Room Acoustical Fields . Fridolin Mechel Room Acoustical Fields FridolinMechel Landhausstraße12 71120Grafenau Germany ISBN978-3-642-22355-6 ISBN978-3-642-22356-3(eBook) DOI10.1007/978-3-642-22356-3 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012952470 # Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Room acoustics in practice is a difficult job. Sound fields are to be predicted and quantitatively described in large rooms with complicated shapes. They are generated by complex-shaped sound sources, received and estimated by an aestheticallysensitiveaudience,andfurthertheyareweightedbyalotofaesthetical and psychological criteria and standards. Architects and acoustic consultants eagerlycompetewitheachothertodesignsuchrooms. Nevertheless,thebackground,thesoundfield,isaphysicalwavephenomenon. This fact should be kept in mind when reading some of the room acoustical literature, in which the rules for the distribution of sound resemble the rules of a billiardgameonabilliardtablewithabizarreshape.Orthesoundsourceresembles a snow cannon ejecting the “sound particles” in a random manner, whirling in a huge number through the room until, finally, a small basket at the position of a presumedlistenercatchesafewofthemtobecountedandselectedaccordingtheir timeanddirectionofarrival. The reasoning behind such sometimes grotesque imaginations and procedures mostlyarederivedfromthelong-establishedmirrorsourcemethod.Theargumen- tation, on the other hand, suggests that the mirror source method is of no use in practicebecausethenumbersofrequiredmirrorsourcesallegedlyareonastronom- ical orders of magnitude, and therefore one cannot evaluate sound fields with the mirrorsourcemethod.Suchstatementsarosebecauseeitherthecuboidroomshape, which is singular-behaving, formed the basis of the derivation, and/or elementary exclusion rules for mirror sources were neglected (such as: “somebody standing behindamirrorcannotexpecttoproduceamirrorimageofhimself”). Itisoneoftheprincipalaimsofthepresentbooktofreethemirrorsourcemodel from its reputation as being useless. It shall be applied to rooms which could function as concert halls. As such they should be equipped also with an orchestra pit,forexample.Thisgoalwillnotbereached,ifnotattheendanalgorithmfora mirrorsourceprogramcanbepresented.Atleastanoutlineofsuchaprogramalong whichthereadercanwritehisownprogramshouldbedescribed. v vi Preface Ontheotherhand,itcannotbedeniedthatexactevaluationsofroomacoustical fields are practically impossible (except in rooms with very simple elementary shapes). One will always be restricted by the necessity to apply approximations. This, however, cannot be an excuse for using methods conflicting with physics. Approximations, when they are used, should follow from analytical derivations and/or they should be tested with “calibration objects”. In this way “higher methods” can be derived and their precision can be estimated. They should offer some numerical advantages, such as, for example, the “mirror point method”, instead of the “mirror source method”, in which the field point is reflected at the wallsinsteadofthesources(themirrorpointmethodisadvantageousinconnection with sources having directional distributions). Another development allowing a higherlevelofperformanceisthecreationof“cornersources”whichrepresentthe soundfieldwhichthecollectionofallmirrorsourcesatacoupleofwallsforminga wedge-shapedspacewouldproduce.Theconceptofcornersourceshastheadvan- tageovertheconventionalmirrorsourcemethodinthatthepositionofthecorner sourcemustnotbesearchedaswiththemirrorsources,butitspositioncanbeeasily constructed.Afurtherstepinthatdirectionofdevelopmentisananalyticalsolution ofthescatteringtaskofasoundsourceinawedge-shapedspace.Suchsolutionsare fundamentally important in convex corners (angular range between the flanking wallswiderthanp),becausethenthesimplemirrorsourcemethodtotallyfails. Exception: one reduces the scattering at a convex corner to a superposition of scattered fields at two concave corners. Suchprinciples ofsuperposition will play animportantroleinthisbook. A further fundamental difference between the sound field descriptions in the presentbookandinthetraditionalroomacousticalliteraturecanbeseeninthefact that the latter deal, if not with sound particles, nearly exclusively with the sound energy, or, more precisely, with the real part of the sound intensity, i.e. with the squared magnitude of a field quantity. Here, however, the sound field will be describedbyits(complex)soundpressurepandparticlevelocityv,i.e.withlinear magnitudeswhichcontaintheinformationofphase.Thisinformationisnecessary for the evaluation of sound interference and standing waves between different sound field components. One consequence of this difference is the fact that the reflection coefficient |r|2 or the absorption coefficient a ¼ 1–|r|2 of a wall is not sufficient for the characterisation of the sound absorption of a wall. Instead, we mostlyusethewalladmittanceG ¼v⊥/p,i.e.theratiooftheparticlevelocityv⊥ normaltothewallandthesoundpressurep.ThequantityGisintroducedintothe field evaluation by the boundary conditions at the flanking walls. There does not exist an equivalent boundary condition for |r|2 or a . In a correct evaluation the sound intensity I ¼ p·v is evaluated in a post-procedure from the sound field quantitiespandv. Thephase-sensitivemirrorsourcemethodoffersthequantityofsoundinterfer- ence to the modern room acoustician who tries to characterise the near field structureofasoundfieldinandaroundafield point.Thisphenomenoncannotbe describedwithsoundparticlesorwithsoundintensityrays. Preface vii Thepresentauthorisconsciousofthefactthatwiththisbookheonlycangivean impetustofurtherevaluationsofroomacousticalfieldsusingamoresolidphysical base. Some details will be roughly sketched only, for example, repetitions of procedures in the evaluation of a frequency response. Also the evaluation of a wall admittance rarely will be described; here the application of existing parallel literature or of available computer programs is supposed. In order to avoid misunderstandings:theroomspresentedasillustrationsofcomputingmethodsare not meant to be specially qualified pieces of architecture, but they simply are exercise objects. The primary goal of the illustration of methods also explains the extensive use of two-dimensional objects. They are much easier to draw and toexplainthanthree-dimensionalobjects,andoftenthetransferofthemethodinto threedimensionsismucheasierthandirectlydescribingthemethoddevelopmentin threedimensions. A typical reader of the present book is a colleague working in room acoustics who possibly is interested in writing a computer program for sound fields in auditoria or in other halls. Therefore he is interested and experienced in room acoustics.Heisfamiliarwiththefundamentalsofacousticsandwiththestandard literature, e.g. with [1–7]. So elementary acoustics will be explained here only in the context of completing an evaluation scheme ; the details and derivation of fundamentals mostly will be dealt with by reference to literature. Some newer topicsinthepresentbookwerepreviouslydescribedinanarticle,[8],inaformula collection, [9], and in a German version (“Raumakustische Felder”) of the book, [62].Significantmodificationsandenlargements,ascomparedwiththatbook,are applied here in Chaps. 22–24 about the sound scattering in and at wedge-shaped spaces, where now numerical tests with formulas and methods from the literature will demonstrate that many of them are useless for numerical evaluations. Minor corrections,modificationsandenlargementsareappliedinanumberofplaces. The author is indepted to Mrs. Laura Morgan for her valuable help in reading through the English text in order to make it as fluent as it is possible with a text includingsomanyformulas. Grafenau,Germany FridolinMechel . Contents 1 SoundSources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 MonopoleLineSourceorPointSource. . . . . . . . . . . . . . . . . . . . . 3 2.1 TransformationofLineSourceResultstoPointSourceResults... 3 3 MirrorSources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1 LineSourceAboveanAbsorbentPlane. . . . . . . . . . . . . . . . . . 5 3.2 NumericalTestsoftheMirrorSourceApproximationfor aLineSource. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 PointSourceAboveanAbsorbentPlane. . . . . . . . . . . . . . . . . . 15 3.4 NumericalExamplesforFieldsofaPointSourceAbovean AbsorbentPlane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 NumericalSurveyofthePrecisionoftheMirrorSource ApproximationforaPointSource. . . . . . . . . . . . . . . . . . . . . . 20 4 ModifiedMirrorSources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Hard–SoftSuperposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6 CubicRoom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.1 PointSourceinaCubewithHardWallsonAllSides,Classical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 PointSourceinaCubewithAbsorbentWallsonAllSides, ClassicalSolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.3 TheCubewithUnsymmetricalAbsorptionatOppositeWalls. . 59 7 ZoneSolutioninaCube.. . . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . 61 7.1 TheOne-SidedNiche. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.2 TheAll-SidesAbsorbentCube,ZoneSolution. . . . . . . . . . . . . 71 7.3 CubewithSoundExcitationinaWall. . . . . . . . . . . . . . . . . . . 77 8 FieldinaRectangularReverberantRoom. . . . . .. . . . . . . . . . . . . 85 8.1 StationaryFieldwithOneAbsorbentWall. . . . . . . . . . . . . . . . 86 8.2 ReverberationfromStationaryExcitation. . . . . . . . . . . . . . . . . 110 ix x Contents 8.3 ReverberantRoomwithTransversalZones. . . . . . . . . . . . . . . . 121 8.4 ReverberationintheRoomwithTransversalZones. . . . . . . . . . 129 8.4.1 DecayExponentfromgmz andPhaseVelocitycph;mz . . . 129 8.4.2 DecayExponentfromgm andSoundVelocityc0. . . . . 135 z 8.4.3 DecayExponentfromgm andSoundVelocityc0but z withoutmz ¼ 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.4.4 DecayExponentfromem. . . . . . . . . . . . . . . . . . . . . . . 138 8.4.5 “Twittering”Reverberation. . . . . . . . . . . . . . . . . . . . . . 142 8.5 StationaryFieldwithReducedAbsorberArea. . . . . . . . . . . . . . 151 9 FlatRooms. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . .. . . . 157 9.1 ScatteringCylinderinaFlatRoom. . . . . . . . . . . . . . . . . . . . . . 158 9.2 ScatteringCylinderExcitedbyaLineSource. . . . . . . . . . . . . . 161 9.3 RectangularFlatRoomwithAppendedChamber. . . . . . . . . . . 163 9.4 RectangularFlatRoomwithStageRoom. . . . . . . . . . . . . . . . . 170 10 WedgeRooms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10.1 WedgeRoomwithHardFlanks. . . . . . . . . . . . . . . . . . . . . . . 177 10.1.1 ScatteringofaCylindricalWave. . . . . . . . . . . . . . . . 178 10.1.2 ScatteringofaPlaneWave. . . . . . . . . . . . . . . . . . . . 181 10.2 WedgeModeswithAbsorbentFlanks. . . . . . . . . . . . . . . . . . . 185 10.2.1 NumericalSolutionoftheEigenValueEquation. . . . 187 10.2.2 AzimuthalEigenValues. . . . . . . . . . . . . . . . . . . . . . 189 10.2.3 ApproximationtotheWaveEquation. . . . . . . . . . . . 193 10.2.4 FictiveModes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.2.5 TheRemainderr(ϑ)asSuperpositionofFictiveModes 197 10.2.6 SeparateSolutionfortheRemainderr(r)oftheWave Equation. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 199 10.3 WedgeRoomwithSteppedAbsorbentFlank. .. . . . . . . . .. . . 203 10.3.1 ConvergentWedge. . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.3.2 DivergentWedge. . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.4 WedgeRoomwithSteppedWallAdmittance. . . . . . . . . . . . . 214 10.4.1 ExcitationbyaLineSourceQ. . . . . . . . . . . . . . . . . . 215 10.4.2 ExcitationbyaPlaneWave. . . . . . . . . . . . . . . . . . . . 222 10.5 NumericalResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11 VaultRooms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11.1 BarrelArch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11.2 Circular-CylindricalRoom. . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.3 Ring-ShapedWhisperingGallery. . . . . . . . . . . . . . . . . . . . . . 240 11.4 EllipticalCylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 11.5 EllipticalCylinderwithPointSource. . . . . . . . . . . . . . . . . . . 257 12 Cupola-ShapedRoom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263