Applications to simple cases 3977.3.6. Fermat’s principle 4037.3.7. Equation of parabolic waves 405Chapter 8. Introduction to Sound Radiation and Transparency of Walls 4098.1. Waves in membranes and plates 4098.1.1. Longitudinal and quasi-longitudinal waves. 4108.1.2. Transverse shear waves 4128.1.3. Flexural waves 4138.2. Governing equation for thin, plane, homogeneous and isotropic plate in transverse motion 4198.2.1. Equation of motion of membranes 4198.2.2. Thin, homogeneous and isotropic plates in pure bending 4208.2.3. Governing equations of thin plane walls 4248.3. Transparency of infinite thin, homogeneous and isotropic walls 4268.3.1. Transparency to an incident plane wave 4268.3.2. Digressions on the influence and nature of the acoustic field on both sides of the wall 4318.3.3. Transparency of a multilayered system: the double leaf system 4348.4. Transparency of finite thin, plane and homogeneous walls: modal theory 4388.4.1. Generally 4388.4.2. Modal theory of the transparency of finite plane walls 4398.4.3. Applications: rectangular plate and circular membrane 4448.5. Transparency of infinite thick, homogeneous and isotropic plates 4508.5.1. Introduction 4508.5.2. Reflection and transmission of waves at the interface fluid-solid 4508.5.3. Transparency of an infinite thick plate 4578.6. Complements in vibro-acoustics: the Statistical Energy Analysis (SEA) method 4618.6.1. Introduction 4618.6.2. The method 4618.6.3. Justifying approach 463Chapter 9. Acoustics in Closed Spaces 4659.1. Introduction 4659.2. Physics of acoustics in closed spaces: modal theory 4669.2.1. Introduction 4669.2.2. The problem of acoustics in closed spaces 4689.2.3. Expression of the acoustic pressure field in closed spaces 4719.2.4. Examples of problems and solutions 4779.3. Problems with high modal density: statistically quasi-uniform acoustic fields 4839.3.1. Distribution of the resonance frequencies of a rectangular cavity with perfectly rigid walls 4839.3.2. Steady state sound field at “high” frequencies 4879.3.3. Acoustic field in transient regime at high frequencies 4949.4. Statistical analysis of diffused fields 4979.4.1. Characteristics of a diffused field 4979.4.2. Energy conservation law in rooms 4989.4.3. Steady-state radiation from a punctual source 5009.4.4. Other expressions of the reverberation time 5029.4.5. Diffused sound fields 5049.5. Brief history of room acoustics 508Chapter 10. Introduction to Non-linear Acoustics, Acoustics in Uniform Flow, and Aero-acoustics 51110.1. Introduction to non-linear acoustics in fluids initially at rest 51110.1.1. Introduction 51110.1.2. Equations of non-linear acoustics: linearization method 51310.1.3. Equations of propagation in non-dissipative fluids in one dimension, Fubini’s solution of the implicit equations 52910.1.4. Bürger’s equation for plane waves in dissipative (visco-thermal) media 53610.2. Introduction to acoustics in fluids in subsonic uniform flows 54710.2.1. Doppler effect 54710.2.2. Equations of motion 54910.2.3. Integral equations of motion and Green’s function in a uniform and constant flow 55110.2.4. Phase velocity and group velocity, energy transfer – case of the rigid-walled guides with constant cross-section in uniform flow 55610.2.5. Equation of dispersion and propagation modes: case of the rigid-walled guides with constant cross-section in uniform flow 56010.2.6. Reflection and refraction at the interface between two media in relative motion (at subsonic velocity) 56210.3. Introduction to aero-acoustics 56610.3.1. Introduction 56610.3.2. Reminder about linear equations of motion and fundamental sources 56610.3.3. Lighthill’s equation 56810.3.4. Solutions to Lighthill’s equation in media limited by rigid obstacles: Curle’s solution 57010.3.5. Estimation of the acoustic power of quadrupolar turbulences 57410.3.6. Conclusion 574Chapter 11. Methods in Electro-acoustics 57711.1. Introduction 57711.2. The different types of conversion 57811.2.1. Electromagnetic conversion 57811.2.2. Piezoelectric conversion (example) 58311.2.3. Electrodynamic conversion 58811.2.4. Electrostatic conversion 58911.2.5. Other conversion techniques 59111.3. The linear mechanical systems with localized constants 59211.3.1. Fundamental elements and systems 59211.3.2. Electromechanical analogies 59611.3.3. Digression on the one-dimensional mechanical systems with distributed constants: longitudinal motion of a beam 60111.4. Linear acoustic systems with localized and distributed constants 60411.4.1. Linear acoustic systems with localized constants 60411.4.2. Linear acoustic systems with distributed constants: the cylindrical waveguide 61111.5. Examples of application to electro-acoustic transducers 61311.5.1. Electrodynamic transducer 61311.5.2. The electrostatic microphone 61911.5.3. Example of piezoelectric transducer 624Chapter 11: Appendix 626A.1 Reminder about linear electrical circuits with localized constants 626A.2 Generalization of the coupling equations 628Bibliography 631Index 633