Room Acoustical Fields

600 Pages·2013·31.09 MB·English

by  Fridolin Mechel (auth.)

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
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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