Modeling Of Horns And Enclosures For Loudspeakers (Gavin Richard Putland).pdf

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Modelingof
HornsandEnclosures
forLoudspeakers
by
GavinRichardPutland,BE(Qld)
DepartmentofElectricalandComputerEngineering
UniversityofQueensland
SubmittedforthedegreeofDoctorofPhilosophy
December23,1994
RevisedNovember1995
AcceptedFebruary6,1996.
ii
Tomyparents,
FrankandDelPutland
Acknowledgments
MyinterestinhorntheorywasinspiredbyDrE.R.Geddes,whosepaper“Acoustic
WaveguideTheory”[18]raisedthequestionsaddressedinChapters3to5ofthis
thesis,suggestedsomeoftheanswers,andprovidedmanyoftherelatedreferences.
ThesedebtsarenotdiminishedbymydisagreementwithGeddes’analysisofthe
oblatespheroidalwaveguide,whichhesubsequentlyrevised[19].Mypaperonone-
parameterwavesandWebster’sequation[43]wasimprovedasaresultofapersonal
communicationfromDrGeddes(October10,1992),whichalertedmetotheexis-
tenceofFresneldi‹ractionfringeswithinthecoverageanglesoffinitehornsathigh
frequencies.
IwishtothanktheExecutiveEditorandReviewBoardofthe Journalofthe
AudioEngineeringSociety fortheirpatienthandlingofthecomplexexchangeof
correspondencethatfollowedmy“Commentson‘AcousticWaveguideTheory’”[42].
Iamindebtedtoananonymousreviewerofmyarticle“AcousticalPropertiesof
AirversusTemperatureandPressure”[44]forcorrectingmyexplanationofthee‹ect
ofhumidityonabsorption,andfordrawingmyattentiontoANSIS1.26-1978[41],
whichiscitedfrequentlyinthefinalversionofthearticleandinChapter9.
Mysupervisor,DrLarrySkattebol,providedcriticalfeedbackonmythreepubli-
cationsandonnumerouspartialdraftsofthisthesis.Ithankhimforhisflexibilityin
acceptingmyresearchproposal,andforhissubsequentprudenceincurbingmyten-
dencytopursuenewlinesofinquiry;withouthisguidance,thisalreadyprotracted
projectwouldhavetakenconsiderablylonger.IalsothankProf.TomDownsand
DrNickShuley,colleaguesofDrSkattebolintheDept.ofElectricalandComputer
Engineering,UniversityofQueensland,fortheircommentsonreference[42].
TheItaliantextofSomigliana’sletter[51](paraphrasedinAppendixA)was
translatedbyBr.AlanMoss,whowasthenagraduatestudentintheDepartment
ofStudiesinReligion,UniversityofQueensland.
TheJapanesetextofSections1to3ofArai’spaper[2]wastranslatedbyAdrian
TreloaroftheDepartmentofJapaneseandChineseStudies,UniversityofQueens-
land.
Thetitlesandauthorsofthemajorpublic-domainsoftwareusedinthisproject
areasfollows.EquivalentcircuitsimulationswereperformedusingSPICE3byTom
Quarles,withthe nutmeg userinterfacebyWayneChristopher.Programswere
compiledbytheGNUprojectCcompiler.Mostdiagramsweredrawnusing xfig
bySupojSutanthavibuletal.,convertedtoL A T E X picture commandsusing fig2dev
byMicahBecketal.,andfinallyeditedastextfiles.Allsourcefileswereedited
with jove byJonathanPayne,andspell-checkedwith ispell byPaceWillissonet
al.ThisdocumentwastypesetandprintedwithintheDepartmentofElectricaland
ComputerEngineering,UniversityofQueensland,usingL A T E X2 " byLeslieLamport.
ThisprojectwassupportedbyanAustralianPostgraduateResearchAward.
iii
StatementofOriginality
Ideclarethatthecontentofthisthesisis,tothebestofmyknowledgeandbelief,
originalexceptasacknowledgedinthetextandfootnotes,andthatnopartofithas
beenpreviouslysubmittedforadegreeatthisUniversityoratanyotherinstitution.
Inathesisdrawingonsuchdiversesubjectsasmechanics,thermodynamics,
circuittheory,di‹erentialgeometry,vectoranalysisandSturm-Liouvilletheory,the
authormustmakeexaggeratede‹ortstomaintainsometheoreticalcoherence.This
mayinvolvederivingwell-knownresultsinamannerappropriatetothecontext;
examplesincludethehierarchyofformsoftheequationsofmotionandcompression
(Chapter2),andthederivationofadmittancesandGreen’sfunctionsfromWebster’s
equation(Chapter3).Furthermore,whenathesiscontainsresultsatvariancewith
earlierresultsintheliterature,theauthorwillbeexpectedtojustifyhisfindingswith
exceptionalthoroughness.Inparticular,hemaybeobligedtoconductmathematical
argumentsatamorefundamentallevelthanwouldnormallybeappropriate;an
exampleismydetailedsolutionoftheheatequationtodeterminethebasicthermal
timeconstantoftheair-fibersystem(Chapter8).Insuchcases,thecontextwill
indicatethattheresultisincludedforthesakeofclarity,cohesionorrigor,andnot
necessarilybecauseofnovelty.
Toavoidexcessiverelianceon“thecontext”,Io‹erthefollowingsummaryof
whatIbelievetobemyprincipaloriginalcontributionsandtheirdependenceon
theworkofearlierresearchers.Thissummaryalsoservesasanextendedabstract.
InChapter2,Ihaveshownhowthenumerousfamiliarequationsrelatedto
theinertiaandcomplianceofaircanbeunderstoodasalternativeformsof
twobasicequations,whichIcalltheequationsofmotionandcompression.
Thehierarchyofformseliminatesredundancyinthederivationsandclearly
showswhatsimplifyingassumptionsareinvolvedineachform.The“one-
parameter”or“1P”formsapplywhentheexcesspressure p dependsona
singlespatialcoordinate ,whichmeasuresarclengthnormaltotheisobaric
surfaces.Theseformsareexpressedwithunprecedentedgenerality,andare
criticaltothemathematicalargumentofsubsequentchapters.Otherforms
leadtothefamiliarelectricalanalogsforacousticmassandcompliance(both
lumpedanddistributed).
Afterareviewofpreviousliterature,Ihaveshownthatthe“Webster”horn
equation,whichisusuallypresentedasaplane-waveapproximation,follows
exactly fromthe1Pformsoftheequationsofmotionandcompression,without
anyexplicitassumptionconcerningthewavefrontshape.The coordinateis
theaxialcoordinateofthehornwhile S ( )istheareaofaconstant- surface
segmentboundedbyatubeoforthogonaltrajectoriestoalltheconstant-
surfaces;suchtubes(andnoothers)arepossibleguidingsurfaces.
iv
v
IhaveshownthattheHelmholtzequationadmitssolutionsdependingona
singlespatialcoordinate u ifandonlyif jruj and r 2 u arefunctionsof u alone.
The jruj conditionallows u tobetransformedtoanothercoordinate ,which
measuresarclengthalongtheorthogonaltrajectoriestotheconstant- sur-
faces.Hence,inthedefinitionofa“1P”pressurefield,thenormal-arc-length
conditionisredundant.UsinganexpressionfortheLaplacianofa1Ppres-
surefield,Ihaveshownthatthewaveequationreduces exactly toWebster’s
equation;thisisasecondgeometry-independentderivation.Ihavealsoshown
thattheterm“1Pacousticfield”canbedefinedintermsofpressure,velocity
potentialorvelocity,andthatallthreedefinitionsareequivalent.
Ihaveexpressedthe1Pexistenceconditionsintermsofcoordinatescalefac-
torsandfoundthat intheelevencoordinatesystemsthatareseparablewith
respecttotheHelmholtzequation ,theonlycoordinatesadmitting1Psolutions
arethosewhoselevelsurfacesareplanes,circularcylindersandspheres(Chap-
ter5).Geddes[18]reportedin1989thatWebster’sequationisexactinthe
samelistofcoordinates,butdidnotmaketheconnectionbetweenWebster’s
equationand1Pwaves.
Withoutusingseparablecoordinates ,Somigliana[51]showedthatthereare
onlythree1Pwavefrontshapesallowingparallelwavefrontsandrectilinear
propagation;thepermittedshapesareplanar,circular-cylindricalandspheri-
cal.Ihaveshown(Chapter5)thattheconditionsofparallelwavefrontsand
rectilinearpropagationareimplicitinthe1Passumption,sothatthethree
geometriesobtainedbySomiglianaaretheonlypossiblegeometriesfor1P
waves.Thisresulthasthepracticalimplicationthatnonew1Phorngeome-
triesremaintobediscovered.
Ihaveproducedanannotatedparaphrase,inmodernnotation,ofSomigliana’s
proof(AppendixA),andadaptedhisproofsoastotakeadvantageofthe
1Pexistenceconditions(Chapter5).InthetheoremsofChapter5andthe
footnotestoAppendixA,IhavefilledinseveralmissingstepsinSomigliana’s
argument.Ifounditmostconvenienttoprovetheseresultsindependently,
althoughrelatedresultsexistintheliteratureondi‹erentialgeometry.
Workingfromthepermitted1Pwavefrontshapesandtheexactderivationsof
Webster’sequation,Ihavegivenwideconditionsunderwhichthatequationis
approximately true,sothattraditionalapproximatederivationsoftheequation
canbereplacedbythemoregeneral1Ptheory.
Ihaveextendedthefinite-di‹erenceequivalent-circuit(FDEC)methodpro-
posedin1960byArai[2].InChapter6,Ihaveshownthatafinite-di‹erence
approximationtoWebster’sequationyieldsthenodalequationsofan L-C
latternetwork(confirmingArai’sunprovenassertionthathisone-dimensional
methodcanbeadaptedforhorns),whileasimilarapproximationtothewave
equationingeneralcurvilinearorthogonalcoordinatesyieldsthenodalequa-
tionsofathree-dimensional L-C network.Ihaveobtainedthesamecircuits
fromtheequationsofmotionandcompressioninordertoshowthat“current”
intheequivalentcircuitisvolumeflux,asexpected.Ihaveshownhowthe
networkshouldbetruncatedattheboundariesofthemodelandterminated
withadditionalcomponentstorepresentarangeofboundaryconditions.

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