Formulas of Acoustics

26925 Pages·2008·609.67 MB·English

by  Fridolin P. Mechel Prof. (eds.)

Formulas of Acoustics
F.P.Mechel (Ed.)
Formulas of Acoustics
SecondEdition
Withcontributionsby:
M.L.Munjal,M.Vorla¨nder,P.Ko¨ltzsch,M.Ochmann,A.Cummings,
W.Maysenho¨lder,W.Arnold
123
Prof.Dr.FridolinP.Mechel
Landhausstraße12
71120Grafenau
Germany
LibraryofCongressControlNumber:2008922894
ISBN:978-3-540-76832-6
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Preface to the first edition,
abbreviated
Modern acoustics is more and more based on computations,and computations are
based onformulas.Such workneeds previousandcontemporary results.It consumes
much time and effort to search needed formulas during the actual work.Therefore,
fundamentalsandresultsofacousticsthatcanbeexpressedasformulaswillbecollected
inthisbook.
Theformulacollectionissubdividedintofieldsofacoustics(Chapters).Forsomefields,
in which this author is not expert enough,he invitedco-authors to contribute.Most
colleaguescontactedforpossiblecontributionswereconvincedoftheprojectandagreed
spontaneously.
The material within a field of acoustics is subdivided in Sections which deal with a
defined task.Some overlap of Sections should be tolerated; but the subdivision into
well-definedSectionswillbehelpfultothereadertofindaparticulartopicofinterest.
The present formula collection should not be considered a textbook in a condensed
form.Derivationsofapresentedresultwillbedescribedonlyasfarastheyarehelpful
inunderstandingtheproblem;themoreinterestedreaderisreferredtothe“source”of
theresult.Usefulprinciplesandcomputationalprocedureswillalsobeincluded,evenif
theyneed moredescribingtext.SymbolsandquantitieswillbedefinedintheSection,
andwhereverusefulasketchwillhelptoexplaintheobjectandthetask.
Oneoftheadvantagesofaformulacollectionisseeninuniformdefinitions,notations
andsymbolsforquantities.Astrictuniformityintheformofacentrallistofsymbols
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
collectedinacentrallistofsymbols(seeConventions);othersymbolsaredefinedinthe
relevantchapter.
Most Sections contain, below their title or in the text, a reference to the literature.
It cannot be the task and intention of this bookeither to indicate time priorities of
publicationsconcerningatopicortogiveasurveyoftheexistingliterature.Thereference
quotedisthesourceofmoreinformation,whichtheauthorhasused.
Highertranscendentalfunctionsusedintheformulaswillbeexplainedbyreferenceto
mathematical literature,ifnecessary.Iffunctionsareusedwithdifferentdefinitionsin
theliterature,thedefinitionappliedherewillbepresented.
The authors think that the bookin its present formcontains most of traditionaland
modern results of both fundamental and special character so that the book can be
VI Preface
helpful to researchers and engineers in the fields of physical acoustics,noise control,
androomacoustics.
Themanuscriptwaswritteninacamerareadyform(inordertoavoidproofreading).
Soprintingerrorsaretheresponsibilityoftheeditingauthor.Hewouldbegratefulfor
indicationsofsucherrors.
Theauthorgratefullyacknowledgesthesupportgiventotheprojectbytheco-authors
andbythepublisher.
Grafenau,October2001
Preface to the second edition
Thebookwasoutofprintin2004.Theneedofreprintgaveafirstopportunitytoapply
some corrections to (rather harmless) misprints and to a few more serious formula
errors (the positions of the errors are marked by a footnote∗)).Some of the shown
diagramsweregeneratedbythecomputingprogramMathematica(cid:2);thisprogramun-
fortunatelyhaslostitsabilitytowriteaxesandplotlabelssothattheycanbeunderstood
byreceivingtextprograms.Thereforetransscriptionstoplotlabelsareenumeratednear
the diagrams, where necessary. This second edition is moderately enlarged by some
additionaltopicsinnewSections.
Grafenau,May2008
Contents
Prefacetothefirstedition,abbreviated . . . . . . . . . . . . . . . . . . . . . . . . . V
Prefacetothesecondedition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX
A Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
B GeneralLinearFluidAcoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
F.P.Mechel
B.1 FundamentalDifferentialEquations . . . . . . . . . . . . . . . . . . . . . 5
B.2 MaterialConstantsofAir. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
B.3 GeneralRelationforFieldAdmittanceandIntensity. . . . . . . . . . . . 11
B.4 IntegralRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
B.5 Green’sFunctionsandFormalism . . . . . . . . . . . . . . . . . . . . . . 13
B.6 OrthogonalityofModesinaDuctwithLocallyReactingWalls . . . . . 19
B.7 OrthogonalityofModesinaDuctwithBulkReactingWalls . . . . . . . 20
B.8 SourceConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
B.9 Sommerfeld’sCondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
B.10 PrinciplesofSuperposition . . . . . . . . . . . . . . . . . . . . . . . . . . 22
B.11 Hamilton’sPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
B.12 AdjointWaveEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B.13 VectorandTensorFormulationofFundamentals . . . . . . . . . . . . . 26
B.14 BoundaryConditionataMovingBoundary . . . . . . . . . . . . . . . . 38
B.15 BoundaryConditionsinLiquidsandSolids . . . . . . . . . . . . . . . . . 39
B.16 CornerConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
B.17 SurfaceWaveatLocallyReactingPlane . . . . . . . . . . . . . . . . . . . 41
B.18 SurfaceWaveAlongaLocallyReactingCylinder . . . . . . . . . . . . . . 42
B.19 PeriodicStructures,AdmittanceGrid . . . . . . . . . . . . . . . . . . . . 44
B.20 PlaneWallwithWideGrooves . . . . . . . . . . . . . . . . . . . . . . . . . 48
B.21 ThinGridonHalf-InfinitePorousLayer . . . . . . . . . . . . . . . . . . . 50
B.22 GridofFiniteThicknesswithNarrowSlits
onHalf-InfinitePorousLayer . . . . . . . . . . . . . . . . . . . . . . . . . 53
B.23 GridofFiniteThicknesswithWideSlits
onHalf-InfinitePorousLayer . . . . . . . . . . . . . . . . . . . . . . . . . 56
VIII TableofContents
C EquivalentNetworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
F.P.Mechel
C.1 FundamentalsofEquivalentNetworks . . . . . . . . . . . . . . . . . . . . 59
C.2 DistributedNetworkElements . . . . . . . . . . . . . . . . . . . . . . . . 65
C.3 ElementswithConstrictions. . . . . . . . . . . . . . . . . . . . . . . . . . 71
C.4 SuperpositionofMultipleSourcesinaNetwork . . . . . . . . . . . . . . 73
C.5 ChainCircuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
C.6 PartitionImpedanceofOrifices . . . . . . . . . . . . . . . . . . . . . . . . 74
D ReflectionofSound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
F.P.Mechel
D.1 PlaneWaveReflectionataLocallyReactingPlane . . . . . . . . . . . . . 127
D.2 PlaneWaveReflectionatanInfinitelyThickPorousLayer . . . . . . . . 129
D.3 PlaneWaveReflectionataPorousLayerofFiniteThickness . . . . . . . 130
D.4 PlaneWaveReflectionataMultilayerAbsorber . . . . . . . . . . . . . . 132
D.5 DiffuseSoundReflectionataLocallyReactingPlane . . . . . . . . . . . 133
D.6 DiffuseSoundReflectionataBulkReactingPorousLayer . . . . . . . . 136
D.7 SoundReflectionandScatteringatFinite-SizeLocalAbsorbers . . . . . 136
D.8 Uneven,LocalAbsorberSurface. . . . . . . . . . . . . . . . . . . . . . . . 141
D.9 ScatteringattheBorderofanAbsorbentHalf-Plane . . . . . . . . . . . . 142
D.10 AbsorbentStripinaHardBaffleWall,withFarFieldDistribution. . . . 144
D.11 AbsorbentStripinaHardBaffleWall,asaVariationalProblem . . . . . 146
D.12 AbsorbentStripinaHardBaffleWall,withMathieuFunctions . . . . . 148
D.13 AbsorptionofFinite-SizeAbsorbers,asaProblemofRadiation . . . . . 153
D.14 AMonopoleLineSourceAboveanInfinite,PlaneAbsorber;
IntegrationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
D.15 AMonopoleLineSourceAboveanInfinite,PlaneAbsorber;
withPrincipleofSuperposition . . . . . . . . . . . . . . . . . . . . . . . . 162
D.16 AMonopolePointSourceAboveaBulkReactingPlane,
ExactForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
D.17 AMonopolePointSourceAboveaLocallyReactingPlane,
ExactForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
D.18 AMonopolePointSourceAboveaLocallyReactingPlane,
ExactSaddlePointIntegration . . . . . . . . . . . . . . . . . . . . . . . . 170
D.19 AMonopolePointSourceAboveaLocallyReactingPlane,
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
D.20 AMonopolePointSourceAboveaBulkReactingPlane,
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
E ScatteringofSound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
F.P.Mechel
E.1 PlaneWaveScatteringatCylinders . . . . . . . . . . . . . . . . . . . . . . 185
E.2 PlaneWaveScatteringatCylindersandSpheres . . . . . . . . . . . . . . 188
E.3 MultipleScatteringatCylindersandSpheres . . . . . . . . . . . . . . . . 198
E.4 CylindricalWaveScatteringatCylinders. . . . . . . . . . . . . . . . . . . 199
TableofContents IX
E.5 CylindricalorPlaneWaveScatteringataCornerSurrounded
byaCylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
E.6 PlaneWaveScatteringataHardScreen . . . . . . . . . . . . . . . . . . . 208
E.7 CylindricalorPlaneWaveScatteringataScreenwithan
EllipticalCylinderAtop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
E.8 UniformScatteringatScreensandDams . . . . . . . . . . . . . . . . . . 214
E.9 ScatteringataFlatDam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
E.10 ScatteringataSemicircularAbsorbingDam
onAbsorbingGround. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
E.11 ScatteringinRandomMedia,General . . . . . . . . . . . . . . . . . . . . 230
E.12 FunctionTablesforMonotypeScattering . . . . . . . . . . . . . . . . . . 238
E.13 SoundAttenuationinaForest . . . . . . . . . . . . . . . . . . . . . . . . . 242
E.14 MixedMonotypeScatteringinRandomMedia . . . . . . . . . . . . . . 244
E.15 MultipleTriple-TypeScatteringinRandomMedia . . . . . . . . . . . . . 248
E.16 PlaneWaveScatteringatElasticCylindricalShell . . . . . . . . . . . . . 260
E.17 PlaneWaveBackscatteringbyaLiquidSphere . . . . . . . . . . . . . . . 263
E.18 SphericalWaveScatteringataPerfectlyAbsorbingWedge . . . . . . . . 264
E.19 ImpulsiveSphericalWaveScatteringataHardWedge . . . . . . . . . . . 266
E.20 SphericalWaveScatteringataHardScreen . . . . . . . . . . . . . . . . . 268
E.21 SphericalWaveScatteringataCone . . . . . . . . . . . . . . . . . . . . . 270
E.22 PolarModeNumbersataSoftCone . . . . . . . . . . . . . . . . . . . . . 275
E.23 PolarModeNumbersataHardCone . . . . . . . . . . . . . . . . . . . . 279
E.24 ScatteringataConewithAxialSoundIncidence . . . . . . . . . . . . . . 282
F RadiationofSound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
F.P.Mechel
F.1 DefinitionofRadiationImpedanceandEndCorrections . . . . . . . . . 287
F.2 SomeMethodstoEvaluatetheRadiationImpedance . . . . . . . . . . . 289
F.3 SphericalRadiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
F.4 CylindricalRadiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
F.5 PistonRadiatoronaSphere . . . . . . . . . . . . . . . . . . . . . . . . . . 297
F.6 Strip-ShapedRadiatoronCylinder . . . . . . . . . . . . . . . . . . . . . . 299
F.7 PlanePistonRadiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
F.8 UniformEndCorrectionofPlanePistonRadiators . . . . . . . . . . . . 309
F.9 NarrowStrip-Shaped,Field-ExcitedRadiator . . . . . . . . . . . . . . . . 309
F.10 WideStrip-Shaped,Field-ExcitedRadiator . . . . . . . . . . . . . . . . . 311
F.11 WideRectangular,Field-ExcitedRadiator . . . . . . . . . . . . . . . . . . 313
F.12 EndCorrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
F.13 PistonRadiatingIntoaHardTube . . . . . . . . . . . . . . . . . . . . . . 327
F.14 OscillatingMassofaFenceinaHardTube . . . . . . . . . . . . . . . . . 328
F.15 ARing-ShapedPistoninaBaffleWall . . . . . . . . . . . . . . . . . . . . 329
F.16 MeasuresofRadiationDirectivity . . . . . . . . . . . . . . . . . . . . . . 330
F.17 DirectivityofRadiatorArrays . . . . . . . . . . . . . . . . . . . . . . . . . 330
F.18 RadiationofFiniteLengthCylinder . . . . . . . . . . . . . . . . . . . . . 335
F.19 MonopoleandMultipoleRadiators . . . . . . . . . . . . . . . . . . . . . . 337
F.20 PlaneRadiatorinaBaffleWall . . . . . . . . . . . . . . . . . . . . . . . . 339
X TableofContents
F.21 RatioofRadiationandExcitationEfficienciesofPlates . . . . . . . . . . 344
F.22 RadiationofPlateswithSpecialExcitations . . . . . . . . . . . . . . . . . 344
G PorousAbsorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
F.P.Mechel
G.1 StructureParametersofPorousMaterials . . . . . . . . . . . . . . . . . . 347
G.2 TheoryoftheQuasi-homogeneousMaterial . . . . . . . . . . . . . . . . 350
G.3 RayleighModelwithRoundCapillaries . . . . . . . . . . . . . . . . . . . 352
G.4 ModelwithFlatCapillaries . . . . . . . . . . . . . . . . . . . . . . . . . . 354
G.5 LongitudinalFlowResistivityinParallelFibres. . . . . . . . . . . . . . . 356
G.6 LongitudinalSoundinParallelFibres . . . . . . . . . . . . . . . . . . . . 358
G.7 TransversalFlowResistivityinParallelFibres. . . . . . . . . . . . . . . . 360
G.8 TransversalSoundinParallelFibres . . . . . . . . . . . . . . . . . . . . . 370
G.9 EffectiveWaveMultipleScatteringinTransversalFibreBundle . . . . . 381
G.10 Biot’sTheoryofPorousAbsorbers . . . . . . . . . . . . . . . . . . . . . . 385
G.11 EmpiricalRelationsforCharacteristicValues
ofFibreAbsorbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
G.12 CharacteristicValuesfromTheoreticalModelsFitted
toExperimentalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
H CompoundAbsorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
F.P.Mechel
H.1 AbsorberofFlatCapillaries . . . . . . . . . . . . . . . . . . . . . . . . . . 404
H.2 PlatewithNarrowSlits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
H.3 PlatewithWideSlits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
H.4 DissipationlessSlitResonator . . . . . . . . . . . . . . . . . . . . . . . . . 415
H.5 ResonanceFrequenciesandRadiationLossofSlitResonators . . . . . . 419
H.6 SlitArraywithViscousandThermalLosses . . . . . . . . . . . . . . . . . 420
H.7 SlitResonatorwithViscousandThermalLosses . . . . . . . . . . . . . . 426
H.8 FreePlatewithanArrayofCircularHoles,withLosses . . . . . . . . . . 429
H.9 ArrayofHelmholtzResonatorswithCircularNecks . . . . . . . . . . . . 435
H.10 SlitResonatorArraywithPorousLayerintheVolume,Fields . . . . . . 437
H.11 SlitResonatorArraywithPorousLayer
intheVolume,Impedances . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
H.12 SlitResonatorArraywithPorousLayeronBackOrifice. . . . . . . . . . 449
H.13 SlitResonatorArraywithPorousLayeronFrontOrifice . . . . . . . . . 452
H.14 ArrayofSlitResonatorswithSubdividedNeckPlate . . . . . . . . . . . 456
H.15 ArrayofSlitResonatorswithSubdividedNeckPlate
andFloatingFoilintheGap . . . . . . . . . . . . . . . . . . . . . . . . . . 457
H.16 ArrayofSlitResonatorsCoveredwithaFoil . . . . . . . . . . . . . . . . 462
H.17 Poro-elasticFoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
H.18 FoilResonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
H.19 RingResonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
H.20 Wide-AngleAbsorber,ScatteredFarField . . . . . . . . . . . . . . . . . . 474
H.21 Wide-AngleAbsorber,NearFieldandAbsorption . . . . . . . . . . . . . 480
H.22 TightPanelAbsorber,RigorousSolution . . . . . . . . . . . . . . . . . . 485
TableofContents XI
H.23 TightPanelAbsorber,Approximations . . . . . . . . . . . . . . . . . . . . 493
H.24 PorousPanelAbsorber,RigorousSolution. . . . . . . . . . . . . . . . . . 496
I SoundTransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
F.P.Mechel
I.1 “NoiseBarriers” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
I.2 SoundTransmissionthroughaSlitinaWall . . . . . . . . . . . . . . . . 506
I.3 SoundTransmissionthroughaHoleinaWall . . . . . . . . . . . . . . . 511
I.4 HoleTransmissionwithEquivalentNetwork . . . . . . . . . . . . . . . . 515
I.5 SoundTransmissionthroughLinedSlitsinaWall . . . . . . . . . . . . . 517
I.6 ChamberedJoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
I.7 “NoiseSluice” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
I.8 SoundTransmissionßindexsoundtransmissionthroughplatesthrough
Plates,SomeFundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 525
I.9 SoundTransmissionthroughaSimplePlate . . . . . . . . . . . . . . . . 532
I.10 InfiniteDouble-ShellWallwithAbsorberFill . . . . . . . . . . . . . . . . 538
I.11 Double-ShellWallwithThinAirGap . . . . . . . . . . . . . . . . . . . . . 540
I.12 PlatewithAbsorberLayerBehind . . . . . . . . . . . . . . . . . . . . . . 541
I.13 SandwichPanels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
I.14 Finite-SizePlate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
I.15 SinglePlateacrossaFlatDuct . . . . . . . . . . . . . . . . . . . . . . . . . 555
I.16 SinglePlateinaWallNiche . . . . . . . . . . . . . . . . . . . . . . . . . . 559
I.17 Strip-ShapedWallinInfiniteBaffleWall . . . . . . . . . . . . . . . . . . . 564
I.18 Finite-SizePlatewithaFrontSideAbsorberLayer . . . . . . . . . . . . . 567
I.19 Finite-SizePlatewithaBackSideAbsorberLayer . . . . . . . . . . . . . 570
I.20 Finite-SizeDoubleWallwithanAbsorberCore. . . . . . . . . . . . . . . 571
I.21 PlenumModes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
I.22 SoundTransmissionthroughSuspendedCeilings . . . . . . . . . . . . . 577
I.23 OfficeFences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
I.24 OfficeFences,withSecondPrincipleofSuperposition . . . . . . . . . . 584
I.25 InfinitePlateBetweenTwoDifferentFluids . . . . . . . . . . . . . . . . . 587
I.26 SandwichPlatewithanElasticCore . . . . . . . . . . . . . . . . . . . . . 589
I.27 WallofMultipleSheetswithAirInterspaces . . . . . . . . . . . . . . . . 591
J DuctAcoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
F.P.Mechel
J.1 FlatCapillarywithIsothermalBoundaries . . . . . . . . . . . . . . . . . 595
J.2 FlatCapillarywithAdiabaticBoundaries . . . . . . . . . . . . . . . . . . 598
J.3 CircularCapillarywithIsothermalBoundary. . . . . . . . . . . . . . . . 598
J.4 LinedDucts,General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
J.5 ModesinRectangularDuctswithLocallyReactingLining . . . . . . . . 605
J.6 LeastAttenuatedModeinRectangular,LocallyLinedDucts . . . . . . . 607
J.7 SetsofModeSolutionsinRectangular,LocallyLinedDucts . . . . . . . 613
J.8 FlatDuctwithaBulkReactingLining . . . . . . . . . . . . . . . . . . . . 619
J.9 FlatDuctwithanAnisotropic,BulkReactingLining . . . . . . . . . . . . 621
J.10 ModeSolutionsinaFlatDuctwithBulkReactingLining . . . . . . . . . 623