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1197 Pages·2004·4.12 MB·English
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Formulas of Acoustics 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 1s t ed. 2002. Corr. 2nd printing With approximate 620 Figures and 70 Tables ~ Springer Editor: Prof. Dr. Fridolin P. Mechel Landhausstr. 12 71120 Grafenau Germany ISBN 978-3-662-07298-1 ISBN 978-3-662-07296-7 (eBook) DOI 10.1007/978-3-662-07296-7 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 http://dnb.ddb.de 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-Ve rlag Berlin Heidelberg 2002, 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1s t 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-5 4 3 2 1 O v Formulas of Acoustics 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 contac ted 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 sec tions 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. Deri vations 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 prin ciples 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 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 prin ting 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 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 ......................................... 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.IO Principles of superposition ............................................................................................. 20 B.ll Hamilton's principle ....................................................................................................... 23 B.12. Adjoined wave equation .................................................................................................. 24 B.13 Vector and tensor formulation offundamentals .............................................................. 25 B.14 Boundary condition at a moving boundary ..................................................................... 37 B.IS Boundary conditions at liquids and solids ...................................................................... 38 Formulas of Acoustics VIII Contents B.16 Corner conditions ............................................................................................................ 39 B.17 Surface wave at locally reacting plane ............................................................................. 39 B.18 Surface wave along a locally reacting cylinder ................................................................. 41 B.19 Periodic structures, admittance grid ................................................................................ 42 B.20 Plane wall with wide grooves .......................................................................................... 47 8.21 Thin grid on half-infinite porous layer ............................................................................ 49 B.22 Grid of finite thickness with narrow slits on half-infinite porous layer ......................... 52 8.23 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 CA 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 0.2 Plane wave reflection at an infinitely thick porous layer ................................................ 75 D.3 Plane wave reflection at a porous layer of finite thickness ............................................. 76 DA Plane wave reflection at a multiple layer absorber. ......................................................... 78 D.5 Diffuse sound reflection at a locally reacting plane ......................................................... 79 0.6 Diffuse sound reflection at a bulk reacting porous layer ................................................. 81 D.7 Sound reflection and scattering at finite-size absorbers .................................................. 82 D.8 Uneven, local absorber surface ........................................................................................ 86 0.9 Scattering at the border of an absorbent half-plane ......................................................... 88 Formulas of Acoustics IX Contents 0.10 Absorbent strip in a hard baffle wall, with far field distribution .................................... 89 0.11 Absorbent strip in a hard baffle wall, as variational problem ......................................... 91 0.12 Absorbent strip in a hard baffle wall, with Mathieu functions ....................................... 94 0.13 Absorption of finite-size absorbers as a problem of radiation ........................................ 98 0.14 A monopole line source above an infinite, plane absorber; integration method ........................................................................................................... 99 0.15 A monopole line source above an infinite, plane absorber; with principle of superposition .................................................................................... 107 0.16 A monopole point source above a bulk reacting plane, exact forms ............................. 109 0.17 A monopole point source above a locally reacting plane, exact forms .......................... 112 0.18 A monopole point source above a locally reacting plane, exact saddle point integration ........................................................................................ 114 0.19 A monopole point source above a locally reacting plane, approximations ................... 117 0.20 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 comer 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.lO Scattering at a semicircular absorbing dam on absorbing ground ................................... 169 E.ll Scattering in random media, generaL ............................................................................ 173 x Formulas of Acoustics Contents E.12 Function tables for monotype scattering ...................................................................... 180 E.13 Sound attenuation in a forest ......................................................................................... 184 E.l4 Mixed monotype scattering in random media ............................................................... 186 E.l5 Multiple triple-type scattering in random media .......................................................... 191 E. I 6 Plane wave scattering at elastic cylindrical shell ........................................................... 202 E.17 Plane wave backscattering by a liquid sphere ............................................................... 205 E.l8 Spherical wave scattering at a perfectly absorbing wedge ............................................. 206 E.l9 Impulsive spherical wave scattering at a hard wedge .................................................... 208 E.20 Spherical wave scattering at a hard screen ..................................................................... 210 F. Radiation of Sound F.I 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.II Wide rectangular, field excited radiator .......................................................................... 240 F .12 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

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