Formulas of Acoustics

Transcription

1 Formulas of Acoustics

2 F. P. Mechel (Ed.) Formulas of Acoustics With contributions by M.1. Munjal. M. VorHinder. P. Koitzsch. M. Ochmann A. Cummings W. Maysenholder. W. Arnold o. V. Rudenko 1 st ed Corr. 2nd printing With approximate 620 Figures and 70 Tables ~ Springer

3 Editor: Prof. Dr. Fridolin P. Mechel Landhausstr Grafenau Germany ISBN ISBN (ebook) DOI / Cataloging-in-Publication Data applied for Bibliographic information is published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche N ationalbibliografie; detailed bibliographic data is available in the Internet at This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting. reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law ofseptember 9, 1965, in its current version, and permis sion for use must always be obtained from Springer-Verlag Berliu Heidelberg GmbH. Violations are liable to prosecution act under German Copyright Law. springeronline.com Springer-V erlag Berlin Heidelberg 2002, 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1 st edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera ready by authors Cover-Design: deblik, Berlin Printed on acid free paper 62/3020/kk O

4 Formulas of Acoustics v Editorial A Editorial A.1 Preface : Modem acoustics is more and more based on computations, and computations are based on formulas. Such work needs previous and contemporary results. It consumes much time and effort to search needed formulas during the actual work. Therefore, fundamentals and results of acoustics that can be expressed as formulas will be collected in this book. The author has compiled a private formula collection during all his professional life. This personal collection gave the stimulus to expand it and to make it available to colleagues. The present formula collection is subdivided into fields of acoustics. For some fields, in which this author is not expert enough, he invited co-authors to contribute. Most colleagues contacted for possible contributions were convinced of the project and agreed spontaneously. The material within a field of acoustics is subdivided in sections which deal with a defmed task. Some overlap of sections has to be tolerated; but the subdivision into well-defined sections will be helpful to the reader to find a particular topic of interest. The present formula collection should not be considered a textbook in a condensed form. Derivations of a presented result will be described only as far as they are helpful in understanding the problem; the more interested reader is referred to the "source" of the result. Useful principles and computational procedures will also be included, even if they need more description. Symbols and quantities will be well defined, and wherever useful a sketch will help to explain the object and the task. One of the advantages of a formula collection is seen in uniform definitions, notations and symbols for quantities. A strict uniformity in the form of a central list of symbols used never works, according to this author's observation. Therefore, only commonly used symbols (such as medium density, speed of sound, circular frequency, etc.) are collected in a central list of symbols; other symbols are defined in the relevant chapter. Many sections contain, below their title or in the text, a reference to the literature. It cannot be the task and intention of this book either to indicate time priorities of publications concerning

5 Formulas of Acoustics VI Editorial a topic or to give a survey of the existing literature. The reference quoted is the source of more information, which the author has used. Computation and formulas are a form of mathematics. Higher transcendental functions are used in the mathematics of acoustics. Functions will be explained by reference to mathematical literature, if necessary. If functions are used with different definitions in the literature, the definition used will be presented. For more details of higher transcendental functions the reader is referred to mathematical formula collections and handbooks. To the author's knowledge, this is the first formula collection existing in acoustics, and, like every first realisation of an idea, it is an experiment. Also for the publisher, who has limited the first edition to about 1200 pages. This frame is one of the reasons why not all important results from the literature are contained in the book; another reason is the fact that the author(s) had no knowledge of such results. If this book will be accepted by the "acoustics community" further editions surely will follow without page number limitation. Colleagues who wish to find formulas in a future edition which not yet are included are kindly invited to give suitable indications to the author. Nevertheless the authors think that the book in its present form contains most of traditional and modem results of both fundamental and special character so that the book can be helpful to researchers and engineers in the fields of physical acoustics, noise control, and room acoustics. The manuscript was written in a camera ready form (in order to avoid proof reading). So printing errors are the responsibility of the author. He would be grateful for indications of such errors. The author gratefully acknowledges the support given to the project by the co-authors and by the publisher. Grafenau, October 2001

6 Formulas of Acoustics VII Contents A.2 Contents A. Editorials A.I Preface... V A.2 Contents... VII A.3 Conventions... XXI B. General Linear Fluid Acoustics B.I 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.S Green's functions and formalism... 9 B.6 Orthogonality of modes in a duct with locally reacting walls B.7 Orthogonality of modes in a duct with laterally reacting walls B.8 Source conditions B.9 Sommerfeld's condition B.IO Principles of superposition B.ll Hamilton's principle B.12. Adjoined wave equation B.13 Vector and tensor formulation offundamentals B.14 Boundary condition at a moving boundary B.IS Boundary conditions at liquids and solids... 38

7 Formulas of Acoustics VIII Contents B.16 Corner conditions B.17 Surface wave at locally reacting plane B.18 Surface wave along a locally reacting cylinder B.19 Periodic structures, admittance grid B.20 Plane wall with wide grooves Thin grid on half-infinite porous layer B.22 Grid of finite thickness with narrow slits on half-infinite porous layer Grid of finite thickness with wide slits on half-infinite porous layer c. Equivalent Networks C.l Fundamentals of equivalent networks C.2 Distributed network elements C.3 Elements with constrictions CA Superposition of multiple sources in a network C.5 Chain circuit D. Reflection of Sound D.l Plane wave reflection at a locally reacting plane Plane wave reflection at an infinitely thick porous layer D.3 Plane wave reflection at a porous layer of finite thickness DA Plane wave reflection at a multiple layer absorber D.5 Diffuse sound reflection at a locally reacting plane Diffuse sound reflection at a bulk reacting porous layer D.7 Sound reflection and scattering at finite-size absorbers D.8 Uneven, local absorber surface Scattering at the border of an absorbent half-plane... 88

8 Formulas of Acoustics IX Contents 0.10 Absorbent strip in a hard baffle wall, with far field distribution Absorbent strip in a hard baffle wall, as variational problem Absorbent strip in a hard baffle wall, with Mathieu functions Absorption of finite-size absorbers as a problem of radiation A monopole line source above an infinite, plane absorber; integration method A monopole line source above an infinite, plane absorber; with principle of superposition A monopole point source above a bulk reacting plane, exact forms A monopole point source above a locally reacting plane, exact forms A monopole point source above a locally reacting plane, exact saddle point integration A monopole point source above a locally reacting plane, approximations A monopole point source above a bulk reacting plane, approximations E. Scattering of Sound E.l Plane wave scattering at cylinders E.2 Plane wave scattering at cylinders and spheres E.3 Multiple scattering at cylinders and spheres E.4 Cylindrical wave scattering at cylinders E.5 Cylindrical or plane wave scattering at a comer surrounded by a cylinder E.6 Plane wave scattering at a hard screen E.7 Cylindrical or plane wave scattering at a screen with an elliptical cylinder atop E.8 Uniform scattering at screens and dams E.9 Scattering at a flat dam E.lO Scattering at a semicircular absorbing dam on absorbing ground E.ll Scattering in random media, general

9 Formulas of Acoustics x Contents E.12 Function tables for monotype scattering E.13 Sound attenuation in a forest E.l4 Mixed monotype scattering in random media E.l5 Multiple triple-type scattering in random media E. I 6 Plane wave scattering at elastic cylindrical shell E.17 Plane wave backscattering by a liquid sphere E.l8 Spherical wave scattering at a perfectly absorbing wedge E.l9 Impulsive spherical wave scattering at a hard wedge E.20 Spherical wave scattering at a hard screen F. Radiation of Sound F.I Definition of radiation impedance and end corrections F.2 Some methods to evaluate the radiation impedance F.3 Spherical radiators F.4 Cylindrical radiators F.5 Piston radiator on a sphere F.6 Strip-shaped radiator on cylinder F.7 Plane piston radiators F.8 Uniform end correction of plane piston radiators F.9 Narrow strip-shaped, field excited radiator F.10 Wide strip-shaped, field excited radiator F.II Wide rectangular, field excited radiator F.12 End corrections F.13 Piston radiating into a hard tube F.14 Oscillating mass of a fence in a hard tube F.15 A ring-shaped piston in a baffle wall

10 Formulas of Acoustics XI Contents F.16 Measures of radiation directivity F.17 Directivity of radiator arrays F.18 Radiation of finite length cylinder F.19 Monopole and multi pole radiators F.20 Plane radiator in a baffle wall F.2l Ratio of radiation and excitation efficiencies of plates F.22 Radiation of plates with special excitations G. Porous Absorbers G.l Structure parameters of porous materials G.2 Theory ofthe quasi-homogeneous material G.3 RAYLEIGH model with round capillaries G.4 Model with flat capillaries G.5 Longitudinal flow resistivity in parallel fibres G.6 Longitudinal sound in parallel fibres G.7 Transversal flow resistivity in parallel fibres G.8 Transversal sound in parallel fibres G.9 Effective wave multiple scattering in transversal fibre bundle G.IO BlOT's theory of porous absorbers G.11 Empirical relations for characteristic values of fibre absorbers G.l2 Characteristic values from theoretical models fitted to experimental data H. Compound Absorbers H.l Absorber of flat capillaries H.2 Plate with narrow slits H.3 Plate with wide slits

11 Formulas of Acoustics XII Contents HA Dissipationless slit resonator H.5 Resonance frequencies and radiation loss of slit resonators H.6 Slit array with viscous and therrnallosses H.7 Slit resonator with viscous and therrnallosses H.8 Free plate with an array of circular holes, with losses H.9 Array of Helmholtz resonators with circular necks H.IO Slit resonator array with porous layer in the volume, fields H.ll Slit resonator array with porous layer in the volume, impedances H.12 Slit resonator array with porous layer on back orifice H.13 Slit resonator array with porous layer on front orifice H.14 Array of slit resonators with subdivided neck plate H.15 Array of slit resonators with subdivided neck plate and floating foil in the gap H.l6 Array of slit resonators covered with a foil H.17 Poro-elastic foils H.18 Foil resonator H.19 Ring resonator H.20 Wide-angle absorber, scattered far field H.21 Wide-angle absorber, near field and absorption H.22 Tight panel absorber, rigorous solution H.23 Tight panel absorber, approximations H.24 Porous panel absorber, rigorous solution I. Sound Transmission 1.1 "Noise barriers" Sound transmission through a slit in a wall I.3 Sound transmission through a hole in a wall

12 Formulas of Acoustics XIII Contents 1.4 Hole transmission with equivalent network Sound transmission through lined slits in a wall Chambered joint "Noise sluice" I.8 Sound transmission through plates, some fundamentals I.9 Sound transmission through a simple plate Infinite double-shell wall with absorber fill Double-shell wall with thin air gap Plate with absorber layer I.13 Sandwich panels I.14 Finite size plate I.15 Single plate across a flat duct I.16 Single plate in a wall niche I.17 Strip-shaped wall in infinite baffle wall Finite size plate with front side absorber layer Finite size plate with back side absorber layer I.20 Finite size double-shell wall with an absorber core Plenum modes I.22 Sound transmission through suspended ceilings I.23 Office fences I.24 Office fences, with 2 nd principle of superposition I.25 Infinite plate between two different fluids I.26 Sandwich plate with elastic core I.27 Wall of multiple sheets with air interspaces

13 Formulas of Acoustics XIV Contents J. Duct Acoustics J.l Flat capillary with isothermal boundaries Flat capillary with adiabatic boundaries J.3 Circular capillary with isothermal boundary J.4 Lined ducts, general J.5 Modes in rectangular ducts with locally reacting lining J.6 Least attenuated mode in rectangular, locally lined ducts J.7 Sets of mode solutions in rectangular, locally lined ducts Flat duct with bulk reacting lining J.9 Flat duct with anisotropic, bulk reacting lining J.1O Mode solutions in a flat duct with bulk reacting lining Flat duct with unsymmetrical, locally reacting lining J.12 Flat duct with unsymmetrical, bulk reacting lining J.13 Round duct with locally reacting lining J.14 Admittance of annular absorbers approximated with flat absorbers J.15 Round duct with bulk reacting lining J.16 Annular ducts J.17 Duct with cross-layered lining J.18 Single step of duct height and/or duct lining Sections and cascades of silencers without feedback J.20 A section with feedback between sections without feedback Concentrated absorber in an otherwise homogeneous lining J.22 Wide splitter type silencer with locally reacting splitters J.23 Splitter type silencer with locally reacting splitters in a hard duct Splitter type silencer with simple porous layers as bulk reacting splitters

14 Formulas of Acoustics xv Contents J.25 Splitter type silencer with splitters of porous layers covered with a foil Lined duct corners and junctions Sound radiation from lined duct orifice J.28 Conical duct transitions; special case: hard walls J.29 Lined conical duct transition, evaluated with stepping duct sections J.30 Lined conical duct transition, evaluated with stepping admittance sections Mode mixtures Mode excitation coefficients CREMER's admittance CREMER's admittance with parallel resonators J.35 Influence of flow on attenuation Influence of temperature on attenuation Stationary flow resistance of splitter silencers Nonlinearities by amplitude and/or flow Flow-induced nonlinearity of perforated sheets J 040 Reciprocity at duct joints Turning-vane splitter silencer K. Acoustic Mufflers K.O Conventions in the present chapter K.l Acoustic power in a flow duct K.2 Radiation from the open end of a flow ducl K.3 Transfer matrix representation KA Muffler performance parameters K.5 Uniform tube with flow and viscous losses K.6 Sudden area changes

15 Formulas of Acoustics XVI Contents K.7 Extended inlet/outlet K.8 Conical tube K.9 Exponential horn K. 10 Hose K.II Two-duct perforated elements K.12 Three-duct perforated elements K.13 Three-duct perforated elements with extended perforations K. 14 Three-pass (or four-duct) perforated elements K.15 Catalytic converter elements K.16 Helmholtz resonator K.17 In-line cavity K.18 Bellows K.19 Pod silencer K.20 Quincke tube K.21 Annular airgap lined duct K.22 Micro-perforated Helmholtz panel parallel baffle muffler K.23 Acoustically lined circular duct K.24 Parallel baffle muffler (Multi-pass lined duct} L. Capsules and Cabins L.l The energetic approximation for the efficiency of capsules L.2 Absorbent sound source in a capsule L.3 Semicylindrical source and capsule LA Hemispherical source and capsule L.5 Cabins, semicylindrical model L.6 Cabin with plane walls

16 Formulas of Acoustics XVII Contents L.7 Cabin with rectangular cross section M. Room Acoustics M.I Eigenfunctions in parallelepipeds M.2 Density of eigenfrequencies in rooms M.3 Geometrical room acoustics in paralle\epipeds M.4 Statistical room acoustics M.5 The mirror source model M.6 Ray tracing models M.7 Room impulse responses, decay curves, and reverberation times M.8 Other room acoustical parameters N. Flow Acoustics N.I Concepts and notations in fluid mechanics, in connection with the field of aeroacoustics N.2 Some tools in fluid mechanics and aeroacoustics N.3 The basic equations of fluid motion NA The equations of linear acoustics N.5 The inhomogeneous wave equation, Lighthill's acoustic analogy N.6 Acoustic Analogy with source terms using the pressure N.7 Acoustic analogy with mean flow effects, in form of convective inhomogeneous wave equation N.8 Acoustic analogy in terms of vorticity, wave operators for enthalpy N.9 Acoustic analogy with effects of solid boundaries N.l 0 Acoustic analogy in terms of entropy, heat sources as sound sources, sound generation by turbulent two-phase flow N.11 Acoustics of moving sources

17 Formulas of Acoustics XVIII Contents N.12 Aerodynamic sound sources in practice N.13 Power law of the aerodynamic sound sources O. Analytical and Numerical Methods in Acoustics 0.1 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 P. Variational Principles in Acoustics P.l Eigenfrequencies of a rigid-walled cavity and modal cut-on frequencies of a uniform flat-oval duct with zero mean fluid flow P.2 Sound propagation in a uniform narrow tube of arbitrary cross-section with zero mean fluid flow 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 P.4 Sound propagation in a uniform, rigid-walled, rectangular flow duct containing an anisotropic bulk-reacting wall lining or baffles P.5 Sound propagation in a uniform, rigid-walled, flow duct of arbitrary cross-section, with an inhomogeneous, anisotropic bulk lining 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 P.7 Sound propagation in a rectangular section duct with four flexible walls, an anisotropic bulk lining and no mean gas flow

18 Formulas of Acoustics XIX Contents Q. Elasto-Acoustics Q.l Fundamental equations of motion Q.2 Anisotropy and isotropy Q.3 Interface conditions, reflection and refraction of plane waves Q.4 Material damping Q.5 Energy Q.6 Random media Q.7 Periodic media Q.8 Homogenization Q.9 Plane waves in unbounded homogeneous media Q.I0 Waves in bounded media Q.l1 Moduli of isotropic materials and related quantities Q.12 Modes of rectangular plates Q.13 Partition impedance of plates Q.14 Partition impedance of shells Q.15 Density of eigenfrequencies in plates, bars, strings, membranes Q.16 Foot point impedances of forces Q.17 Transmission loss at steps, joints, comers Q.18 Cylindrical shell Q.19 Similarity relations for spherical shells Q.20 Sound radiation from plates R. Ultrasound Absorption in Solids R.l Generation of ultrasound R.2 Ultrasonic attenuation

19 Formulas of Acoustics xx Contents R.3 Absorption and dispersion in solids due to dislocations RA Absorption due to the thermoelastic effects, phonon scattering and related effects 1132 R.5 Interaction of ultrasound with electrons in metals R.6 Wave propagation in piezoelectric semiconducting solids R.7 Absorption in amorphous solids and glasses R.8 Relation of ultrasonic absorption to internal friction R.9 Gases and liquids R.10 Kramers-Kroning relation s. Nonlinear Acoustics S.l General formulas S.2 Riemann waves S.3 Plane nonlinear waves in a dissipative medium SA One-dimensional nonlinear waves in a dissipative medium Index 1153

20 Formulas of Acoustics XXI Conventions A.3 Conventions The following conventions will be valid in the book. Exceptions will be clearly noted. Time factor The time factor for harmonic oscillations and waves is e j 00 t. This choice implies that the imaginary part of impedances with mass reaction are positive, with spring reaction negative, and the imaginary part of admittances with mass reaction are negative, with spring reaction positive. If not stated differently, the time function is assumed to be e joot ; the time factor then is dropped, mostly. Impedance and admittance "Impedance" is used for the ratio of sound pressure p to the vector component in some direction of particle velocity v: Z=p/v. "Mechanical impedance" is used for the ratio of vector components of force F to particle velocity v: Zm=F/v. "Flow impedance" is used for the ratio of sound pressure p to volume flow q=s v through a surface S, with v the velocity component normal to S. "Admittance" is the ratio of the vector Y of particle velocity to sound pressure p. It is a true vector. "Mechanical admittance" is the reciprocal of mechanical impedance. "Flow admittance" is the ratio of the flow vector q to sound pressure; it is a true vector. Sound intensity and power Sound intensity is the vector 1= p. y * (where the asterisk indicates the complex conjugate); I stands for the momentary sound power in the direction of y through a unit surface. The effective intensity is the real part thereof in the time average. (The formally possible defmition j = p *.y = I * would produce conflicts at sound sources.)

21 Formulas of Acoustics XXII Conventions Sound power is the integral of the scalar product of sound intensity with the surface element vector ds over a surface S: n = f Y. ds. Dimensions S Where necessary, the dimension of a quantity is indicated in brackets [... ). Complex quantities Field quantities, such as sound pressure p, particle velocity v, oscillating parts of density p, and of temperature T, etc. are mostly complex. If you record such a quantity in an oscillogram, you may take either the real or the imaginary part of a complex expression, after multiplication with the dropped time factor. If you record the amplitude, this corresponds to taking the magnitude of the complex quantity. Symbol "decorations" Unnecessary symbol decorations, such as hats for amplitudes, underbars for complex quantities, etc. are avoided. If necessary in the local context, an arrow indicates a vector v, a star is used for the complex conjugate p *,primes are used either for the derivative of functions, f'(x),f"(x), or (where no ambiguity is possible) for the real and imaginary parts of complex quantities. Commonly used symbols These symbols are commonly used in most sections of the book. If a section uses the same symbol with a different definition, it will be noted. co f adiabatic sound speed in the medium (e.g. in air) [m/s); frequency [Hz); ="(-1) imaginary unit; ko =<.O/cO free field wave number ofa plane wave [11m); 2 p sound pressure [Palm ); q volume flow [I/(m s»); time [sc]; v velocity [m/s];

22 Formulas of Acoustics XXIII Conventions Zo =poco wave impedance offree plane wave [Pa s/m]; sound absorption coefficient; K adiabatic exponent of the medium; wavelength [m]; AO P PO wavelength of free plane wave; mass density [kg/m \ mass density of the medium; =211:/f angular (or circular) frequency [I/s]; polar angle of cylindrical and spherical co-ordinates; azimuthal angle of spherical co-ordinates; n power; 2 L'lp/(d v) flow resistivity of porous material [Pa s/m ] (flow resistance per unit thickness d); Numbering of equations Equations are numbered, beginning with number (1) in each Section. Reference to an equation is made as, e.g., "eq.(x)" to the equation with number (x) in the same Section, or as, e.g., "equ.(k.y.x)" to the equation with number (x) in the Section with number y of the Chapter K.