Fabien Anselmet is Professor at Ecole Centrale de Marseille in France. His research focuses on fluid turbulence and its numerous applications in the fields of industry and the environment. Pierre-Olivier Mattei is a CNRS researcher at LMA in Marseille, France. He leads research activities that aim towards a better understanding of the physics of acoustic radiation and fluid-structure interaction.

• Preface xiChapter 1 A Bit of History 11.1. The production of sound 11.2. The propagation of sound 41.3. The reception of sound 61.4. Aeroacoustics 7Chapter 2 Elements of Continuum Mechanics 92.1. Mechanics of deformable media 92.1.1. Continuum 92.1.2. Kinematics of deformable media 102.1.3. Deformation tensor (or Green’s tensor) 122.2. Conservation laws 132.2.1. Conservation of mass 132.2.2. Conservation of momentum 142.2.3. Conservation of energy 152.3. Constitutive laws 152.3.1. Elasticity 162.3.2. Thermoelasticity and effects of temperature variations 192.3.3. Viscoelasticity 212.3.4. Fluid medium 282.4. Hamilton principle 292.5. Characteristics of materials 29Chapter 3 Small Mathematics Travel Kit 313.1. Measure theory and Lebesgue integration 323.1.1. Boolean algebra 323.1.2. Measure on a σ-algebra 333.1.3. Convergence and integration of measurable functions 333.1.4. Functional space – functional 353.1.5. Measure as linear functional 363.2. Distributions 373.2.1. The space D of test functions 373.2.2. Distributions definition 373.2.3. Operations on distributions 393.2.4. N-dimensional generalization 433.2.5. Distributions tensor product 473.3. Convolution 483.3.1. Definition and first properties 483.3.2. Convolution algebra and Green’s function 503.4. Modal methods 523.4.1. Eigenmodes of a conservative system 523.4.2. Eigenmodes of a non-conservative system 55Chapter 4 Fluid Acoustics 654.1. Acoustics equations 664.1.1. Conservation equations 664.1.2. Establishment of general equations 674.1.3. Establishment of the wave equation 684.1.4. Velocity potential 694.2. Propagation and general solutions 694.2.1. One-dimensional motion 694.2.2. Three-dimensional motion 704.3. Permanent regime: Helmholtz equation 714.3.1. General solutions 724.3.2. Green’s kernels 764.3.3. Wave group, phase velocity and group velocity 784.4. Discontinuity equations 804.4.1. Interface between two propagating media 804.4.2. Interface between a propagating and a non-propagating medium 824.5. Impedance: measurement and model 834.5.1. Kundt’s tube 834.5.2. Delany–Bazley model 854.6. Homogeneous anisotropic medium 874.7. Medium with a slowly varying celerity 884.8. Media in motion 894.8.1. Homogeneous medium in uniform motion 894.8.2. Plane interface between media in motion 904.8.3. Cylindrical interface between media in motion 924.8.4. Acoustic radiation of a moving surface 94Chapter 5 Radiation, Diffraction, Enclosed Space 1055.1. Acoustic radiation 1065.1.1. A simple example 1065.2. Acoustic radiation of point sources 1075.2.1. Multipolar sources in a harmonic regime 1075.2.2. Far-field 1115.3. Radiation of distributed sources 1115.3.1. Layer potentials 1115.3.2. Green’s representation of pressure and introduction to the theory of diffraction 1145.4. Acoustic radiation of a piston in a plane 1195.4.1. Far-field radiation of a circular piston: directivity 1225.4.2. Radiation along the axis of a circular piston 1255.5. Acoustic radiation of a rectangular baffled structure 1265.6. Acoustic radiation of moving sources 1315.6.1. Compact and non-compact sources 1315.6.2. Sources in uniform and non-uniform motion 1355.7. Sound propagation in a bounded medium 1385.7.1. Eigenfrequencies and resonance frequencies 1385.7.2. The Helmholtz resonator 1395.7.3. Example in dimension 1 1405.7.4. Example in dimension 3 1415.7.5. Propagation of pure sound in a circular enclosure 1435.8. Basics of room acoustics 1495.8.1. The concept of acoustic power 1495.8.2. Directivity index 1495.8.3. Reverberation duration 1505.8.4. Reverberant fields 1535.8.5. Pressure level in rooms 1545.8.6. Crossover frequency and the reverberation distance 1555.9. Sound propagation in a wave guide 1565.9.1. General solution in a wave guide 1565.9.2. Physical interpretation and theory of modes 1575.9.3. Green’s function 1605.9.4. Section change 1615.9.5. Propagation in a conduit in the presence of flow 164Chapter 6 Wave Propagation in Elastic Media 1676.1. Equation of mechanical wave propagation 1686.2. Free waves 1696.2.1. Volumic waves 1696.2.2. Plane wave case 1706.2.3. Surface waves 1716.3. Green’s kernels in a harmonic regime 1766.4. Thin body approximation for plannar structures 1776.4.1. Straight beams 1786.4.2. Plane plates 1866.5. Thin body approximation for cylindrical structures 1986.5.1. Cylinder 1986.5.2. Ring 212Chapter 7 Vibrations of Thin Structures 2197.1. Beam vibrations 2197.1.1. Beam compression vibrations 2197.1.2. Beam bending vibrations 2237.2. Plate vibrations 2337.2.1. Infinite plate 2337.2.2. Finite plate 2397.2.3. Plate of arbitrary shape 2567.3. Cylindrical shell vibrations 2607.3.1. Infinite shell 2607.3.2. Finite shell 264Chapter 8 Acoustic Radiation of Thin Plates 2758.1. First notions of vibroacoustics: a simple example 2768.1.1. Motion equations 2778.1.2. Acoustic radiation 2788.1.3. “Light fluid” approximation 2808.1.4. Sound transmission 2818.1.5. Transient regime 2908.2. Free waves in an infinite plate immersed in a fluid 2948.2.1. Roots of the dispersion equation 2958.2.2. Light fluid approximation 2978.3. Transmission of a plane wave by a thin plate 2998.4. Radiation of an infinite plate under point excitation 3028.4.1 Integro-differential equation with respect to u 3038.4.2 Fourier transform of u 3038.4.3 Calculation of u(r) 3058.4.4. Radiated acoustic pressure 3068.5. Acoustic radiation and vibration of finite plates 3078.5.1. Statement of the problem 3078.5.2. Exact methods 3088.5.3. Light fluid approximation 3138.5.4. Higher order approximations 3198.6. Heavy fluid coupling: resonance estimation 3278.6.1. Clamped rectangular plate coupled with a heavy fluid 3278.6.2. Location of resonances of a coupled plate 3438.7. Vibrations of a thin plate in a turbulent flow 3468.7.1. Interspectral density: simple models 3478.7.2. Green’s representation of a coupled plate 3508.8. Aeroelastic coupling and sloshing 3548.8.1. Sloshing 3548.8.2. Convective instability 3568.8.3. Kelvin–Helmholtz instability 360Chapter 9 Basic Theoretical Aeroacoustics Models 3639.1. Preamble 3639.2. Lighthill’s equation and some of the generalizations that have followed 3659.3. Reminder of some notions on turbulence which will be useful here 3769.4. The Proudman model for homogeneous and isotropic turbulence 3819.5. The Lilley model for homogeneous and isotropic turbulence 3869.6. The recent models and a few experimental validations 3879.7. The Powell–Howe equation for vorticity-generated sound 397Chapter 10 A Few Situations Closer to Reality 40310.1. The Ribner model for jets 40310.2. Problems and approaches specific to boundary layers 41610.3. Flame-generated noise 42610.4. Noise generated by blades 43210.4.1. Noise generated by a solid body in motion, in the temporal domain 43310.4.2. Noise generated by a set of rotating blades and fixed cascading blades, in the frequency domain 44010.4.3. Noise generated by blade–vortex interaction, using the vortex sound generation method 44910.5. Noise generated and propagation in the outer atmosphere: accounting for the thermal stratification and for likely obstacles 45410.5.1. Characteristic properties of the atmospheric boundary layer and impacts on sound propagation 45510.5.2. Models of sound wave propagation in the atmosphere 464Chapter 11 Implementation and Usage of Numerical Simulations 47511.1. Hybrid methods 47611.2. Direct numerical simulations/large eddy simulations 47811.3. Conclusion 488Bibliography 491Index 507

Fabien Anselmet is Professor at Ecole Centrale de Marseille in France. His research focuses on fluid turbulence and its numerous applications in the fields of industry and the environment. Pierre-Olivier Mattei is a CNRS researcher at LMA in Marseille, France. He leads research activities that aim towards a better understanding of the physics of acoustic radiation and fluid-structure interaction.

• Preface xiChapter 1 A Bit of History 11.1. The production of sound 11.2. The propagation of sound 41.3. The reception of sound 61.4. Aeroacoustics 7Chapter 2 Elements of Continuum Mechanics 92.1. Mechanics of deformable media 92.1.1. Continuum 92.1.2. Kinematics of deformable media 102.1.3. Deformation tensor (or Green’s tensor) 122.2. Conservation laws 132.2.1. Conservation of mass 132.2.2. Conservation of momentum 142.2.3. Conservation of energy 152.3. Constitutive laws 152.3.1. Elasticity 162.3.2. Thermoelasticity and effects of temperature variations 192.3.3. Viscoelasticity 212.3.4. Fluid medium 282.4. Hamilton principle 292.5. Characteristics of materials 29Chapter 3 Small Mathematics Travel Kit 313.1. Measure theory and Lebesgue integration 323.1.1. Boolean algebra 323.1.2. Measure on a σ-algebra 333.1.3. Convergence and integration of measurable functions 333.1.4. Functional space – functional 353.1.5. Measure as linear functional 363.2. Distributions 373.2.1. The space D of test functions 373.2.2. Distributions definition 373.2.3. Operations on distributions 393.2.4. N-dimensional generalization 433.2.5. Distributions tensor product 473.3. Convolution 483.3.1. Definition and first properties 483.3.2. Convolution algebra and Green’s function 503.4. Modal methods 523.4.1. Eigenmodes of a conservative system 523.4.2. Eigenmodes of a non-conservative system 55Chapter 4 Fluid Acoustics 654.1. Acoustics equations 664.1.1. Conservation equations 664.1.2. Establishment of general equations 674.1.3. Establishment of the wave equation 684.1.4. Velocity potential 694.2. Propagation and general solutions 694.2.1. One-dimensional motion 694.2.2. Three-dimensional motion 704.3. Permanent regime: Helmholtz equation 714.3.1. General solutions 724.3.2. Green’s kernels 764.3.3. Wave group, phase velocity and group velocity 784.4. Discontinuity equations 804.4.1. Interface between two propagating media 804.4.2. Interface between a propagating and a non-propagating medium 824.5. Impedance: measurement and model 834.5.1. Kundt’s tube 834.5.2. Delany–Bazley model 854.6. Homogeneous anisotropic medium 874.7. Medium with a slowly varying celerity 884.8. Media in motion 894.8.1. Homogeneous medium in uniform motion 894.8.2. Plane interface between media in motion 904.8.3. Cylindrical interface between media in motion 924.8.4. Acoustic radiation of a moving surface 94Chapter 5 Radiation, Diffraction, Enclosed Space 1055.1. Acoustic radiation 1065.1.1. A simple example 1065.2. Acoustic radiation of point sources 1075.2.1. Multipolar sources in a harmonic regime 1075.2.2. Far-field 1115.3. Radiation of distributed sources 1115.3.1. Layer potentials 1115.3.2. Green’s representation of pressure and introduction to the theory of diffraction 1145.4. Acoustic radiation of a piston in a plane 1195.4.1. Far-field radiation of a circular piston: directivity 1225.4.2. Radiation along the axis of a circular piston 1255.5. Acoustic radiation of a rectangular baffled structure 1265.6. Acoustic radiation of moving sources 1315.6.1. Compact and non-compact sources 1315.6.2. Sources in uniform and non-uniform motion 1355.7. Sound propagation in a bounded medium 1385.7.1. Eigenfrequencies and resonance frequencies 1385.7.2. The Helmholtz resonator 1395.7.3. Example in dimension 1 1405.7.4. Example in dimension 3 1415.7.5. Propagation of pure sound in a circular enclosure 1435.8. Basics of room acoustics 1495.8.1. The concept of acoustic power 1495.8.2. Directivity index 1495.8.3. Reverberation duration 1505.8.4. Reverberant fields 1535.8.5. Pressure level in rooms 1545.8.6. Crossover frequency and the reverberation distance 1555.9. Sound propagation in a wave guide 1565.9.1. General solution in a wave guide 1565.9.2. Physical interpretation and theory of modes 1575.9.3. Green’s function 1605.9.4. Section change 1615.9.5. Propagation in a conduit in the presence of flow 164Chapter 6 Wave Propagation in Elastic Media 1676.1. Equation of mechanical wave propagation 1686.2. Free waves 1696.2.1. Volumic waves 1696.2.2. Plane wave case 1706.2.3. Surface waves 1716.3. Green’s kernels in a harmonic regime 1766.4. Thin body approximation for plannar structures 1776.4.1. Straight beams 1786.4.2. Plane plates 1866.5. Thin body approximation for cylindrical structures 1986.5.1. Cylinder 1986.5.2. Ring 212Chapter 7 Vibrations of Thin Structures 2197.1. Beam vibrations 2197.1.1. Beam compression vibrations 2197.1.2. Beam bending vibrations 2237.2. Plate vibrations 2337.2.1. Infinite plate 2337.2.2. Finite plate 2397.2.3. Plate of arbitrary shape 2567.3. Cylindrical shell vibrations 2607.3.1. Infinite shell 2607.3.2. Finite shell 264Chapter 8 Acoustic Radiation of Thin Plates 2758.1. First notions of vibroacoustics: a simple example 2768.1.1. Motion equations 2778.1.2. Acoustic radiation 2788.1.3. “Light fluid” approximation 2808.1.4. Sound transmission 2818.1.5. Transient regime 2908.2. Free waves in an infinite plate immersed in a fluid 2948.2.1. Roots of the dispersion equation 2958.2.2. Light fluid approximation 2978.3. Transmission of a plane wave by a thin plate 2998.4. Radiation of an infinite plate under point excitation 3028.4.1 Integro-differential equation with respect to u 3038.4.2 Fourier transform of u 3038.4.3 Calculation of u(r) 3058.4.4. Radiated acoustic pressure 3068.5. Acoustic radiation and vibration of finite plates 3078.5.1. Statement of the problem 3078.5.2. Exact methods 3088.5.3. Light fluid approximation 3138.5.4. Higher order approximations 3198.6. Heavy fluid coupling: resonance estimation 3278.6.1. Clamped rectangular plate coupled with a heavy fluid 3278.6.2. Location of resonances of a coupled plate 3438.7. Vibrations of a thin plate in a turbulent flow 3468.7.1. Interspectral density: simple models 3478.7.2. Green’s representation of a coupled plate 3508.8. Aeroelastic coupling and sloshing 3548.8.1. Sloshing 3548.8.2. Convective instability 3568.8.3. Kelvin–Helmholtz instability 360Chapter 9 Basic Theoretical Aeroacoustics Models 3639.1. Preamble 3639.2. Lighthill’s equation and some of the generalizations that have followed 3659.3. Reminder of some notions on turbulence which will be useful here 3769.4. The Proudman model for homogeneous and isotropic turbulence 3819.5. The Lilley model for homogeneous and isotropic turbulence 3869.6. The recent models and a few experimental validations 3879.7. The Powell–Howe equation for vorticity-generated sound 397Chapter 10 A Few Situations Closer to Reality 40310.1. The Ribner model for jets 40310.2. Problems and approaches specific to boundary layers 41610.3. Flame-generated noise 42610.4. Noise generated by blades 43210.4.1. Noise generated by a solid body in motion, in the temporal domain 43310.4.2. Noise generated by a set of rotating blades and fixed cascading blades, in the frequency domain 44010.4.3. Noise generated by blade–vortex interaction, using the vortex sound generation method 44910.5. Noise generated and propagation in the outer atmosphere: accounting for the thermal stratification and for likely obstacles 45410.5.1. Characteristic properties of the atmospheric boundary layer and impacts on sound propagation 45510.5.2. Models of sound wave propagation in the atmosphere 464Chapter 11 Implementation and Usage of Numerical Simulations 47511.1. Hybrid methods 47611.2. Direct numerical simulations/large eddy simulations 47811.3. Conclusion 488Bibliography 491Index 507

Fabien Anselmet is Professor at Ecole Centrale de Marseille in France. His research focuses on fluid turbulence and its numerous applications in the fields of industry and the environment. Pierre-Olivier Mattei is a CNRS researcher at LMA in Marseille, France. He leads research activities that aim towards a better understanding of the physics of acoustic radiation and fluid-structure interaction.

• Preface xiChapter 1 A Bit of History 11.1. The production of sound 11.2. The propagation of sound 41.3. The reception of sound 61.4. Aeroacoustics 7Chapter 2 Elements of Continuum Mechanics 92.1. Mechanics of deformable media 92.1.1. Continuum 92.1.2. Kinematics of deformable media 102.1.3. Deformation tensor (or Green’s tensor) 122.2. Conservation laws 132.2.1. Conservation of mass 132.2.2. Conservation of momentum 142.2.3. Conservation of energy 152.3. Constitutive laws 152.3.1. Elasticity 162.3.2. Thermoelasticity and effects of temperature variations 192.3.3. Viscoelasticity 212.3.4. Fluid medium 282.4. Hamilton principle 292.5. Characteristics of materials 29Chapter 3 Small Mathematics Travel Kit 313.1. Measure theory and Lebesgue integration 323.1.1. Boolean algebra 323.1.2. Measure on a σ-algebra 333.1.3. Convergence and integration of measurable functions 333.1.4. Functional space – functional 353.1.5. Measure as linear functional 363.2. Distributions 373.2.1. The space D of test functions 373.2.2. Distributions definition 373.2.3. Operations on distributions 393.2.4. N-dimensional generalization 433.2.5. Distributions tensor product 473.3. Convolution 483.3.1. Definition and first properties 483.3.2. Convolution algebra and Green’s function 503.4. Modal methods 523.4.1. Eigenmodes of a conservative system 523.4.2. Eigenmodes of a non-conservative system 55Chapter 4 Fluid Acoustics 654.1. Acoustics equations 664.1.1. Conservation equations 664.1.2. Establishment of general equations 674.1.3. Establishment of the wave equation 684.1.4. Velocity potential 694.2. Propagation and general solutions 694.2.1. One-dimensional motion 694.2.2. Three-dimensional motion 704.3. Permanent regime: Helmholtz equation 714.3.1. General solutions 724.3.2. Green’s kernels 764.3.3. Wave group, phase velocity and group velocity 784.4. Discontinuity equations 804.4.1. Interface between two propagating media 804.4.2. Interface between a propagating and a non-propagating medium 824.5. Impedance: measurement and model 834.5.1. Kundt’s tube 834.5.2. Delany–Bazley model 854.6. Homogeneous anisotropic medium 874.7. Medium with a slowly varying celerity 884.8. Media in motion 894.8.1. Homogeneous medium in uniform motion 894.8.2. Plane interface between media in motion 904.8.3. Cylindrical interface between media in motion 924.8.4. Acoustic radiation of a moving surface 94Chapter 5 Radiation, Diffraction, Enclosed Space 1055.1. Acoustic radiation 1065.1.1. A simple example 1065.2. Acoustic radiation of point sources 1075.2.1. Multipolar sources in a harmonic regime 1075.2.2. Far-field 1115.3. Radiation of distributed sources 1115.3.1. Layer potentials 1115.3.2. Green’s representation of pressure and introduction to the theory of diffraction 1145.4. Acoustic radiation of a piston in a plane 1195.4.1. Far-field radiation of a circular piston: directivity 1225.4.2. Radiation along the axis of a circular piston 1255.5. Acoustic radiation of a rectangular baffled structure 1265.6. Acoustic radiation of moving sources 1315.6.1. Compact and non-compact sources 1315.6.2. Sources in uniform and non-uniform motion 1355.7. Sound propagation in a bounded medium 1385.7.1. Eigenfrequencies and resonance frequencies 1385.7.2. The Helmholtz resonator 1395.7.3. Example in dimension 1 1405.7.4. Example in dimension 3 1415.7.5. Propagation of pure sound in a circular enclosure 1435.8. Basics of room acoustics 1495.8.1. The concept of acoustic power 1495.8.2. Directivity index 1495.8.3. Reverberation duration 1505.8.4. Reverberant fields 1535.8.5. Pressure level in rooms 1545.8.6. Crossover frequency and the reverberation distance 1555.9. Sound propagation in a wave guide 1565.9.1. General solution in a wave guide 1565.9.2. Physical interpretation and theory of modes 1575.9.3. Green’s function 1605.9.4. Section change 1615.9.5. Propagation in a conduit in the presence of flow 164Chapter 6 Wave Propagation in Elastic Media 1676.1. Equation of mechanical wave propagation 1686.2. Free waves 1696.2.1. Volumic waves 1696.2.2. Plane wave case 1706.2.3. Surface waves 1716.3. Green’s kernels in a harmonic regime 1766.4. Thin body approximation for plannar structures 1776.4.1. Straight beams 1786.4.2. Plane plates 1866.5. Thin body approximation for cylindrical structures 1986.5.1. Cylinder 1986.5.2. Ring 212Chapter 7 Vibrations of Thin Structures 2197.1. Beam vibrations 2197.1.1. Beam compression vibrations 2197.1.2. Beam bending vibrations 2237.2. Plate vibrations 2337.2.1. Infinite plate 2337.2.2. Finite plate 2397.2.3. Plate of arbitrary shape 2567.3. Cylindrical shell vibrations 2607.3.1. Infinite shell 2607.3.2. Finite shell 264Chapter 8 Acoustic Radiation of Thin Plates 2758.1. First notions of vibroacoustics: a simple example 2768.1.1. Motion equations 2778.1.2. Acoustic radiation 2788.1.3. “Light fluid” approximation 2808.1.4. Sound transmission 2818.1.5. Transient regime 2908.2. Free waves in an infinite plate immersed in a fluid 2948.2.1. Roots of the dispersion equation 2958.2.2. Light fluid approximation 2978.3. Transmission of a plane wave by a thin plate 2998.4. Radiation of an infinite plate under point excitation 3028.4.1 Integro-differential equation with respect to u 3038.4.2 Fourier transform of u 3038.4.3 Calculation of u(r) 3058.4.4. Radiated acoustic pressure 3068.5. Acoustic radiation and vibration of finite plates 3078.5.1. Statement of the problem 3078.5.2. Exact methods 3088.5.3. Light fluid approximation 3138.5.4. Higher order approximations 3198.6. Heavy fluid coupling: resonance estimation 3278.6.1. Clamped rectangular plate coupled with a heavy fluid 3278.6.2. Location of resonances of a coupled plate 3438.7. Vibrations of a thin plate in a turbulent flow 3468.7.1. Interspectral density: simple models 3478.7.2. Green’s representation of a coupled plate 3508.8. Aeroelastic coupling and sloshing 3548.8.1. Sloshing 3548.8.2. Convective instability 3568.8.3. Kelvin–Helmholtz instability 360Chapter 9 Basic Theoretical Aeroacoustics Models 3639.1. Preamble 3639.2. Lighthill’s equation and some of the generalizations that have followed 3659.3. Reminder of some notions on turbulence which will be useful here 3769.4. The Proudman model for homogeneous and isotropic turbulence 3819.5. The Lilley model for homogeneous and isotropic turbulence 3869.6. The recent models and a few experimental validations 3879.7. The Powell–Howe equation for vorticity-generated sound 397Chapter 10 A Few Situations Closer to Reality 40310.1. The Ribner model for jets 40310.2. Problems and approaches specific to boundary layers 41610.3. Flame-generated noise 42610.4. Noise generated by blades 43210.4.1. Noise generated by a solid body in motion, in the temporal domain 43310.4.2. Noise generated by a set of rotating blades and fixed cascading blades, in the frequency domain 44010.4.3. Noise generated by blade–vortex interaction, using the vortex sound generation method 44910.5. Noise generated and propagation in the outer atmosphere: accounting for the thermal stratification and for likely obstacles 45410.5.1. Characteristic properties of the atmospheric boundary layer and impacts on sound propagation 45510.5.2. Models of sound wave propagation in the atmosphere 464Chapter 11 Implementation and Usage of Numerical Simulations 47511.1. Hybrid methods 47611.2. Direct numerical simulations/large eddy simulations 47811.3. Conclusion 488Bibliography 491Index 507