Fundamentals of Acoustics

by Unknown

• ISBN13: 9781905209255
• ISBN10: 1905209258
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-06-23
• Publisher: Wiley-ISTE

Summary

The central theme of the chapters is acoustic propagation in fluid media, dissipative or non-dissipative, homogeneous or nonhomogeneous, infinite or limited, placing particular emphasis on the theoretical formulation of the problems considered.

Author Biography

Michel Bruneau is a professor emeritus at the Université du Maine. He is the founder of the Laboratoire d'Acoustique de l'Université du Maine and a former director of postgraduate studies in acoustics at the Centre National de la Recherche Scientifique. Thomas Scelo has worked in the department of mechanical engineering at the University of Auckland.

Table of Contents

Preface	13	(2)
Chapter 1. Equations of Motion in Non-dissipative Fluid	15	(35)
1.1. Introduction	15	(5)
1.1.1. Basic elements	15	(1)
1.1.2. Mechanisms of transmission	16	(1)
1.1.3. Acoustic motion and driving motion	17	(1)
1.1.4. Notion of frequency	17	(1)
1.1.5. Acoustic amplitude and intensity	18	(1)
1.1.6. Viscous and thermal phenomena	19	(1)
1.2. Fundamental laws of propagation in non-dissipative fluids	20	(13)
1.2.1. Basis of thermodynamics	20	(5)
1.2.2. Lagrangian and Eulerian descriptions of fluid motion	25	(2)
1.2.3. Expression of the fluid compressibility: mass conservation law	27	(2)
1.2.4. Expression of the fundamental law of dynamics: Euler's equation	29	(1)
1.2.5. Law of fluid behavior: law of conservation of thermomechanic energy	30	(1)
1.2.6. Summary of the fundamental laws	31	(1)
1.2.7. Equation of equilibrium of moments	32	(1)
1.3. Equation of acoustic propagation	33	(9)
1.3.1. Equation of propagation	33	(1)
1.3.2. Linear acoustic approximation	34	(4)
1.3.3. Velocity potential	38	(2)
1.3.4. Problems at the boundaries	40	(2)
1.4. Density of energy and energy flow, energy conservation law	42	(8)
1.4.1. Complex representation in the Fourier domain	42	(1)
1.4.2. Energy density in an "ideal" fluid	43	(2)
1.4.3. Energy flow and acoustic intensity	45	(3)
1.4.4. Energy conservation law	48	(2)
Chapter 1: Appendix. Some General Comments on Thermodynamics	50	(5)
A.1. Thermodynamic equilibrium and equation of state	50	(1)
A.2. Digression on functions of multiple variables (study case of two variables)	51	(4)
A.2.1. Implicit functions	51	(2)
A.2.2. Total exact differential form	53	(2)
Chapter 2. Equations of Motion in Dissipative Fluid	55	(38)
2.1. Introduction	55	(1)
2.2. Propagation in viscous fluid: Navier-Stokes equation	56	(14)
2.2.1. Deformation and strain tensor	57	(5)
2.2.2. Stress tensor	62	(2)
2.2.3. Expression of the fundamental law of dynamics	64	(6)
2.3. Heat propagation: Fourier equation	70	(2)
2.4. Molecular thermal relaxation	72	(5)
2.4.1. Nature of the phenomenon	72	(2)
2.4.2. Internal energy, energy of translation, of rotation and of vibration of molecules	74	(1)
2.4.3. Molecular relaxation: delay of molecular vibrations	75	(2)
2.5. Problems of linear acoustics in dissipative fluid at rest	77	(16)
2.5.1. Propagation equations in linear acoustics	77	(4)
2.5.2. Approach to determine the solutions	81	(3)
2.5.3. Approach of the solutions in presence of acoustic sources	84	(1)
2.5.4. Boundary conditions	85	(8)
Chapter 2: Appendix. Equations of continuity and equations at the thermomechanic discontinuities in continuous media	93	(18)
A.1. Introduction	93	(4)
A.1.1. Material derivative of volume integrals	93	(3)
A.1.2. Generalization	96	(1)
A.2. Equations of continuity	97	(10)
A.2.1. Mass conservation equation	97	(1)
A.2.2. Equation of impulse continuity	98	(1)
A.2.3. Equation of entropy continuity	99	(1)
A.2.4. Equation of energy continuity	99	(8)
A.3. Equations at discontinuities in mechanics	107
A.3.1. Introduction	102	(1)
A.3.2. Application to the equation of impulse conservation	103	(3)
A.3.3. Other conditions at discontinuities	106	(1)
A.4. Examples of application of the equations at discontinuities in mechanics: interface conditions	106	(5)
A.4.1. Interface solid – viscous fluid	107	(1)
A.4.2. Interface between perfect fluids	108	(1)
A.4.3 Interface between two non-miscible fluids in motion	109	(2)
Chapter 3. Problems of Acoustics in Dissipative Fluids	111	(58)
3.1. Introduction	111	(1)
3.2. Reflection of a harmonic wave from a rigid plane.	111	(7)
3.2.1. Reflection of an incident harmonic plane wave	111	(4)
3.2.2. Reflection of a harmonic acoustic wave	115	(3)
3.3. Spherical wave in infinite space: Green's function	118	(7)
3.3.1. Impulse spherical source	118	(3)
3.3.2. Green's function in three-dimensional space	121	(4)
3.4. Digression on two- and one-dimensional Green's functions in non-dissipative fluids	125	(6)
3.4.1. Two-dimensional Green's function	125	(3)
3.4.2. One-dimensional Green's function	128	(3)
3.5. Acoustic field in "small cavities" in harmonic regime	131	(5)
3.6. Harmonic motion of a fluid layer between a vibrating membrane and a rigid plate, application to the capillary slit	136	(5)
3.7. Harmonic plane wave propagation in cylindrical tubes: propagation constants in "large" and "capillary" tubes	141	(7)
3.8. Guided plane wave in dissipative fluid	148	(3)
3.9. Cylindrical waveguide, system of distributed constants	151	(3)
3.10. Introduction to the thermoacoustic engines (on the use of phenomena occurring in thermal boundary layers)	154	(8)
3.11. Introduction to acoustic gyrometry (on the use of the phenomena occurring in viscous boundary layers)	162	(7)
Chapter 4. Basic Solutions to the Equations of Linear Propagation in Cartesian Coordinates	169	(48)
4.1. Introduction	169	(4)
4.2. General solutions to the wave equation	173	(5)
4.2.1. Solutions for propagative waves	173	(3)
4.2.2. Solutions with separable variables	176	(2)
4.3. Reflection of acoustic waves on a locally reacting surface	178	(9)
4.3.1. Reflection of a harmonic plane wave	178	(5)
4.3.2. Reflection from a locally reacting surface in random incidence	183	(1)
4.3.3. Reflection of a harmonic spherical wave from a locally reacting plane surface	184	(1)
4.3.4. Acoustic field before a plane surface of impedance Z under the load of a harmonic plane wave in normal incidence	185	(2)
4.4. Reflection and transmission at the interface between two different fluids	187	(6)
4.4.1. Governing equations	187	(2)
4.4.2. The solutions	189	(1)
4.4.3. Solutions in harmonic regime	190	(2)
4.4.4. The energy flux	192	(1)
4.5. Harmonic waves propagation in an infinite waveguide with rectangular cross-section	193	(13)
4.5.1. The governing equations	193	(2)
4.5.2. The solutions	195	(2)
4.5.3. Propagating and evanescent waves	197	(3)
4.5.4. Guided propagation in non-dissipative fluid	200	(6)
4.6. Problems of discontinuity in waveguides	206	(4)
4.6.1. Modal theory	206	(1)
4.6.2. Plane wave fields in waveguide with section discontinuities	207	(3)
4.7. Propagation in horns in non-dissipative fluids	210	(7)
4.7.1. Equation of horns	210	(4)
4.7.2. Solutions for infinite exponential horns	214	(3)
Chapter 4: Appendix. Eigenvalue Problems, Hilbert Space	217	(10)
A.1. Eigenvalue problems	217	(4)
A.1.1. Properties of eigenfunctions and associated eigenvalues	217	(3)
A.1.2. Eigenvalue problems in acoustics	220	(1)
A.1.3. Degeneracy	220	(1)
A.2. Hilbert space	221	(6)
A.2.1. Hilbert functions and 2 space	221	(1)
A.2.2. Properties of Hilbert functions and complete discrete ortho-normal basis	222	(1)
A.2.3. Continuous complete ortho-normal basis	223	(4)
Chapter 5. Basic Solutions to the Equations of Linear Propagation in Cylindrical and Spherical Coordinates	227	(50)
5.1. Basic solutions to the equations of linear propagation in cylindrical coordinates	227	(18)
5.1.1. General solution to the wave equation	227	(4)
5.1.2. Progressive cylindrical waves: radiation from an infinitely long cylinder in harmonic regime	231	(5)
5.1.3. Diffraction of a plane wave by a cylinder characterized by a surface impedance	236	(2)
5.1.4. Propagation of harmonic waves in cylindrical waveguides	238	(7)
5.2. Basic solutions to the equations of linear propagation in spherical coordinates	245	(32)
5.2.1. General solution of the wave equation	245	(5)
5.2.2. Progressive spherical waves	250	(8)
5.2.3. Diffraction of a plane wave by a rigid sphere	258	(4)
5.2.4. The spherical cavity	262	(4)
5.2.5. Digression on monopolar, dipolar and 2n-polar acoustic fields	266	(11)
Chapter 6. Integral Formalism in Linear Acoustics	277	(80)
6.1. Considered problems	277	(19)
6.1.1. Problems	277	(1)
6.1.2. Associated eigenvalues problem	278	(1)
6.1.3. Elementary problem: Green's function in infinite space	279	(1)
6.1.4. Green's function in finite space	280	(14)
6.1.5. Reciprocity of the Green's function	294	(2)
6.2. Integral formalism of boundary problems in linear acoustics	296	(13)
6.2.1. Introduction	296	(1)
6.2.2. Integral formalism	297	(3)
6.2.3. On solving integral equations	300	(9)
6.3. Examples of application	309	(48)
6.3.1. Examples of application in the time domain	309	(9)
6.3.2. Examples of application in the frequency domain	318	(39)
Chapter 7. Diffusion, Diffraction and Geometrical Approximation	357	(52)
7.1. Acoustic diffusion: examples	357	(5)
7.1.1. Propagation in non-homogeneous media	357	(3)
7.1.2. Diffusion on surface irregularities	360	(2)
7.2. Acoustic diffraction by a screen	362	(23)
7.2.1. Kirchhoff-Fresnel diffraction theory	362	(2)
7.2.2. Fraunhofer's approximation	364	(2)
7.2.3. Fresnel's approximation	366	(3)
7.2.4. Fresnel's diffraction by a straight edge	369	(2)
7.2.5. Diffraction of a plane wave by a semi-infinite rigid plane: introduction to Sommerfeld's theory.	371	(5)
7.2.6. Integral formalism for the problem of diffraction by a semi-infinite plane screen with a straight edge	376	(3)
7.2.7. Geometric Theory of Diffraction of Keller (GTD)	379	(6)
7.3. Acoustic propagation in non-homogeneous and non-dissipative media in motion, varying "slowly" in time and space: geometric approximation	385	(24)
7.3.1. Introduction	385	(1)
7.3.2. Fundamental equations	386	(2)
7.3.3. Modes of perturbation	388	(4)
7.3.4. Equations of rays	392	(5)
7.3.5. Applications to simple cases	397	(6)
7.3.6. Fermat's principle	403	(2)
7.3.7. Equation of parabolic waves	405	(4)
Chapter 8. Introduction to Sound Radiation and Transparency of Walls	409	(56)
8.1. Waves in membranes and plates	409	(10)
8.1.1. Longitudinal and quasi-longitudinal waves	410	(2)
8.1.2. Transverse shear waves	412	(1)
8.1.3. Flexural waves	413	(6)
8.2. Governing equation for thin, plane, homogeneous and isotropic plate in transverse motion	419	(7)
8.2.1. Equation of motion of membranes	419	(1)
8.2.2. Thin, homogeneous and isotropic plates in pure bending	420	(4)
8.2.3. Governing equations of thin plane walls	424	(2)
8.3. Transparency of infinite thin, homogeneous and isotropic walls	426	(12)
8.3.1. Transparency to an incident plane wave	426	(5)
8.3.2. Digressions on the influence and nature of the acoustic field on both sides of the wall	431	(3)
8.3.3. Transparency of a multilayered system: the double leaf system	434	(4)
8.4. Transparency of finite thin, plane and homogeneous walls: modal theory	438	(12)
8.4.1. Generally	438	(1)
8.4.2. Modal theory of the transparency of finite plane walls	439	(5)
8.4.3. Applications: rectangular plate and circular membrane	444	(6)
8.5. Transparency of infinite thick, homogeneous and isotropic plates	450	(11)
8.5.1. Introduction	450	(1)
8.5.2. Reflection and transmission of waves at the interface fluid-solid	450	(7)
8.5.3. Transparency of an infinite thick plate	457	(4)
8.6. Complements in vibro-acoustics: the Statistical Energy Analysis (SEA) method	461	(4)
8.6.1. Introduction	461	(1)
8.6.2. The method	461	(2)
8.6.3. Justifying approach	463	(2)
Chapter 9. Acoustics in Closed Spaces	465	(46)
9.1. Introduction	465	(1)
9.2. Physics of acoustics in closed spaces: modal theory	466	(17)
9.2.1. Introduction	466	(2)
9.2.2. The problem of acoustics in closed spaces	468	(3)
9.2.3. Expression of the acoustic pressure field in closed spaces	471	(6)
9.2.4. Examples of problems and solutions	477	(6)
9.3. Problems with high modal density: statistically quasi-uniform acoustic fields	483	(14)
9.3.1. Distribution of the resonance frequencies of a rectangular cavity with perfectly rigid walls	483	(4)
9.3.2. Steady state sound field at "high" frequencies	487	(7)
9.3.3. Acoustic field in transient regime at high frequencies	494	(3)
9.4. Statistical analysis of diffused fields	497	(11)
9.4.1. Characteristics of a diffused field	497	(1)
9.4.2. Energy conservation law in rooms	498	(2)
9.4.3. Steady-state radiation from a punctual source	500	(2)
9.4.4. Other expressions of the reverberation time	502	(2)
9.4.5. Diffused sound fields	504	(4)
9.5. Brief history of room acoustics	508	(3)
Chapter 10. Introduction to Non-linear Acoustics, Acoustics in Uniform Flow, and Aero-acoustics	511	(66)
10.1. Introduction to non-linear acoustics in fluids initially at rest	511	(36)
10.1.1. Introduction	511	(2)
10.1.2. Equations of non-linear acoustics: linearization method	513	(16)
10.1.3. Equations of propagation in non-dissipative fluids in one dimension, Fubini's solution of the implicit equations	529	(7)
10.1.4. Burger's equation for plane waves in dissipative (visco-thermal) media	536	(11)
10.2. Introduction to acoustics in fluids in subsonic uniform flows	547	(19)
10.2.1. Doppler effect	547	(2)
10.2.2. Equations of motion	549	(2)
10.2.3. Integral equations of motion and Green's function in a uniform and constant flow	551	(5)
10.2.4. Phase velocity and group velocity, energy transfer — case of the rigid-walled guides with constant cross-section in uniform flow	556	(4)
10.2.5. Equation of dispersion and propagation modes: case of the rigid-walled guides with constant cross-section in uniform flow	560	(2)
10.2.6. Reflection and refraction at the interface between two media in relative motion (at subsonic velocity)	562	(4)
10.3. Introduction to aero-acoustics	566	(11)
10.3.1. Introduction	566	(1)
10.3.2. Reminder about linear equations of motion and fundamental sources	566	(2)
10.3.3. Lighthill's equation	568	(2)
10.3.4. Solutions to Lighthill's equation in media limited by rigid obstacles: Curle's solution	570	(4)
10.3.5. Estimation of the acoustic power of quadrupolar turbulences	574	(1)
10.3.6. Conclusion	574	(3)
Chapter 11. Methods in Electro-acoustics	577	(49)
11.1. Introduction	577	(1)
11.2. The different types of conversion	578	(14)
11.2.1. Electromagnetic conversion	578	(5)
11.2.2. Piezoelectric conversion (example)	583	(5)
11.2.3. Electrodynamic conversion	588	(1)
11.2.4. Electrostatic conversion	589	(2)
11.2.5. Other conversion techniques	591	(1)
11.3. The linear mechanical systems with localized constants	592	(12)
11.3.1. Fundamental elements and systems	592	(4)
11.3.2. Electromechanical analogies	596	(5)
11.3.3. Digression on the one-dimensional mechanical systems with distributed constants: longitudinal motion of a beam	601	(3)
11.4. Linear acoustic systems with localized and distributed constants	604	(9)
11.4.1. Linear acoustic systems with localized constants	604	(7)
11.4.2. Linear acoustic systems with distributed constants: the cylindrical waveguide	611	(2)
11.5. Examples of application to electro-acoustic transducers	613	(13)
11.5.1. Electrodynamic transducer	613	(6)
11.5.2. The electrostatic microphone	619	(5)
11.5.3. Example of piezoelectric transducer	624	(2)
Chapter 11: Appendix	626	(5)
A.1 Reminder about linear electrical circuits with localized constants	626	(2)
A.2 Generalization of the coupling equations	628	(3)
Bibliography	631	(2)
Index	633