Formulas of Acoustics

Transcription

1 F. P. Mechel (Ed.) Formulas of Acoustics With contributions by M. L. Munjal M. Vorländer P. Költzsch M. Ochmann A. Cummings W. Maysenhölder W. Arnold O. V. Rudenko With approximate 620 Figures and 70 Tables JS Springer

2 Formulas of Acoustics VII Contents A.2 Contents A. Editorials A.l Preface V A.2 Contents VII A.3 Conventions XXI B. General Linear Fluid Acoustics B.l Fundamental differential equations 1 B.2 Material constants of air 4 B.3 General relation for field admittance and intensity 7 B.4 Integral relations 8 B.5 Green's functions and formalism 9 B.6 Orthogonality of modes in a duct with locally reacting walls 17 B.7 Orthogonality of modes in a duct with laterally reacting walls 18 B.8 Source conditions 18 B.9 Sommerfeld's condition 20 B.10 Principles of superposition 20 B.ll Hamilton's principle 23 B.12. Adjoined wave equation 24 B.13 Vector and tensor formulation of fundamentals 25 B.14 Boundary condition at a moving boundary 37 B.15 Boundary conditions at liquids and solids 38

3 Formulas of Acoustics VIII Contents B.16 Corner conditions 39 B.17 Surface wave at locally reacting plane 39 B.l 8 Surface wave along a locally reacting cylinder 41 B.19 Periodic structures, admittance grid 42 B.20 Plane wall with wide grooves 47 B.21 Thin grid on half-infinite porous layer 49 B.22 Grid of finite thickness with narrow slits on half-infinite porous layer 52 B.23 Grid of finite thickness with wide slits on half-infinite porous layer 54 C. Equivalent Networks C.l Fundamentals of equivalent networks 59 C.2 Distributed network elements 65 C.3 Elements with constrictions 70 C.4 Superposition of multiple sources in a network 71 C.5 Chain circuit 71 D. Reflection of Sound D.l Plane wave reflection at a locally reacting plane 74 D.2 Plane wave reflection at an infinitely thick porous layer 75 D.3 Plane wave reflection at a porous layer of finite thickness 76 D.4 Plane wave reflection at a multiple layer absorber 78 D.5 Diffuse sound reflection at a locally reacting plane 79 D.6 Diffuse sound reflection at a bulk reacting porous layer 81 D.7 Sound reflection and scattering atfinite-sizeabsorbers 82 D.8 Uneven, local absorber surface 86 D.9 Scattering at the border of an absorbent half-plane 88

4 Formulas of Acoustics IX Contents D.10 Absorbent strip in a hard baffle wall, with far field distribution 89 D.l 1 Absorbent strip in a hard baffle wall, as variational problem 91 D.12 Absorbent strip in a hard baffle wall, with Mathieu functions 94 D.13 Absorption of finite-size absorbers as a problem of radiation 98 D. 14 A monopole line source above an infinite, plane absorber; integration method 99 D. 15 A monopole line source above an infinite, plane absorber; with principle of superposition 107 D.16 A monopole point source above a bulk reacting plane, exact forms 109 D. 17 A monopole point source above a locally reacting plane, exact forms 112 D.l8 A monopole point source above a locally reacting plane, exact saddle point integration 114 D.19 A monopole point source above a locally reacting plane, approximations 117 D.20 A monopole point source above a bulk reacting plane, approximations 124 E. Scattering of Sound E.l Plane wave scattering at cylinders 129 E.2 Plane wave scattering at cylinders and spheres 132 E.3 Multiple scattering at cylinders and spheres 142 E.4 Cylindrical wave scattering at cylinders 143 E.5 Cylindrical or plane wave scattering at a corner surrounded by a cylinder 145 E.6 Plane wave scattering at a hard screen 152 E.7 Cylindrical or plane wave scattering at a screen with an elliptical cylinder atop 153 E.8 Uniform scattering at screens and dams 158 E.9 Scattering at a flat dam 167 E.10 Scattering at a semicircular absorbing dam on absorbing ground 169 E.l 1 Scattering in random media, general 173

5 Formulas of Acoustics X Contents E.12 Function tables for monotype scattering 180 E.13 Sound attenuation in a forest 184 E.14 Mixed monotype scattering in random media 186 E.15 Multiple triple-type scattering in random media 191 E.16 Plane wave scattering at elastic cylindrical shell 202 E.17 Plane wave backscattering by a liquid sphere 205 E.18 Spherical wave scattering at a perfectly absorbing wedge 206 E.19 Impulsive spherical wave scattering at a hard wedge 208 E.20 Spherical wave scattering at a hard screen 210 F. Radiation of Sound F.l Definition of radiation impedance and end corrections 214 F.2 Some methods to evaluate the radiation impedance 216 F.3 Spherical radiators 218 F.4 Cylindrical radiators 222 F.5 Piston radiator on a sphere 224 F.6 Strip-shaped radiator on cylinder 226 F.7 Plane piston radiators 227 F.8 Uniform end correction of plane piston radiators 236 F.9 Narrow strip-shaped, field excited radiator 236 F.10 Wide strip-shaped, field excited radiator 238 F.ll Wide rectangular, field excited radiator 240 F.l2 End corrections 243 F.13 Piston radiating into a hard tube 253 F.14 Oscillating mass of a fence in a hard tube 253 F. 15 A ring-shaped piston in a baffle wall 254

6 Formulas of Acoustics XI Contents F.16 Measures of radiation directivity 255 F.17 Directivity of radiator arrays 256 F.18 Radiation of finite length cylinder 260 F.19 Monopole and multipole radiators 261 F.20 Plane radiator in a baffle wall 264 F.21 Ratio of radiation and excitation efficiencies of plates 269 F.22 Radiation of plates with special excitations 269 G. Porous Absorbers G.l Structure parameters of porous materials 272 G.2 Theory of the quasi-homogeneous material 276 G.3 RAYLEIGH model with round capillaries 277 G.4 Model with flat capillaries 280 G.5 Longitudinal flow resistivity in parallel fibres 281 G.6 Longitudinal sound in parallel fibres 283 G.7 Transversal flow resistivity in parallel fibres 286 G.8 Transversal sound in parallel fibres 293 G.9 Effective wave multiple scattering in transversal fibre bundle 304 G.10 BlOT's theory of porous absorbers 309 G.l 1 Empirical relations for characteristic values of fibre absorbers 319 G.12 Characteristic values from theoretical models fitted to experimental data 324 H. Compound Absorbers H.l Absorber of flat capillaries 330 H.2 Plate with narrow slits 333 H.3 Plate with wide slits 336

7 Formulas of Acoustics XII Contents H.4 Dissipationless slit resonator 340 H.5 Resonance frequencies and radiation loss of slit resonators 344 H.6 Slit array with viscous and thermal losses 346 H.7 Slit resonator with viscous and thermal losses 351 H.8 Free plate with an array of circular holes, with losses 354 H.9 Array of Helmholtz resonators with circular necks 359 H.10 Slit resonator array with porous layer in the volume, fields 362 H.ll Slit resonator array with porous layer in the volume, impedances 369 H.12 Slit resonator array with porous layer on back orifice 375 H.13 Slit resonator array with porous layer on front orifice 377 H.14 Array of slit resonators with subdivided neck plate 381 H.15 Array of slit resonators with subdivided neck plate and floating foil in the gap 383 H.16 Array of slit resonators covered with a foil 387 H.17 Poro-elastic foils 390 H.18 Foil resonator 395 H.19 Ring resonator 397 H.20 Wide-angle absorber, scattered far field 401 H.21 Wide-angle absorber, near field and absorption 407 H.22 Tight panel absorber, rigorous solution 412 H.23 Tight panel absorber, approximations 420 H.24 Porous panel absorber, rigorous solution 423 I. Sound Transmission 1.1 "Noise barriers" Sound transmission through a slit in a wall Sound transmission through a hole in a wall 439

8 Formulas of Acoustics XIII Contents 1.4 Hole transmission with equivalent network Sound transmission through lined slits in a wall Chambered joint "Noise sluice" Sound transmission through plates, some fundamentals Sound transmission through a simple plate Infinite double-shell wall with absorber fill Double-shell wall with thin air gap Plate with absorber layer Sandwich panels Finite size plate Single plate across a flat duct Single plate in a wall niche Strip-shaped wall in infinite baffle wall Finite size plate with front side absorber layer Finite size plate with back side absorber layer Finite size double-shell wall with an absorber core Plenum modes Sound transmission through suspended ceilings Office fences Office fences, with 2 nd principle of superposition Infinite plate between two different fluids Sandwich plate with elastic core Wall of multiple sheets with air interspaces 521

9 Formulas of Acoustics XIV Contents J. Duct Acoustics J.l Flat capillary with isothermal boundaries 527 J.2 Flat capillary with adiabatic boundaries 530 J.3 Circular capillary with isothermal boundary 531 J.4 Lined ducts, general 534 J.5 Modes in rectangular ducts with locally reacting lining 538 J.6 Least attenuated mode in rectangular, locally lined ducts 541 J.7 Sets of mode solutions in rectangular, locally lined ducts 545 J.8 Flat duct with bulk reacting lining 552 J.9 Flat duct with anisotropic, bulk reacting lining 554 J.10 Mode solutions in a flat duct with bulk reacting lining 555 J. 11 Flat duct with unsymmetrical, locally reacting lining 558 J. 12 Flat duct with unsymmetrical, bulk reacting lining 560 J.13 Round duct with locally reacting lining 561 J. 14 Admittance of annular absorbers approximated with flat absorbers 575 J. 15 Round duct with bulk reacting lining 577 J. 16 Annular ducts 580 J. 17 Duct with cross-layered lining 583 J.18 Single step of duct height and/or duct lining 591 J. 19 Sections and cascades of silencers without feedback 602 J.20 A section with feedback between sections without feedback 603 J.21 Concentrated absorber in an otherwise homogeneous lining 607 J.22 Wide splitter type silencer with locally reacting splitters 611 J.23 Splitter type silencer with locally reacting splitters in a hard duct 614 J.24 Splitter type silencer with simple porous layers as bulk reacting splitters 620

10 Formulas of Acoustics XV Contents J.25 Splitter type silencer with splitters of porous layers covered with a foil 623 J.26 Lined duct corners and junctions 625 J.27 Sound radiation from lined duct orifice 630 J.28 Conical duct transitions; special case: hard walls 634 J.29 Lined conical duct transition, evaluated with stepping duct sections 637 J.30 Lined conical duct transition, evaluated with stepping admittance sections 644 J.31 Mode mixtures 647 J.32 Mode excitation coefficients 651 J.33 CREMER's admittance 653 J.34 CREMER's admittance with parallel resonators 658 J.35 Influence of flow on attenuation 665 J.36 Influence of temperature on attenuation 673 J.37 Stationary flow resistance of splitter silencers 675 J.38 Nonlinearities by amplitude and/or flow 675 J.39 Flow-induced nonlinearity of perforated sheets 681 J.40 Reciprocity at duct joints 683 J.41 Turning-vane splitter silencer 683 K. Acoustic Mufflers K.O Conventions in the present chapter 689 K.l Acoustic power in a flow duct 690 K.2 Radiation from the open end of a flow duct 691 K.3 Transfer matrix representation 692 K.4 Muffler performance parameters 693 K.5 Uniform tube with flow and viscous losses 695 K.6 Sudden area changes 696

11 Formulas of Acoustics XVI Contents K.7 Extended inlet/outlet 698 K.8 Conical tube 700 K.9 Exponential horn 700 K. 10 Hose 701 K.l 1 Two-duct perforated elements 703 K.12 Three-duct perforated elements 711 K.13 Three-duct perforated elements with extended perforations 717 K. 14 Three-pass (or four-duct) perforated elements, 722 K.15 Catalytic converter elements 726 K.l6 Helmholtz resonator 728 K.l7 In-line cavity 728 K.18 Bellows 729 K.19 Pod silencer 730 K.20 Quincke tube 731 K.21 Annular airgap lined duct 732 K.22 Micro-perforated Helmholtz panel parallel baffle muffler 734 K.23 Acoustically lined circular duct 735 K.24 Parallel baffle muffler (Multi-pass lined duct) 737 L. Capsules and Cabins L.l The energetic approximation for the efficiency of capsules 741 L.2 Absorbent sound source in a capsule 745 L.3 Semicylindrical source and capsule 751 L.4 Hemispherical source and capsule 755 L.5 Cabins, semicylindrical model 760 L.6 Cabin with plane walls 764

12 Formulas of Acoustics XVII Contents L.7 Cabin with rectangular cross section 770 M. Room Acoustics M. 1 Eigenfunctions in parallelepipeds 774 M.2 Density of eigenfrequencies in rooms 777 M.3 Geometrical room acoustics in parallelepipeds 778 M.4 Statistical room acoustics 780 M.5 The mirror source model 784 M.6 Ray tracing models 837 M. 7 Room impulse responses, decay curves, and reverberation times 841 M.8 Other room acoustical parameters 842 N. Flow Acoustics N.l Concepts and notations in fluid mechanics, in connection with the field of aeroacoustics 846 N.2 Some tools in fluid mechanics and aeroacoustics 850 N.3 The basic equations of fluid motion 856 N.4 The equations of linear acoustics 863 N.5 The inhomogeneous wave equation, Lighthill's acoustic analogy 866 N.6 Acoustic Analogy with source terms using the pressure 871 N.7 Acoustic analogy with mean flow effects, in form of convective inhomogeneous wave equation 874 N.8 Acoustic analogy in terms of vorticity, wave operators for enthalpy 880 N.9 Acoustic analogy with effects of solid boundaries 890 N.10 Acoustic analogy in terms of entropy, heat sources as sound sources, sound generation by turbulent two-phase flow 895 N.l 1 Acoustics of moving sources 901

13 Formulas of Acoustics XVIII Contents N.12 Aerodynamic sound sources in practice 908 N.13 Power law of the aerodynamic sound sources 919 O. Analytical and Numerical Methods in Acoustics O.l Computational optimisation of sound absorbers Computing with mixed numeric-symbolic expressions, illustrated with silencer cascades Five standard problems of numerical acoustics The source simulation technique The boundary element method (BEM) The finite element method (FEM) The Cat's Eye model The Orange model 1005 P. Variational Principles in Acoustics P. 1 Eigenfrequencies of a rigid-walled cavity and modal cut-on frequencies of a uniform flat-oval duct with zero mean fluid flow 1026 P.2 Sound propagation in a uniform narrow tube of arbitrary cross-section with zero mean fluid flow 1029 P.3 Sound propagation in a uniform, rigid-walled, duct of arbitrary cross-section with a bulk-reacting lining and no mean fluid flow: low frequency approximation 1034 P.4 Sound propagation in a uniform, rigid-walled, rectangular flow duct containing an anisotropic bulk-reacting wall lining or baffles 1035 P.5 Sound propagation in a uniform, rigid-walled, flow duct of arbitrary cross-section, with an inhomogeneous, anisotropic bulk lining 1038 P.6 Sound propagation in a uniform duct of arbitrary cross-section with one or more plane flexible walls, an isotropic bulk lining and a uniform mean gas flow 1042 P.7 Sound propagation in a rectangular section duct with four flexible walls, an anisotropic bulk lining and no mean gas flow 1046

14 Formulas of Acoustics XIX Contents Q. Elasto-Acoustics Q.l Fundamental equations of motion 1052 Q.2 Anisotropy and isotropy 1054 Q.3 Interface conditions, reflection and refraction of plane waves 1059 Q.4 Material damping 1060 Q.5 Energy 1064 Q.6 Random media 1066 Q.7 Periodic media 1067 Q.8 Homogenization 1070 Q.9 Plane waves in unbounded homogeneous media 1073 Q.10 Waves in bounded media 1077 Q.ll Moduli of isotropic materials and related quantities 1089 Q.12 Modes of rectangular plates 1093 Q.13 Partition impedance of plates 1096 Q.14 Partition impedance of shells 1099 Q. 15 Density of eigenfrequencies in plates, bars, strings, membranes 1102 Q.16 Foot point impedances of forces 1102 Q.17 Transmission loss at steps, joints, corners 1108 Q.18 Cylindrical shell 1110 Q.19 Similarity relations for spherical shells 1114 Q.20 Sound radiation from plates 1115 R. Ultrasound Absorption in Solids R.l Generation of ultrasound 1123 R.2 Ultrasonic attenuation 1124

15 Formulas of Acoustics XX Contents R.3 Absorption and dispersion in solids due to dislocations 1129 R.4 Absorption due to the thermoelastic effects, phonon scattering and related effects 1132 R.5 Interaction of ultrasound with electrons in metals 1134 R.6 Wave propagation in piezoelectric semiconducting solids 1136 R.7 Absorption in amorphous solids and glasses 1136 R.8 Relation of ultrasonic absorption to internal friction 1138 R.9 Gases and liquids 1138 R. 10 Kramers-Kroning relation 1138 S. Nonlinear Acoustics 5.1 General formulas Riemann waves Plane nonlinear waves in a dissipative medium One-dimensional nonlinear waves in a dissipative medium 1150 index 1153