Perforated Panel Absorbers with Viscous Energy Dissipation ... · dissertation presents two new...

Perforated Panel Absorberswith ViscousEnergy Dissipation

Enhancedby Orifice Design

Rolf ToreRandeberg

DOCTORAL THESIS

submittedto theDepartmentof Telecommunications,NorwegianUniversityof ScienceandTechnology,

in partialfulfillment of therequirementsfor thedegreeof doktoringeniør

Trondheim,June2000

Abstract

Currently, thereis a greatinterestin panelabsorberdesignwhereporouscomponentsareexcludedduetoenvironmentalandcleaningconsiderations.Forsuchabsorbers,thechallengeis to increasethenatural,viscouslossesto attainanacceptableabsorptionbandwidth.Thisdissertationpresentstwo new perforatedpanelabsorberconcepts,wheretheviscousenergydissipationhasbeenenhancedby theuseof non-traditionaldesignof theperforations.

The first conceptis a perforatedpanelwherethe perforationshasbeenshapedassmallhorns. The inner part of the hornshave dimensionscomparableto microperforatedpanels.The purposeof the designis to increasethe surfaceareaof the opening,increasethe flowvelocity in the inner part of the horn, andoffer a betterimpedancematchto the incomingwave. Theconcepthasbeeninvestigatedprimarily by calculationsusingtheFiniteDifferenceMethod.Theresultsindicatethata relatively largeabsorptionbandwidthcanbeobtainedfora hornwith wideouterradiusandsmall innerradius.

The secondconceptis a doubleperforatedpanel,consistingof two parallel,perforatedplatesseparatedby a smalldistance,typically 0.1–0 � 3mm. Themainpartof theenergy dis-sipationtakesplacein thesmallgapbetweentheplates.Both perforatedandslittedvariantshavebeeninvestigatedby simulationsandexperiments.For theslittedpanelcase,absorptionbandwidthsequivalentto microperforatedpanelshasbeenobserved. Theslittedvariantcanalsobe designedto be adjustable,allowing the lateraldistancebetweenthe slits in the twoplatesto be varied. This offers two specialfeatures:The maximumabsorptioncoefficientcanbe adjustedfrom unity to almostzero,andthe resonancefrequency canbe shifted. Afrequency shift of oneoctaveat normalsoundincidencehasbeenobtained.

iii

iv ABSTRACT

Preface

This thesissummarizesmy work for the degreeof doktor ingeniør at the Norwegian Uni-versity of ScienceandTechnology. The work hasbeenconductedat the Acousticsgroup,Departmentof Telecommunications,NTNU, andwasfinancedby Hydro Aluminium AS aspartof their strategic doctorateprogram“Aluminium in Buildings” at NTNU. Thework de-scribedin this thesiswasperformedin theperiodFebruary1997to June2000.

Acknowledgements

First, I want to thank my supervisors,ProfessorTor Erik Vigran and ProfessorUlf Kris-tiansen,for theirexcellentsupportto this thesis.I alsowantto thankthemfor their invaluablesupportandinspirationto my experimentalandtheoreticalwork duringtheproject.A specialthankgoesto Tor Erik Vigranfor continuinghis supportalsoafterhis retirement.

I want to thankmy colleaguesat theAcousticsgroupfor sharingthoughtsandcommonfrustrations.In particularI amthankfulto AsbjørnSæbø,with whomI sharedoffice. Hehasalwaysbeenhelpful whenI have hadquestionsaboutEmacs,Linux, LATEX, Perl,MATLABor anything else. I alsowant to thankthe secretaryat theAcousticsgroup,ÅseVikdal, forherhelpfulnesson all occasions.

Leif O. Malvik andthetechniciansat theDepartmentworkshopdeservesaspecialthank.They have provided extensive help with the designand implementationof equipmentandsamplesusedduring the experimentalwork. I specialthankalsogoesto the GNU Project,the Linux Community, DonaldE. Knuth, Leslie Lamport,Larry Wall andMathWorks Inc.for providing excellentsoftwarefor my computationalwork.

I amvery gratefulto my wife Lise andmy daughters,RagnaKristine andIngaAmalie,for their enduranceduringthefrustratingperiodsof my work.

Finally, I want to thankHydro Aluminium, andvice presidentEinar Wathnein partic-ular, for giving me the opportunityto take part in this project. Hydro Aluminium hasalsogenerouslysupportedguidedtoursto someof their productionsiteshomeandabroad.Thesetourshave given thepossibility of sharingtechnicalthoughts,aswell assocializing,acrossthedisciplines.

v

vi PREFACE

Contents

Abstract iii

Preface v

List of Figures ix

List of Tables xi

Nomenclature xiii

1 Intr oduction 1

2 Helmholtz resonators 32.1 Resonancefrequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 SingleHelmholtzresonator. . . . . . . . . . . . . . . . . . . . . . . 42.1.2 DistributedHelmholtzresonators . . . . . . . . . . . . . . . . . . . 9

2.2 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Thermalandviscousresistanceat asurface . . . . . . . . . . . . . . 112.2.2 Viscousresistancein resonatorneck . . . . . . . . . . . . . . . . . . 12

2.3 Impedanceof cylindrical andslit-shapedopenings. . . . . . . . . . . . . . . 132.3.1 Lineardomain,independentperforations . . . . . . . . . . . . . . . 132.3.2 Effectof nonlinearityathigh soundpressurelevels . . . . . . . . . . 142.3.3 Effectof interactionbetweenperforations. . . . . . . . . . . . . . . 15

2.4 Developmentsin panelabsorbergeometries . . . . . . . . . . . . . . . . . . 182.4.1 Non-traditionalaperturegeometries . . . . . . . . . . . . . . . . . . 182.4.2 Helmholtzresonatorscoveredby platesor foils . . . . . . . . . . . . 23

3 Micr operforated panelswith horn-shapedorifices 253.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.1 IntegrationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.2 Finite ElementMethod . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.3 Finite DifferenceMethod. . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Measurementsandsimulations . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Measureddimensionsof samples . . . . . . . . . . . . . . . . . . . 33

vii

viii CONTENTS

3.2.2 Impedancemeasurementsin Kundt’s tube . . . . . . . . . . . . . . . 353.2.3 Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.1 Comparisonof simulationmethods . . . . . . . . . . . . . . . . . . 393.3.2 Variationof horngeometry. . . . . . . . . . . . . . . . . . . . . . . 413.3.3 Comparisonwith measurements. . . . . . . . . . . . . . . . . . . . 463.3.4 Comparisonwith microperforatedpanels . . . . . . . . . . . . . . . 47

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Doublepanel absorbers 494.1 Circularperforations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Measurementsandsimulations. . . . . . . . . . . . . . . . . . . . . 534.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Slit-shapedperforationswith constantseparation . . . . . . . . . . . . . . . 584.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Measurementsandsimulations. . . . . . . . . . . . . . . . . . . . . 624.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Slit-shapedperforationswith adjustableseparation . . . . . . . . . . . . . . 704.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.2 Measurementsandsimulations. . . . . . . . . . . . . . . . . . . . . 714.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4 Adjustableslittedpanelabsorber. . . . . . . . . . . . . . . . . . . . . . . . 894.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.4.2 Measurementsandsimulations. . . . . . . . . . . . . . . . . . . . . 904.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Conclusions 97

Bibliography 99

APPENDIX

A Documentationof functions 105A.1 Generalfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Microhornmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A.3 Perforatedpanelmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113A.4 Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Figures

2.1 SingleHelmholtzresonator. . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Ingard’scalculatedinnermassendcorrectionδi . . . . . . . . . . . . . . . . 72.3 Resonancefrequency asfunctionof cavity depthto width ratio . . . . . . . . 82.4 DistributedHelmholtzresonators. . . . . . . . . . . . . . . . . . . . . . . . 92.5 Thermalandviscousboundarylayerthicknessesdh anddv . . . . . . . . . . 112.6 Effectof interactionon resistiveandreactiveendcorrections . . . . . . . . . 162.7 Comparisonof correctionfactorsfor classicendcorrection . . . . . . . . . . 172.8 Helmholtzresonatorswith slottedneckplates . . . . . . . . . . . . . . . . . 192.9 Half-absorptionbandwidthof MPPasfunctionof perforateconstantx . . . . 212.10 Calculatedandmeasuredabsorptionof a microperforatedfoil . . . . . . . . . 222.11 Calculatedandmeasuredabsorptionof a two-layermicroperforatedfoil . . . 222.12 Helmholtzresonatorscoveredby protectingplate . . . . . . . . . . . . . . . 232.13 Helmholtzresonatorcoveredby foil . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Frontview andcross-sectionof microhornpanelabsorber. . . . . . . . . . . 253.2 Geometryof microhorn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 FDM grid in microhorn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Treatmentof FDM grid pointscloseto curvedsurface. . . . . . . . . . . . . 323.5 Measuredandmodelledradiusof microhornsample. . . . . . . . . . . . . . 343.6 Overview of setupfor impedancemeasurementsin Kundt’s tube . . . . . . . 363.7 Mountingof microhornpanelsamplein Kundt’s tube . . . . . . . . . . . . . 363.8 Setupfor impedancemeasurementsin Kundt’s tube . . . . . . . . . . . . . . 373.9 Flowchartfor FEM simulationsprocess . . . . . . . . . . . . . . . . . . . . 373.10 FEM simulationsof micro-perforationwith porousmaterial. . . . . . . . . . 393.11 FDM andIM simulationsof microhorngeometries.Comparison . . . . . . . 403.12 FDM simulationsof microhorngeometries.Effectof hornshape,I . . . . . . 413.13 FDM simulationsof microhorngeometries.Effectof hornshape,II . . . . . 423.14 FDM simulationsof microhorngeometries.Effectof horndimensions,I . . . 433.15 FDM simulationsof microhorngeometries.Effectof horndimensions,II . . 443.16 FDM simulationsof microhorngeometries.Effectof horndimensions,III . . 453.17 FDM simulationsandmeasurementof microhornpanelsample. . . . . . . . 463.18 FLAG simulationsandmeasurementsof panelabsorbersamples . . . . . . . 48

ix

x LIST OF FIGURES

4.1 Mountingof perforateddoublepanelin Kundt’s tube . . . . . . . . . . . . . 504.2 Geometryof theperforateddoublepanelmodel . . . . . . . . . . . . . . . . 514.3 Equivalentcircuit for perforateddoublepanelmodel . . . . . . . . . . . . . 524.4 CircularKundt’s tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.5 Simulationsandmeasurementsof perforateddoublepanels,I . . . . . . . . . 564.6 Simulationsandmeasurementsof perforateddoublepanels,II . . . . . . . . 564.7 Simulationsandmeasurementsof perforateddoublepanels,III . . . . . . . . 574.8 Simulationsandmeasurementsof perforateddoublepanels,IV . . . . . . . . 574.9 Simulationsandmeasurementsof perforatedsinglepanels . . . . . . . . . . 584.10 Geometryof theslitteddoublepanelresonator. . . . . . . . . . . . . . . . . 594.11 Geometryof FDM modelfor slitteddoublepanelresonator. . . . . . . . . . 604.12 Simulationsandmeasurementsof slitteddoublepanels,I . . . . . . . . . . . 664.13 Simulationsandmeasurementsof slitteddoublepanels,II . . . . . . . . . . . 664.14 Simulationsandmeasurementsof slitteddoublepanels,III . . . . . . . . . . 674.15 Measurementsof slitteddoublepanels.Effectof plateseparation. . . . . . . 684.16 Measurementsof slitteddoublepanels.Effectof platereversal . . . . . . . . 694.17 Measurementsof slitteddoublepanels.Effectof distancebetweenslits . . . . 704.18 Measurementsof slitteddoublepanelsandsimulationsof MPPs . . . . . . . 704.19 Sampleof adjustableslittedpanelabsorber. . . . . . . . . . . . . . . . . . . 724.20 Overview of setupfor impedanceandvibrationmeasurementsin Kundt’s tube 734.21 Setupfor impedanceandvibrationmeasurementsin Kundt’s tube . . . . . . 734.22 Velocityfieldsof adjustableslitteddoublepanels . . . . . . . . . . . . . . . 754.23 Simulationsof doublepanels.Effectof exchangeof plates . . . . . . . . . . 774.24 Simulationsof doublepanels.Effectof platethicknessandslit width . . . . . 784.25 Simulationsof doublepanels.Effectof plateseparation. . . . . . . . . . . . 794.26 Simulationsof doublepanels.Effectof center-centerdistancebetweenslits . 814.27 Simulationsof doublepanels.Effectof separationbetweenfront andrearslits 824.28 Measurementsof panelvibrationvelocity, modulus . . . . . . . . . . . . . . 834.29 Measurementsof panelvibrationvelocity, phase. . . . . . . . . . . . . . . . 844.30 Measurementsof panelvibrationandabsorptioncoefficient. Comparison . . 854.31 Simulationsandmeasurementsof doublepanels,I . . . . . . . . . . . . . . . 864.32 Simulationsandmeasurementsof doublepanels,II . . . . . . . . . . . . . . 874.33 Simulationsandmeasurementsof doublepanels,III. “Open” � “closed” . . 884.34 FLAG simulationsandmeasurementsof panelabsorbersamples . . . . . . . 894.35 Sampleof full scaleadjustableslittedpanelabsorber . . . . . . . . . . . . . 914.36 Full scaleadjustableslittedpanelsin reverberationroom . . . . . . . . . . . 934.37 Measurementsof doublepanelsin diffusesoundfield . . . . . . . . . . . . . 944.38 Simulationsandmeasurementsof doublepanelsin diffusesoundfield . . . . 95

List of Tables

3.1 Geometryparametersusedin FDM andIM simulationsof microhorn,I . . . 383.2 Geometryparametersusedin FDM simulationsof microhorn,II . . . . . . . 38

4.1 Dimensionsof perforatedplatesusedfor measurements. . . . . . . . . . . . 534.2 Dimensionsof measuredandsimulatedperforateddoublepanels,I . . . . . . 554.3 Dimensionsof measuredandsimulatedperforateddoublepanels,III . . . . . 554.4 Dimensionsof slittedplatesusedfor measurements. . . . . . . . . . . . . . 634.5 Dimensionsof measuredslitteddoublepanels,I . . . . . . . . . . . . . . . . 644.6 Dimensionsof measuredslitteddoublepanels,II . . . . . . . . . . . . . . . 644.7 Simulationparametersusedin FDM modelof slitteddoublepanels. . . . . . 654.8 Calculatedeffectivemassfor typical slit andgapwidths . . . . . . . . . . . . 684.9 Dimensionsof measuredadjustable,slitteddoublepanels. . . . . . . . . . . 714.10 Calculatedmaximumvelocitiesin slitteddoublepanel . . . . . . . . . . . . 764.11 Dimensionsof full scaleadjustableslittedpanels . . . . . . . . . . . . . . . 91

xi

xii LIST OF TABLES

Nomenclature

Symbols

A Cross-sectionareaof resonatorcavity volumea Width of resonatorcavity volumeB Lengthof resonatorcavity volumeb Center-centerdistancebetweenresonatoropeningsC AcousticreactanceCP Heatcapacityof air at constantpressure,CP

� 1 � 01J�Kkg � †

c0 Velocityof soundin air, c0� 340m

�s �� †

D Perimeterof resonatoropeningd Thicknessof air layerbehindpanelabsorberdh, dv Thicknessof thermalandviscousboundarylayersf , f0 Frequency, andfrequency at resonancef1, f2 Lowerandupperhalf-absorptionfrequencies,α � α0

�2 for f � f1 � f2

g Gapwidth betweenplatesof doublepanelabsorberh Lengthof outerpartof microhorni An integerdenotingplatenumberJm Bessel’s functionof first kind andmth orderk Wavenumber, k � 2π

�λ

ks Structurefactorof porousmaterialL Acousticinductance

Numberof x- or r-stepsin FiniteDifferencesimulationsL1, L2 Minimum numberof x-stepsin narrowestslit, andminimumtotal stepsl Lengthof resonatorneck

Lengthof innerpartof microhornAn integer

l i , l �i Lengthandreducedlengthof perforationsin platei, l �i � l i � ∆zSi

M Numberof shellsin microhornIntegrationMethodsimulationNumberof z-stepsin microhornFiniteDifferencesimulationNumberof z-stepsin gapin doublepanelFinite DifferencesimulationNumberof discreteanglesin approximationof αst

xiii

xiv NOMENCLA TURE

m An integerN Numberof elementsfacingincomingwave in FiniteElementsimulationn A vectornormalto asurfaceP, P0 Pressure,andambientair pressurePh, Pv Thermalandviscousenergy dissipationat asurfacep, ∆p Soundpressure,andsoundpressuredropacrosstubeor perforationp0 Assumedinputsoundpressurein FiniteDifferencemodels,p0 � 1Papl m Soundpressureat grid point l � m�Q Q-valueof resonance,Q � f0

� �f2 � f1 �

q Lateraldistancebetweencentersof front andrearslitR Acousticresistancer, r Radiusof orifice,andhydraulicradiusof orifice, r � 2S

�D

∆r Radialgrid lengthin Finite Differencesimulationsr0, rh Radiusof outerandinnerpartsof microhornr i Radiusof perforationsin plateir lin , r log Linearandsemi-empiricradiusfunctionfor microhorn,Eqs.(3.26)and(3.25)

rm Radiusof shellm in microhornimpedancecalculationS Cross-sectionareaof resonatoropening∆S Smallareaassociatedwith grid pointSi Numberof z-stepsinto slit i in FiniteDifferencesimulations Distancebetweenmicrophonesin Kundt’s tubeT Totalnumberof z-stepsin FiniteDifferencesimulation,T � S1 � M � S2t Distancebetweenfirst microphoneandsamplein Kundt’s tubeti Thicknessof plateiU Volumeflow rateu Horizontal,lateralor tangentialparticlevelocityu, ul m Particlevelocity, u � ux � vz, andparticlevelocityat grid point l � m�V Volumeof resonatorcavityv, v Vertical,axial or normalparticlevelocity, andaverageaxial velocityWi Numberof x-stepsin slit i in FiniteDifferencesimulationw, wi Width of slits in panelabsorber, andwidth of slits in plateiX Positionof left edgeof rearslit (seeFig. 4.11)x Perforateconstant(acousticReynold’snumber),§ x � r � ωρ0

�µ

∆x Horizontal,lateralor tangentialgrid lengthin FiniteDifferencesimulationsxi Perforateconstantof perforationsin platei, with radiusr � r i

xm Valueof x for shellm in microhornimpedancecalculationxs Perforateconstantfor slits,x � w

2 � ωρ0

�µ

y Ratioof resonatoropeningandcavity dimensionsZ Acousticimpedance,Z � z

�S � p

�vS

z Specificimpedance,z � p�v

NOMENCLA TURE xv

∆z Lengthof shell in microhornintegrationmodelVertical,axial or normalgrid lengthin Finite Differencesimulations

za Characteristicimpedanceof air, za� ρ0c0

α, α0 Absorptioncoefficient,andabsorptioncoefficientat resonanceαst Absorptioncoefficient in a diffusesoundfieldβ , β � Anglesγ Ratioof specificheatsof air, γ � 1 � 4δ Massendcorrectionof resonatornecklength,correctedδ0, δi Massendcorrectionof resonatornecklength,outerandinneraperturesδtot Massendcorrectionof resonatornecklength,bothaperturesδs tot Massendcorrectionof resonatornecklength,bothapertures(slittedpanel)ε Argumentto Fok’s functionψFok, ε � � S

�A

ζ Relative,specificimpedance,ζ � z�Φza

� θ � jωχηm Distanceto surfaceto theright of grid point L m� � m� , in unitsof ∆rθ Relative,specificresistanceθi , θr Relative,specificsystemresistanceandradiationresistanceκ Thermalconductivity of air, κ � 23� 1mW

�Km � †

λ , λ0 Wavelength,λ � c�

f , andwavelengthat resonanceλl m Distanceto surfacebelow Finite Differencegrid point l � m� , in unitsof ∆z

µ Coefficientof viscosityof air, µ � 17� 9 � Ns�m2 �� ††

ρ0 Densityof air, ρ0� 1 � 23kg

�m3 �� ††

ρe Effectivedensityof air in tubeρeff Effectivedensityof air in cylindrical opening,includesviscouslossesρs eff Effectivedensityof air in slit-shapedopening,includesviscouslossesσ Flow resistivity of a porousmaterialσe Effectiveresistivity of tubeΦ Perforationof distributedresonator(perforatedpanel),Φ � S

�A

Φs Perforationof distributedresonator(slittedpanel),Φs� w

�b

φ Porosityof a porousmaterialχ Relative,specificinductanceof perforatedpanelψFok A polynomialfunctionfor interactionendcorrections,usedin Eq. (2.46)ω , ω0 Angularfrequency, ω � 2π f , andangularfrequency at resonance

�at0 � C andatmosphericpressure��at15 � C andatmosphericpressure

†MorseandIngard [1968]††Gerhartetal. [1993]§CraggsandHildebrandt [1984];Maa [1998]

xvi NOMENCLA TURE

Abbreviations

FFT FastFourierTransformFDM FiniteDifferenceMethodFEM FiniteElementMethodIM IntegrationMethodKE Kinetic energyMLS MaximumLengthSequenceMPP MicroperforatedpanelPE Potentialenergy

Chapter 1

Intr oduction

TheHelmholtzresonatorprincipleis anold andverysimpleconcept.Basically, it consistsofanair-filled cavity with arelatively smallopening.It hasbeeninvestigatedfor over100years,mostnotablyby vonHelmholtz,Lord RayleighandIngard [1953]. Helmholtzresonatorsarein commonusein soundabsorberswhereabsorptionat low frequenciesis required[Seee.g.Leeand Swenson, 1992]. The mostcommonimplementationsareasperforatedpanelab-sorbersplaceda distancefrom a backwall, andassilencersin ducts. The requiredvolumeof suchdistributedHelmholtzresonatorsis significantlysmallerthanfor absorbersof porousmaterials.Sucha constructioncanhavea largesoundabsorptioncoefficientat theresonancefrequency. However, theabsorptionbandwidthasusuallyverylimited,dueto thesmallinher-entenergy dissipation.To compensatefor this, thecavity behindthepanelhastraditionallybeenpartly or completelyfilled with porousmaterial,or a resistive layerhasbeenput neartheresonatoropenings[Ingard andLyon, 1953;Ingard, 1954;KristiansenandVigran, 1994;Mechel, 1994b]. For suchconfigurations,the main functionsof the perforatedpanelaretoaddamassreactanceto theimpedanceandto protecttheporousmaterial.

Today, thereis atrendtowardapanelabsorberandsilencerdesignwhereporousmaterialsareexcluded.This is primarily dueto theenvironmentaldisadvantagesof porousmaterials.Porousmaterialsof thefibroustypemayreleasefibersinto theair, andarenoteasilycleaned.Additionally, porousmaterialsmaynot be robustenoughin physicallyor chemicallyroughenvironments[Ackermannetal., 1988].Consequently, severalauthorshave investigatednewpanelabsorbersconcepts.By clever designof thepanels,the inherentviscousenergy dissi-pationat thesurfacesmaybeincreasedcomparedto thetraditionalperforatedpanels.

The simplestof thesedesignsaremicroperforatedpanels(MPP), which areperforatedpanelwith sub-millimeterperforations.Thesmallperforationsincreasestheviscousenergydissipation,andhencetheabsorptionbandwidth,significantlycomparedto traditionalperfo-ratedpanels[Maa, 1987,1998;Fuchsetal., 1999].Theabsorptionbandwidthmaybefurtherextendedby combiningseveralMPPsseparatedby adistance,asdescribedby ZhangandGu[1998]; Kang et al. [1998]. The MPP conceptis also the basisfor commercialproductslike MICROSORBERR

�, wherethe “panels” arethin, perforatedfoils [Fuchset al., 1999].

Anotherpanelabsorberconceptwithout addedporousmaterialsis discussedby Frommholdet al. [1994] andby Mechel [1994a]. It is a grid of Helmholtzresonatorscoveredby a thin

1

2 Intr oduction

plateor a foil. The foil is separatedfrom the resonatorsby a small air gap. The purposeof thesekind of absorbersis to increasethe numberof possibleresonances,in additiontotheHelmholtzresonance.By optimaldesign,theflow in the thin gapmaycausesignificantenergy dissipation.Mechelhasalsopresentedanotherconcept,wherethepanelis laterallydivided in two. The small separationbetweenthe two partsintroducesadditionalpossibleresonances.A foil mayalsobeput betweenthetwo partsof thepanelto increasetheflow inthesmallgapbetweenthetwo parts,andhenceincreasetheenergy dissipation.

As a continuationof the trend describedabove, this thesispresentsmy work on twodifferentperforatedpanelabsorberswheretheviscousenergy dissipationhasbeenenhancedby theuseof non-traditionaldesignsof thepanelopenings.

Chapter2 containsanintroductoryoverview of thetheoryof Helmholtzresonators.Thisincludestheory for the calculationof resonancefrequency and energy dissipationof res-onators,aswell asthe impedanceof resonatoropenings.The chapteralsoincludesa shortreview of relevantprior work onnon-fibrousabsorbers.

Chapter3 presentsthework with themicrohornconcept.This is amicroperforatedpanelwheretheorificesarehorn-shaped.Thechapterpresentsthreedifferentmodels.Oneof these,theFiniteDifferenceMethod,hasbeenusedto simulateseveralmicrohornsgeometrieswherethehornwidth, lengthandshapehasbeenvaried. Theresultsof thesesimulations,andoneexperimenton a microhornpanelsample,arepresented.

Chapter4 presentsthework on thedoublepanelconcept.Thechapterhasfour sections,correspondingto thefour differentdoublepanelconceptsthathasbeentestedexperimentally.Two differentmodelsarepresented,onefor doublepanelswith circularperforationsandtheotherfor doublepanelswith slit-shapedopenings.

Chapter 2

Helmholtz resonators

TheHelmholtzresonatoris anancientandverysimpleconcept.Yet,it hasbeenthesubjectofinvestigationsfor over100years,perhapsmostnotablyby vonHelmholtz,Lord RayleighandIngard [1953]. ThoseinvestigatorscontributedfundamentalknowledgeabouttheHelmholtzresonatorprinciple,someof which is recapitulatedin whatfollows.

Helmholtz resonatorsare in commonusein applicationssuchas acousticelementsinroomsand in duct silencers.They cantake two principal forms: single resonatorsor dis-tributedresonators.Theperforatedpanelsoftenusedin roomsis anexampleof distributedHelmholtzresonators.Thegeometriesof Helmholtzresonatorsarevery diverse,but they allhave two characteristicfeaturesin common:A cavityanda relatively smallopeningthroughwhich thesoundenergy entersthecavity. In thecaseof thedistributedHelmholtzresonator,thecavity is sharedby theresonatoropenings.For sufficiently largepanels,eachopeningcanbeassociatedwith acavity volumedeterminedby theseparationbetweentheperforations.Inits variousforms,theHelmholtzresonatorhasbothadvantagesanddisadvantagescomparedto the commonlyusedporousor fibrousabsorbers.By varying the volumeof the cavity orthe sizeof the opening,the resonatorcan be tunedto absorbsoundat a given frequency.Thus,absorptionin thelow frequency rangecanbeachievedwithout increasingthedepthoftheabsorberconstruction,aswould berequiredfor any porousabsorbent.Oneobviousdis-advantageof theHelmholtzresonator, comparedto porousabsorbers,is that the absorptionbandwidthis usuallyrelatively small. This is becausetheresonatorsystemin itself haslowenergy dissipation.The low bandwidthhastraditionallybeencompensatedfor by partly orcompletelyfilling the cavity with porousmaterial,or by covering the inner aperturesby athin, resistive layer. Theeffectof this hasbeendiscussedto someextentin theliterature[seee.g. Ingard andLyon, 1953;Ingard, 1954;Brouard et al., 1993;Mechel, 1994b;KristiansenandVigran, 1994].This will not bediscussedfurtherhere.

For alimited frequency range,aHelmholtzresonatoris analogousto asimplemechanicaldampedresonatorsystem.Theresilienceof theair in thecavity makesit similar to a spring.Themassof theair in andaroundtheorifice is equivalentto a mechanicalmass.Thevalueof the “spring constant”andthe massis what mainly determinesthe resonancefrequency,asshown in thenext section.Theanalogousmechanicalresistanceof thesystemis mainlydeterminedby viscousenergy dissipationat thesurfacesof theresonator(seeSec.2.2).

3

4 Helmholtz resonators

V

l

S

Figure 2.1: A Helmholtzresonator. The openingof the resonatoris often referred to asthe neck. Thecavity volumeV, the neck length l and the cross-sectionarea S of the neckdeterminetheresonancefrequencyof theresonator, byEq.(2.11).

2.1 Resonancefr equency

Due to the inherentsmall bandwidthof Helmholtz resonators,a small shift in resonancefrequency canresultin asignificantdecreasein absorptioncoefficientat thetargetfrequency.Thecalculationof absorptionfrequency shouldthereforebeasaccurateaspossible.Severalauthorshaveinvestigatedthefrequency dependency of thegeometryof Helmholtzresonators,andhave found that thegeometryhasa significantinfluence.For low valuesof the systemresistance,its effecton theresonancefrequency canbeignored,asis donebelow.

2.1.1 SingleHelmholtz resonator

The original resonatorstudiedby von Helmholtzwasa very simpleone; A rigid cavity ofvolumeV, with small dimensionscomparedto the wavelengthof the incidentsound. Thecavity had a small circular orifice of radius r. When small dimensionsare assumed,thecomplex problemof how wavespropagatein the cavity canbe ignored. The resonatorcanthen be modeledas a mass-spring-resistanceproblemas describedabove. An alternativeis to model the resonatorby a “lumped-circuit” electricalmodel,wheresoundpressureisanalogousto voltageandvolumeflow rateis analogousto current.Eitherway theresonancefrequency of vonHelmholtz’ resonatoris foundto be[MorseandIngard, 1968,Eq.9.1.30]

f0� c0

2π

�2rV� c0

2π

�4SVD � (2.1)

wherec0 is thevelocity of soundin air, S is thecross-sectionareaof theorifice andD is theorifice perimeter. This equationcanalsobe usedfor non-circularorificesof regular shape,i. e. not verywideor long.

Equation(2.1)doesnotcontainany referenceto thelengthof thecavity opening.Theres-onancefrequency of a cavity with anopeningof lengthl , asshown in Fig. 2.1,canbedevel-opedusingthemechanicalmass-springanalogymentionedabove. Themechanicalstiffnessof the air in the cavity, km, canbe derived from the adiabatic,perfectgasequationapplied

2.1Resonancefr equency 5

on acylindrical volumeV, which is compressedanamount∆V � S∆x by soundpressure∆Pappliedto theresonatoropening.It is assumedthatthecavity walls arerigid.

PVγ � constant (2.2)

gives,by derivation,

∆PVγ � � γPVγ � 1∆V � (2.3)

whereP � P0 is theambientpressureandγ � 1 � 4 is the ratio of specificheatsof air. Since∆P � ∆F

�S, wehave

∆FS� � γP0

S∆xV

(2.4)

Therefore,

km� � ∆F

∆x� γP0S2

V� ρ0c2

0S2

V � (2.5)

whereγP0� ρ0c2

0, andρ0 is thedensityof air. Thisderivationassumessoundpressuressmallcomparedto theambientpressure,∆P � P0, andadiabaticcompressionof theair [MorseandIngard, 1968,Ch.6.1]. Themechanicalmassof theair in theopeningis equivalentto thatofa tubeof cross-sectionareaSandlengthl , andis givenby

mm� ρ0Sl (2.6)

The mechanicalstiffnessand massof the resonatoris relatedto acousticcapacitanceandinductance,respectively, by a factorS2. Thus,thecapacitance

C � Vρ0c2

0� (2.7)

andtheinductance

L � ρ0l

S(2.8)

Theresonatorsystemis thereforeequivalentlydescribedby a mechanicalimpedance

Zm� jωmm � km

jω � Rm � (2.9)

or by anacousticimpedance,

Z � jωL � 1jωC � R� (2.10)

whereω � 2π f . The calculationof the systemresistanceR � Rm�S2 may involve several

energydissipationmechanisms,andis describedfurtherin Sec.2.2.Theresonancefrequencyof thesystemis givenby

f0� ω0

2π� 1

2π

�km

mm

� 12π

1�LC

� c0

2π

�S

Vl(2.11)


Thisequationis acoarseapproximation,but canbeusedwith reasonableaccuracy onvariousgeometriesaslong asthedimensionsaresmallcomparedto thewavelength.

Thenecklengthl in Eq.(2.11)mustin mostcasesbecorrectedwith anaddedlength.Thisis becausetheflow of air throughtheneckaffectstheair closeto theinnerandouterapertures.This nearbyair will take part in theflow andthuscontributeto thetotal resonatormass.Forholeswith circular cross-section,Rayleighproposedan end correctionδ0 that shouldbeaddedto l in Eq. (2.11)[Ingard, 1953;Chanaud, 1994].Theproposedendcorrection,

δ0� 8r

3π� 8

3π

�Sπ � (2.12)

correspondsto the inductive part of the radiationimpedanceof a circular, planepistonofradiusr in aninfinite wall. Thelatterexpressionfor δ0 canbeusedasanapproximationforothershapesof theresonatoropening.Sincethisendcorrectioncorrespondsto asinglepistonflangedin aninfinite wall, caremustbetakenif it is usedfor the inneraperture.For thesamereason,Eq.(2.12)mayalsobeinvalid for theouterapertureif thereareotheropeningsnearby.In lackof abetteralternative,RayleighusedEq.(2.12)for theinnerandouterapertures.Thetotal endcorrection,

δtot� 2δ0

� 16r3π � (2.13)

is thenaddedto l in Eq. (2.11).Thus

f0� c0

2π � SV l � δtot � � c0

2π � S

V�l � 2δ0 � (2.14)

Ingard [1953] did an extensive survey on the topic of resonators,andpresentedexpres-sionsfor the innerendcorrectionfor somesimplecircularandrectangulargeometries.As-sumingflat velocity profile in the resonatorneck, he found expressionsfor the inner endcorrection,shown in Fig. 2.2. For openingsrelative small comparedto the resonatorcavitycross-section,theinnerendcorrectioncanbeapproximatedby

δi� δ0 1 � 1 � 25y� � (2.15)

wherey is theratioof thedimensionsof theopeningandthecavity, andmustbelessthan0.4for the approximationto be valid. Accordingto Ingard,Eq. (2.15) is valid for threediffer-ent geometries:circularopening/cavity, squareopening/cavity, andcircularopening/squarecavity. Allard [1993,Eq.10.18]gaveanotherapproximationfor thelattergeometry:

δi� δ0 1 � 1 � 14y� � (2.16)

Note that in his book,Allard gave the correctionasfunction of ε � � S�A, whereA is the

cavity cross-sectionarea.This is probablyanerror, neglectingthefactor�

π�2 relatingε to

y for this geometry. Theagreementwith curve 2 in Fig. 2.2 is very goodfor Eq. (2.16)asitis writtenhere.ThedifferencebetweenEqs.(2.15)and(2.16)is probablynegligible in mostcases.For small openings,δi approachesδ0, andin this casethe latter canbe usedfor theinnerapertureendcorrectionfor all thegeometries.


Figure2.2: Ingard’scalculatedinnermassendcorrectionδi ofvariousgeometriesasfunctionof openingto cavitydimensionratio y. 1, Circular openingandcavity; 2, circular openingandsquarecavity;3, squareopeningandcavity. Figureadaptedfrom[ Ingard, 1953,Fig. 3].

Pantonand Miller [1975] showed that whenthe assumptionof long wavelengthscom-paredto theresonatorlengthwasdismissed,theresonancefrequency for acylindrical Helm-holtz resonatorcouldbefoundby solvingthetranscendentalequation

lABS

kB � cot kB� � (2.17)

whereB is the lengthof the cavity, andk � 2π�λ is the wavenumber. A is assumedcon-

stantthroughthe cavity length. They showed thatby usingthe two first termsin the seriesexpansionof cot kB� , theresonancefrequency wasapproximatelygivenby

f0� c0

2π � S

V l � δtot � � B2S3� (2.18)

whereV � AB, andδtot hasbeenaddedto the necklength. The classicexpressionfor theresonancefrequency, Eq. (2.14),is similarly foundby usingonly thefirst termin theseriesexpansionof thecotangent.Theclassicexpressionwasshown to beaccurateonly for cavitylengthsB � πλ

�16. PantonandMiller alsonotedthatEq. (2.14)conformslesswith exper-

imentaldatawhenIngard’s endcorrectionfor the innerapertureis used,i. e. δtot� δ0 � δi ,

thanif theclassicalendcorrectionis used,δtot� 2δ0. On theotherhand,Ingard’s endcor-

rectiongivesthebestcorrespondencewith measurementswhenusedwith Eq. (2.18). Theyexplainedthis by the fact that thecorrectiontermsin Eqs.(2.15)and(2.18)arein oppositedirections.

Chanaud[1994, 1997] investigatedthe effect of extremesin cavity geometry(deeporshallow), andof theshape,dimensionandplacementof theopening.A transcendentalequa-tion for the resonancefrequency, andexpressionsfor the inner endcorrections,werealso


Figure2.3: Calculatedresonancefrequencyas a functionof cavity depth,for a Helmholtzresonatorwith a square-facedcavity of constantvolume1000cm3 and a circular openingof radius2cm and length0 � 5cm. �� , RayleighequationEq. (2.14); - - -, transcendentalequationEq.(2.17)with l � l � 2δ0; —,Chanaud’sanalysis.Notethat in thefigure, d is thecavitylengthB, anda is thecavitywidth. Figure from[Chanaud, 1994,Fig. 5].

presentedfor parallelepipedicandcylindrical cavities. Chanaudassumedno internalresis-tancein the cavity, a flat velocity profile in the resonatoropening,and long wavelengthsrelative to thedimensionsof theopening. Themainresultswere:

– The resonancefrequency calculatedfrom the Rayleighequation,Eq. (2.14),deviatessignificantlyfrom theonepredictedby Chanaudexceptfor cubiccavities,asillustratedin Fig. 2.3.

– Pantonand Miller’ s transcendentalEq. (2.17), with l substitutedby l � 2δ0, corre-spondswith Chanaud’s equationsfor deepcavities, but not for wide cavities. This isalsoshown in Fig. 2.3.

– Variationin theopeningpositiongivesthegreatestdeviation of theRayleighequationrelative to Chanaud’sequations.

– Variationof theopeningshapegivesnosignificantdeviationbetweentheequations.

Chanaud[1997] alsonotedthat Ingard’s innerendcorrection,Eq. (2.15),deviatedfrom theendcorrectionfoundby Chanaudfor y outsidethe interval 0.22–0.52. It is not stated,butseemsclear that constantcavity volumehasbeenassumedfor this comparison.For wideopenings,Ingardlet y � 0 � 4 betheupperlimit of validity, but indicatedno lower limit on yfor wideandshallow cavities. Otherstudieshaveanalyticallyandnumericallyconfirmedthesignificanceof theshape(i. e. lengthto width ratio) of theresonatorvolumein determiningtheresonatorfrequency [seee.g. Selametet al., 1995,1997;Dickey andSelamet, 1996].


d

A

l

S

Figure2.4: A perforatedpanel.Thisis anexampleof distributedHelmholtzresonators. Eachpanelopeningwith cross-sectionareaShasan associatedcavityvolumeAd. Theresonatorneck lengthl equalsthethicknessof thepanel.

2.1.2 Distrib uted Helmholtz resonators

The resonancefrequency of distributedresonators,asshown in Fig. 2.4, canbe calculatedby Eq. (2.14) if the panelopeningscanbe consideredindependent.This is the casewhenthe openingcross-sectionareaS is considerablysmallerthan the cavity cross-sectionareaA associatedwith eachopening[Morseand Ingard, 1968,Ch. 9.1]. Whenthis is the case,Eq. (2.14)canberewrittenas

f0� c0

2π � Φd l � δtot � � (2.19)

whereΦ � S�A is thepanelperforation.Thepanelis assumedto beplaceda distanced in

front of a rigid wall.Equation(2.19)canalsobeusedfor slittedpanels.Assumingunit lengthslits,Eqs.(2.7)

and(2.8)arewrittenas

Cs� bd

ρ0c20

(2.20)

and

Ls� ρ0l

w � (2.21)

wherew is the width of the slits andb is the center-centerdistancebetweenthe slits. Theperforationin Eq.(2.19)is thenΦs

� w�b. Of course,theendcorrectionin Eq.(2.13)cannot

beusedin this case.SmithsandKosten[1951] have studiedpanelabsorberswith infinitely


long slits. Basedon two differentvelocity profiles(constantvelocityandconstantpressure)which representstheupperandlower limits on theendcorrection,they have developedtheformula

δs tot� � 2w

πln � sin πw

2b !#" (2.22)

for thetotal inductive endcorrectionin theconstantpressurecase.For smallvaluesof b�d,

theendcorrectionin theconstantvelocity casediffersinsignificantlyfrom thevalueof δs totabove. Equation(2.22)wasdevelopedassumingb � λ

�2 and2πd

�b $ 2. For f � 500Hz,

theserequirementsequalsb � 34cmandd $ 10cm, which areusuallynot hardto satisfy.

2.2 Energy dissipation

TheHelmholtzresonatoris bothanabsorberanda scatterer. For distributedHelmholtzres-onators,the absorptioncross-sectionat resonanceis determinedby the ratio of the systemresistanceθi to theouterapertureradiationresistanceθr, andis givenby [Ingard, 1953]

τ � λ 20

2π4θi

�θr�

1 � θi

�θr � 2 � (2.23)

whereλ0 is the resonancewavelength.The absorptioncross-sectionhasa maximum 12π λ 2

0whenθi

� θr. The correspondingabsorptionbandwidthis usuallyvery small. If θi is in-creasedby the additionof someporousmaterialor by specialdesignof the opening,thenthe absorptionbandwidthincreasesat the costof a decreasein theabsorptioncross-sectionarea[seee.g. CremerandMüller, 1978a,pp. 195-197]. For mostgeometries,θr is usuallynegligible comparedto theθi [Ingard andIsing, 1967;Melling, 1973].

Thus,for casesof practicalinterest,thebandwidthis determinedby thesystemresistance,andit is thereforeimportantto predictthiswith reasonableaccuracy. Thesystemresistanceisdeterminedby consideringtheenergy dissipationin theresonatorsystem.Theexactmecha-nismsinvolvedin theenergydissipationin fluidsarequitecomplex [MorseandIngard, 1968,Ch.6.4], andinclude:

– Fluid internalenergy dissipationby% thermalconductivity% fluid viscosity% molecularenergy equipartition

– Surfaceenergy dissipationby% thermalconductivity% viscosity

For the casesdiscussedin this work, energy dissipationin relatively small enclosures,thefluid internalenergy lossis negligible comparedto energy lossat surfaces.

2.2Energy dissipation 11

dhdv

psf

u

Surface

u

Figure2.5: Thermalandviscousboundarylayer thicknessesdh anddv at a surface. Thermaland viscousenergy dissipationat a surfaceoccur mainly insidetheselayers. Thethermalenergy loss,Eq. (2.26),dependson thepressure psf at thesurface. Theviscousenergy loss,Eq. (2.27), is dependenton the tangential particle velocityu at the outsideof the viscousboundarylayer. Theparticlevelocityu is zero at thesurface.

2.2.1 Thermal and viscousresistanceat a surface

Thesurfaceenergy dissipationstake placein (usuallythin) layerscloseto thesurfaces.Thewidthsof theselayers,asgivenby MorseandIngard [1968,Ch.6.4] are

dh� � 2κ

ρ0ωCP & 2 � 4mm� f 'Hz( � (2.24)

for thermallosses,and

dv� � 2µ

ρ0ω & 2 � 2mm� f 'Hz( � (2.25)

for viscouslosses.Hereµ is the coefficient of viscosityof air, κ is the thermalconductiv-ity of air, andCP is the heatcapacityof air at constantpressure.At 100Hz, dh anddv areapproximately0.24and0 � 22mm, respectively. The valuesof dh anddv representthemini-mumdistancefrom a surfacewherethermalandviscousmodescanbeconsiderednegligiblecomparedto thepropagationalmode.

Thethermalenergy dissipationperunit surfaceareais proportionalto thesquaredpres-sureat thesurface:

Ph & γ � 12ρ0c2

0

ωdh )) psf )) 2 � (2.26)

wherepsf is thesoundpressureat thesurface.SeeFig. 2.5. Thesurfaceis assumedto havehigh thermalconductivity comparedto thatof air, sothecompressibilityis isothermalat thesurface.Theviscousenergy dissipationperunit surfaceareais proportionalto thesquareofu, thevelocitycomponenttangentialto thesurfacejust outsidetheviscousboundarylayer:

Pv & 12

ρ0ωdv * u * 2 (2.27)

It is assumedthat the wavelengthλ is muchgreaterthandh anddv, andthat the surfaceisinfinite andplane.Theenergy lossat a surfacecanalsobeexpressedasacousticresistances.


By theelectricalanalogy, thethermalresistanceof asurface∆S, overwhich thepressurep isassumedconstant,is

Rh�,+ γ � 1

ρ0c20

ωdh∆S- � 1 � (2.28)

andtheviscousresistanceof a surface∆S is

Rv� 1

U2 ρ0ωdv * u * 2 ∆S � 1U2 � 2µρ0ω * u * 2 ∆S� (2.29)

whereU is thevolumeflow rateby thesurface.For mostapplications,thermalenergy lossisnegligible comparedto viscousloss[Ingard, 1953;StinsonandShaw, 1985].Theequivalentelectricalcircuit hasa large thermalresistancein parallelwith a smallerviscousresistance.Thetotal resistanceis mainlydeterminedby thelatter.

2.2.2 Viscousresistancein resonatorneck

In Helmholtzresonators,themainpartof theenergy dissipationtakesplacenearandinsidethe necks,wherethe velocity is highest.The amountof energy dissipationdependson thevelocityprofilein andaroundtheresonatorneck.Becausethisvelocityprofilecanbedifficultor impossibleto calculateanalyticallyevenfor quite“normal” geometries,thepredictionofdissipationis not trivial. Undertheassumptionof largeradiusof curvaturecomparedto theviscousboundarylayerthickness(i. e. no sharpedges)anduniform flow in theneck,Ingard[1953] foundanalyticallythatthecombinedacousticresistanceof a circularholeof radiusrandthepanelsurfaceswasgivenby

R � 1πr2 � 2µρ0ω

1r l � r � (2.30)

Thelatterr in theparenthesesrepresentstheenergy dissipationat thepanelsurfaces,andcanbeconsideredto bethetotal resistiveendcorrectionto thenecklengthl . Ingard’sexperimen-tal resultsindicatedthattheendcorrectionshouldratherbe2r insteadof r, thus

R � 1πr2 � 2µρ0ω

1r l � 2r � (2.31)

This wasof coursedue to the fact that the velocity flow is not uniform, and sharpedgesexist. Thereforehigh viscousenergy dissipationtakesplacenearthe sharpedges. Ingardconfirmedthevalidity of Eq. (2.31)for a numberof sampleswith openingslargecomparedto theboundarylayer. Fromtheaboveequation,theresistiveendcorrectionis

Rend� 2

πr2 � 2µρ0ω � 2πr2 ρ0ωdv (2.32)

An expressionfor theslittedpanelcasemayalsobeobtained[KristiansenandVigran, 1994].Written asacousticresistance,andassumingunit lengthslits of width w, it is completelyanalogousto Eq.(2.31):

Rs� 1

w� 2µρ0ω

1w l � 2w� (2.33)

2.3 Impedanceof cylindrical and slit-shapedopenings 13

2.3 Impedanceof cylindrical and slit-shapedopenings

2.3.1 Linear domain, independentperforations

Ingard’sexpressionsfor R arebasedon theresistanceof flow overaninfinite, planesurface,Eq. (2.29). It assumesthat the tangentialvelocity u is calculatedfrom the wave equation,neglectingthe effect of viscosity. Whenthedimensionof theopeningis comparableto thethicknessof theboundarylayers,theprecedingequationsareno longervalid. Viscous(andsometimesthermal)effectsmustthenbeincorporatedinto theequationof motion.

Assumingthat thegasis incompressible(∇ � u � 0), thelinearizedNavier-Stokescanbewrittenas� ∇p � jωρ0u � µ∇2u (2.34)

For simplegeometriesthis equationcanbe usedto find an effective, complex density, ρeff .The imaginarypart of this densityis relatedto the viscousenergy loss. A summaryof thederivation, for the two geometriescircular-cylindrical tubeandinfinite slit, is presentedbyAllard [1993,Ch. 4]. Theprocedureis basedon original work by Kirchoff andlatersimpli-ficationsby Zwikker andKosten[1949,Ch. 2]. Stinson[1991] validatedthesimplificationsfor openingradii r in theverywide range10� 5m � r � f � 3. 2 � 104ms� 3. 2. For axial flow inz-directionin a circular, cylindrical tube,Newton’sequationcanbewrittenas� ∂ p

∂z� jωρeff v � (2.35)

wherev is theaveragevelocityover thetubecross-section.Thenthespecificimpedanceof acirculartubecanbeapproximatedby [seee.g. Maa, 1987;Allard, 1993]

z & ∆pv� jωρeff l

� jωρ0l / 1 � 2x� � j

J1

�x� � j �

J0

�x� � j �10 � 1

(2.36)

Here∆p is thepressuredropacrossthe tubeof length l , Ji is Bessel’s functionof first kindandith order, and

x � r

�ωρ0

µ� � 2

rdv

(2.37)

Theparameterx is proportionalto theratio of tuberadiusandviscousboundarylayerthick-ness.This very importantparameterhasin the literaturebeentermedshearwavenumber,acousticReynolds’numberandperforateconstant[Zwikker andKosten, 1949;Maa, 1998].Justlike theReynolds’ number, x indicatestheimportanceof inertiaforcescomparedto vis-cousforces.With theeffectivedensityof a slit-shapedopening[Allard, 1993,Eq.4.23], theequivalentof Eq.(2.36)for slits is

zs & jωρs eff l� jωρ0l / 1 � tanh

�xs�

j �xs�

j 0 � 1

(2.38)


Heretheparameterxs is givenby

xs� w

2

�ωρ0

µ(2.39)

It shouldbenotedthatEqs.(2.36)and(2.38)arethe impedanceof thetubeor slit only, anddoesnot includeinductiveor resistiveendcorrections.

CraggsandHildebrandt [1984] usedtheFinite ElementMethod(FEM) to solveNavier-Stokesequation,Eq.(2.34)for one-directionalsoundpropagationin tubesof variousshapes.In this caseonly theaxial velocitycomponentneedsto beconsidered:� ∂ p

∂z� jωρ0v � µ + ∂ 2v

∂x2 � ∂ 2v∂y2 - (2.40)

Thepressurewasassumedto bea functionof zonly, while thevelocity v wasassumedto beindependentof z. Equation(2.40)wassolvedfor somesimplecross-sectionshapes,e.g. slit,rectangle,circle andtriangle. The authorsstatedthat the valueof the perforationconstantx, Eq. (2.37),determinehow wavespropagatein the tube. For low valuesof x, i. e. x � 2,the flow is dominatedby viscouseffects, this is termedPoiseuille flow. For high x & 10and above, inertia forcesdominate. This is termedHelmholtzconditions. The numericalresultsconfirmsthis separationinto differentflow regimes. Poiseuilleflow hasa parabolicvelocity profile. For high x, thevelocity profile hasa low, flat partat thecenterof the tube,andhigh peaksnearthe edge.This appliesto circularaswell asrectangularandtriangularcross-sectionshapes.With the velocity v averagedover the tubecross-section,CraggsandHildebrandtwroteNewton’sequationin theform� ∂ p

∂z� jωρev � σev (2.41)

Note that theeffective densityρe andthe effective flow resistivity σe arereal quantities,asopposedto the complex ρeff in Eq. (2.35)above. CraggsandHildebrandtpresentedcalcu-lationsof the dimensionlessvariablesρe

�ρ0 andσer2� µ as functionsof x. The hydraulic

radiusr, definedastwice thecross-sectionareadividedby theperimeter, is usedin thedefi-nition of x. Thedensityratiowasshown to decreasewith increasingx, from values1.2(slit),1.33(circle) and1.44(triangle)at x � 0, to a commonvalue1.15for all cross-sectionshapeswhenx $ 10. Theresistivity ratio wasfoundto beconstantfor smallx, with extremevalues6.5 (triangle)and12 (slit). For largex, the resistivity ratio is proportionalto x [CraggsandHildebrandt, 1986].

2.3.2 Effect of nonlinearity at high soundpressure levels

Thegeometriesto bepresentedin Chs.3 and4 arerelatively complex, andnonlineareffectshavenotbeenincludedin themodelsdescribedin thesechapters.However, it shouldbenotedhow nonlinearityaffectstheimpedanceof a perforatedpanelabsorber. Thenonlinearterms,whichhavebeendiscardedin thebasicequationsabove,becomesignificantwhenthedrivingsoundpressurereachesa certainvalue. Theexactvalueis dependenton thegeometry. Fora typical microperforatedpanels,nonlinearitybecomessignificantfor soundpressurelevels


above 120dB, correspondingto a particlevelocity 2 � 5m�s in theperforations[Maa, 1987].

Ingard [1953] and Ingard and Ising [1967] reportedmeasurementson nonlinearresistanceandreactanceof circularopenings.Theresultsof thelatterauthorsindicatethatthenonlineareffectsbecomessignificantfor particlevelocitiesaround5m

�s. These,andmany otherresults

andtheorieswerereviewedby Melling [1973]. Someof theresultsarepresentedhere.The most important observation is that, for particle velocity v above somethreshold

level, the soundpressurein the openingis proportionalto the squaredvelocity, p ∝ ρ0v2

(Bernoulli’sequation).Theopeningresistanceathighsoundpressurelevel is then,accordingto Ingard andIsing [1967],

Rnl & ρ0v� (2.42)

andis independentof frequency for frequencieswherethe absolutevalueof the reactanceis lessthanRnl. For high soundpressurelevels, the nonlinearresistancegivenabove is thedominatingpartof thesystemresistance,andtheviscouslossesin andneartheopeningcanbeneglected.Theexperimentsof Ingard [1953] indicateda dependency of v1 2 7 insteadof v,while Melling [1973] from ananalyticalapproachderivedthenonlinearresistance

Rnl & 4ρ0

3πv� (2.43)

i. e. with a coefficient roughlyhalf of theoneusedin Eq. (2.42).The nonlinearreactanceof the openingseemsto be difficult to determine.It hasbeen

found to decreasewhenthe soundpressurelevel increases.Ingard [1953] and Ingard andIsing [1967] reportedthedecreaseto be5

�8 and1

�2 of thelinearvalue,ρ0ω l � δtot � , with

δtot givenby Eq. (2.13). Thedecreaseis causedby the formationof turbulenceat the “out-flow” apertureof the opening.The kinetic energy in the turbulentflow doesnot contributeto thereactance,andis dissipated.Accordingto Ingard [1953], thedecreasein reactanceisrelatedto theincreasein resistanceasdescribedabove.

2.3.3 Effect of interaction betweenperforations

The modelspresentedin the precedingsections,with two exceptions(the endcorrectionsin Eq. (2.15)andEq. (2.22)),assumesthat thereis only oneHelmholtzresonatoropening,or, equivalently, that the areaS of the openingsof a perforatedpanelis very much largerthancavity cross-sectionareaA associatedwith eachopening. For somegeometries,thisassumptionmaynotbevalid. In thesecases,theairflow throughtheopeningsareperturbatedby theairflow of theneighboringopenings.Melling [1973]summarizedtheeffectsasfollows.

The reactive endcorrectionis larger, by a factorof�

2, for two separateholesthanforoneholewith areaequalto thesumof thetwo separateholes.Thereactiveandresistive endcorrectionsdecreasewhenperforationsareplacedcloserto eachother. Fig.2.6showsaphys-ical pictureof thesituation.With referenceto calculationsby V. A. Fok [seee. g. Rschevkin,1963,Ch.VII] for a circularopeningin acirculartube,Melling presenteda function

ψFok ε � � �1 � a1ε � a2ε2 � �� 1

(2.44)


Figure2.6: Theeffect of interaction on the resistiveand reactiveendcorrections. a) Twoholesplacedcloseto each otherhavea commonattachedmass.Theattachedmassper holeis lessthanfor a singlehole. b) Two holesplacedcloseto each otherhaveairflow in phase.Theshearregion is reducedor lost. Figure from[Melling, 1973,Fig. 5].

Thefirst polynomialcoefficientsare:

a1� � 1 � 4092� a2

� 0 �a3� 0 � 33818� a4

� 0 �a5� 0 � 06793� a6

� � 0 � 02287�a7� 0 � 03015� a8

� � 0 � 01641

(2.45)

According to Rschevkin, ψFok can be usedto correctthe (reactive) end correctionsof anopeningin a partition in a tubewhenthe openingareaS is not very small comparedto thecavity cross-sectionareaA. Fok’s function is unity for ε � � S

�A � 0, i. e. for a smallhole

or aninfinite partition. Thefunction increasessharplywhenε increases.ThefunctionψFokis usedby Melling to correctthetotal endcorrectionfor thecaseof severalinteractingholes


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y = 2r/b

2δ /

S0.

5

Figure2.7: Comparisonof two correctionfactors for the classicendcorrectionδ0. Totalcorrectedendcorrectionas functionof y � 2r

�b, where r is the perforation radiusand b

is the distancebetweenperforations. , Correctionby Ingard, δ � δ0 1 � 1 � 25y� ; ,CorrectionbyMelling, δ � δ0

�ψFok

�y�

π�2� . CompareMelling’scurvewith Fig. 2.2.

in apanelwith perforationΦ (comparewith Eq. (2.36)):

Z � jωρ0

πr2Φ 3 l � 2δ0

ψFok ε �54 / 1 � 2x� � j

J1

�x� � j �

J0

�x� � j � 0 � 1

(2.46)

NotethatMelling usedtwo differentvaluesfor theviscosity;onefor air nearathermallycon-ductingsurface(usedinsidetheopening),andonefor a non-thermallyconductingmedium(usedfor theendcorrection).For simplicity, thelatterviscosityis usedin Eq. (2.46)for bothcases.Also notethat the total (reactive) endcorrectionlengthfrom Eq. (2.13),2δ0 & 1 � 7r,is hereusedasa correctionfor boththereactiveandtheresistivepartof theimpedance.ThedifferencebetweenthisandIngard’svalue2r in Eq. (2.31)seemsto beinsignificant.

It is quite interestingto notethat,althoughnot statedby Melling, the resultof applyingψFok ε � asa correctiondueto hole interaction,asdoneabove, is practicallyequivalenttousingIngard’s innerendcorrection,Eq. (2.15)for bothaperturesof theholesin a perforatedpanel.With ε writtenasa functionof y � 2r

�b, whereb is theperforationdistance,

ε � y

�π

2 � (2.47)

thetwo correctionscanbecompared,asis doneis Fig. 2.7. Thesimilarity of Fok’s functionandIngard’scalculationsof δi is alsoevidentin thebookby Rschevkin [1963].


Theabove summaryis valid for circularopenings.The total reactive endcorrectionforslits, δs tot in Eq. (2.22),explicitly includestheslit distanceb, andaccountsfor the interac-tion betweentheslits. As notedby SmithsandKosten[1951]: “ �� thedomainof appreciablekinetic energy is certainlynot negligible in comparisonwith thewavelength”. Theendcor-rectionδs tot decreaseswhenb decreases,asit shouldaccordingto thediscussionabove.

2.4 Developmentsin panelabsorbergeometries

It wasmentionedearlierthatporousmaterialsor resistivelayershavetraditionallybeenaddedto Helmholtzresonatorsto increasetheabsorptionbandwidth.Anotherapproachis to couplemoreresonatorsin series,so that the outerneckapertureof the secondresonatoris in thebackwall of thefirst resonatorcavity, andsoon. The“cost” of this solutionis an increasedtotal depth.In aneffort to reducethevolumeandtheneedfor extra materials,variousalter-native geometrieshave beenproposed,all of which try to exploit theinherentviscouslossesoptimally. Two categoriesarepresentedin this section.Thefirst is theuseof non-traditionalshapeandsizeof theopeningandcavity of theresonator. Thesecondis thecombination(andinteraction)of Helmholtzresonancewith theresonancesof platesor foils. Thefirst categoryincludestheconceptsto bediscussedin Chs.3 and4.

2.4.1 Non-traditional aperture geometries

Many authorshave investigatedthe effect of alternative shapesand sizesof the resonatorneckandcavities. Variationsin cavity shapemayhavesignificantinfluenceon theresonancefrequency, asdiscussedin Sec.2.1, but may alsogive high energy lossesin somespecialcases.An exampleof thelatter is the“prefractal” cavities discussedin anarticleby Sapovalet al. [1997]. On the otherhand,variationsin neckshapeandsizemay have a significanteffecton bothresonancefrequency andabsorptionbandwidth.

Helmholtz resonatorswith slottedneck plates

In light of the work to bepresentedin Ch. 4, the mostinterestingnon-traditionalgeometryis the onepresentedby Mechel [1994c]. It is a laterally slotted,distributedHelmholtzres-onator, which consistsof two plateswith slit-shapedperforations,placedin front of a hardwall. SeeFig. 2.8. The platesareseparatedby a small distance,a slot. In contrastto theconceptsdiscussedin detail in Ch.4, theperforationsin thefront plateareplaceddirectly infront of therearplateperforations.Thelateral“side branches”of theneckaddanadditionalimpedanceto the system.Becausethe slot is usually thin, the addedimpedanceis mainlycapacitive in the frequency rangeof interest. Mechelobserved that this addedcapacitanceshiftedthe Helmholtzresonanceto higherfrequency relative to an ordinaryHelmholtzres-onatorwithout the slot. He alsoidentifiedtwo extra resonances,in additionto this shiftedHelmholtzresonance,andtheλ

�4-resonanceof thebackcavity:

– Themass-spring-massresonanceof themassesin thenecksandthespringof theslot

– The spring-mass-springresonanceof the springsin the slot andbackcavity and themassof theinnerneck.

2.4Developmentsin panel absorbergeometries 19

6 66 67 77 78 89 9: :: :;;< < < < < < < < < < < << < < < < < < < < < < <= = = = = = = = = = = == = = = = = = = = = = => > > > > > > > > > > >> > > > > > > > > > > >? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ?

Slit-shaped perforations

Figure 2.8: Helmholtzresonators with slottedneck plates. The two variantsare approxi-matelyequivalentversions.Figureadaptedfrom[Mechel, 1994c,Fig. 1].

Becausetheslot hasa high impedancecomparedto the neck,it is only weaklyexcited. Toincreasethedriving-field impedance,Mechelsuggestedthata freely floating,limp, resistivefoil couldbeput into theslot. This will have two effects:Firstly, it will increasetheoverallsystemresistance.Secondly, it will increasetheimpedancein theneckto a level comparableto the slot impedance,so that the flow in the slot increases.Both effectswill increasetheenergy dissipation,and thereforethe absorptionbandwidth. The addedmassreducestheresonancefrequency. Mechel also notedthat oblique soundincidence,or an asymmetricpartitioningof the slot, result in a splitting of the first of the extra resonanceslisted above.Thesplit resonancesarecloseto eachother, sotheeffect is abroadeningof theresonance.

Micr operforated panels

Microperforatedpanels(MPPs)are increasinglyusedfor reverberationcontrol. It is alsothe basisfor the conceptdiscussedin Ch. 3. As the nameimplies, MPPsare panelsofarbitrarymaterialwith perforationsof very small dimensions.The perforationdiameteristypically lessthana millimeter, exceptfor very low frequency absorbers(LeeandSwenson[1992] used20mm thick panelswith perforationsof diameter2mm for absorptionaround50–60Hz). Perforationsof small radius,approachingtheviscousboundarylayerthickness,have velocity profilesquite different from thoseof “normal” perforations,asdescribedbyCraggsandHildebrandt[1984](seepage14). Thisreducedradiussignificantlyincreasestheviscousenergy lossesin theperforations[MorseandIngard, 1968,Ch.6.4].

Basedon Crandall’ssimplificationsof Eq. (2.36)for high andlow valuesof x,

z �A@ 8µ lr2 � j 4

3ωρ0l if x � 1� 2ρ0µω lr � jωρ0l 1 � 1

r B 2µρ0ω ! if x $ 10� (2.48)

Maa [1987]hasdevelopedanexpressionfor theimpedanceof theopeningsfor intermediatevaluesof x (Thevalueof x is 2.3 for f � 200Hz andperforationdiameter2r � 0 � 5mm). Byaddingtheclassicinductive endcorrection,Eq. (2.13)andIngard’s resistive endcorrection,Eq. (2.32),Maaobtainedanexpressionfor thespecific,relative resistanceandinductanceof


thepanel:

ζ � zzaΦ

� θ � jωχ � (2.49)

whereza� ρ0c0 is thecharacteristicimpedanceof air, Φ is thepanelperforationpercentage,

andtherelative resistanceθ andinductanceχ is givenby

θ �C+ 8µ lΦρ0c0r2 -ED � 1 � x2

32 � � 2xr4l F (2.50)

and

ωχ �C+ ω lΦc-HGI 1 � 1B 9 � 1

2x2� 16r

3π JK (2.51)

Note that Maa’s equationfor θ hadan error. Equation(2.50) is equivalent to an expres-sion given in a later paperby Fuchsand Zha [1995]. Also notethat Ingard’s resistive endcorrectionis usedin Eq. (2.50), despitethe fact that it was only experimentallyvalidatedfor relatively largeopenings[Ingard, 1953], i. e. for Helmholtzconditions, x $ 10. Melling[1973] arguedthat Ingard’s endcorrectionshouldgenerallybevalid, alsofor x � 1. This isbecausetheopeninglengthl is anindependentvariablebothin thegeneralexpressionfor theimpedance,Eq. (2.36),andin theCrandall’s approximateexpressions,Eq. (2.48). Maadidnot includeany correctionof theendcorrectionsdueto hole interaction.For typical dimen-sionsof MPPs,e.g. 2r � 0 � 5mm andb � 5mm, thevalueof ψFok is 1.14,thustheeffect ofholeinteractionneednot beincluded.

The ideal absorberhasθ � 1 (specificresistanceof openingequalscharacteristicresis-tanceof air) andθ

�ωχ L 1 (small inductanceof opening)[Maa, 1987]. The resistanceto

reactanceratio increasesfor smallerr and f . Althoughthereis a lower practicallimit on r,restrictingthesizeof this ratio,microperforatedpanelscanachievea relatively high absorp-tion bandwidth,typically two octaves. In a later paper, Maa [1998] showed that for x lessthanabout1, thebandwidthis mainly determinedby θ , andis practicallyindependentof x.SeeFig. 2.9. Largervaluesof θ giveshigh bandwidth.For largex, theoppositeis true; thebandwidthdecreaseswhenθ increases.The maximumabsorptioncoefficient, α0, is givenby therelative resistanceas

α0� 4θ 1 � θ � 2 (2.52)

The MPPshave also beencombinedin multiple layer configurations. Eachsegment,consistingof oneMPPandanair layer, isequivalenttoanelectricalcircuit segmentconsistingof a resistance,an inductance,anda parallelcapacitance[Kanget al., 1998]. Although anincreasein the total depthof the panelabsorberis inevitable, optimal configurationswillbroadenthe absorptionbandwidthsignificantlycomparedto a singlelayer. Zhangand Gu[1998] showed that the resonanceandanti-resonancefrequenciesof a two-layerMPP did


Figure 2.9: Half-absorptionbandwidthof MPP as a function of perforate constantx forseveral valuesof the relative, specificresistanceθ , as calculatedby Maa. Figure adaptedfrom[Maa, 1998,Fig. 4].

not changesignificantlywhentheresistanceof rearlayerwasassumedzero. Basedon thisassumption,ananalyticalexpressionfor the resonancefrequencieswasdeveloped,andwasshown to agreewith experimentalresults.

If the MPPsare thin, plate vibration may influencethe absorptioncharacteristics[seee.g. Ke et al., 1998;Zhou et al., 1998]. The effect of the platevibration is equivalent toaddingan impedancein parallel to the resistanceandinductanceof the perforation[Kanget al., 1998]. This impedanceconsistsof an inductance,givenby thesurfacedensityof theplate,anda resistance,given by the energy lossescausedby the vibration. The vibrationof MPPsis drivenby the pressuredifferenceover the plate. Tanakaand Takahashi[1999]showed that for large perforations,the vibrationshave little influence. On the otherhand,smallperforationsgiveshighflow resistivity in theholesandlargepressuredifferenceacrossthe plate. This significantlyinfluencesthe MPP absorption.The absorptionpeakis shiftedto lower frequency for increasedmassdensityof the plate. Zhouet al. [1998] notedthatvibrationmodeswith frequenciesbelow theHelmholtzresonanceof theabsorberhave littleinfluenceon theabsorptioncharacteristics.On theotherhand,highervibrationmodesmayor may not have their frequenciesshiftedto higher frequencies,dependingon whethertheimpedanceof the plateis comparableto the impedanceof the perforationsor if it is muchsmallerthantheperforationimpedance.

In additionto metal,themicroperforatedpanelsmayalsobemadeof othermaterials.Arecentdevelopmentis theMICROSORBERR

�concept,which is a transparent,thin (approx.

0 � 1mm), microperforatedfoil madeof plasticslikepolyethyleneor polyester. Theperforationradii canbe assmall as0 � 2mm. Multiple foils canbe combinedto increasethe absorptionbandwidth.Figures2.10and2.11showsmeasurementandsimulationsof absorberconsistingof oneandtwo layersof suchthin foils, respectively.


Figure2.10: Calculatedandmeasured(in reverberation chamber)absorptioncoefficientofmicroperforatedfoil, 0 � 1mm thick, andwith perforationsof diameter0 � 2mmandwith 2mmseparation. Thedistanceto theback wall is 100mm. Figurefrom[Fuchsetal., 1999,Bild 4].

Figure2.11: Calculatedandmeasured(in reverberation chamber)absorptioncoefficientofa two-layermicroperforatedfoil. All foil parameters are as in Fig. 2.10. Front foil placed30mm in front of rear foil. Figureadaptedfrom[Fuchset al., 1999,Bild 5].


Figure2.12:Geometryof absorberconceptdescribedby Frommholdet al. [1994] andAck-ermannet al. [1988]. Helmholtzresonators coveredby protectingplate. 1, Frame;2, cavitywalls; 3, flexible, perforatedplate;4, coverplate;5, supportplate. Figure from[Frommholdet al., 1994,Fig. 1].

2.4.2 Helmholtz resonatorscovered by platesor foils

Severalauthorshave investigatedthecharacteristicsof traditionalHelmholtzresonatorscov-eredby foils or plates.This hastwo implications: Firstly, additionalresonancesareaddedto the system,andmay interactwith the Helmholtzresonance.Secondly, the smoothfrontsurfacemayfunctionasaprotectionin harshenvironments,which is oftendesirable.

Ackermannet al. [1988] presentedan absorbersuitedfor roughenvironments,e.g. asasilencer. The absorberconsistsof several Helmholtz resonatorswith a flexible, perforatedplateastop cover, which in turn is coveredandsealedby a smooth,protecting“membrane”.Mechel pointedout later that the correct term shouldbe foil, becausethere is no tensilestressesinvolved [Mechel, 1994a]. The cover platesarealsoso thin that bendingstiffnesscanbeincludedin aneffectivemassdensityof a limp foil, for frequenciesbelow thecriticalfrequency of theplates.Figure2.12shows theabsorberconcept.Theoutercover platemaybeseparatedfrom theperforatedplatewith athin foamring. Theinnerandoutercoverplates


M M M M M M M M M M MN N N N N N N N N N NLimp foil

Slit

Figure2.13: Helmholtzresonatorcoveredby a limp foil (thefoil is showndotted).Mechel’sarticle discussesasymmetricalair gapsbetweenthefoil andtheslittedresonatorneck plate,therefore the widthsof the gapsneednot be equal. Figure adaptedfrom [Mechel, 1994a,Fig. 1].

aretypically afew tenthsof amillimeter thick,while theopeningsof theHelmholtzresonatorarein theorderof oneor two centimeters.Thenumberof differentdimensionsandmaterialparameters,resultsin a relatively complicatedabsorber.

A detailedexaminationof theabsorberwaspresentedlaterby Frommholdet al. [1994].Theabsorberconceptcannotbedescribedby simplelumpedelementsbecauseof the inter-actionbetweenthe involvedmechanisms.Frommholdet al. identifiedtwo mainabsorptionpeaks.Thefirst is dueto theordinaryHelmholtzresonanceof thecavities coveredwith theperforatedplate.This resonancefrequency is shiftedto lower frequencieswhenthemassoftheoutercoverplateis added.Thesecondpeakis relatedto themainresonanceof theperfo-ratedplate.Thecoverplateis shown to beessentialfor thequalityof theabsorber. Thecoverrisesthemaximumabsorptionof theplateresonancefrom 0.5to about0.8.Frommholdetal.proposedseveralexplanationsfor theeffect of thecover plate: Firstly, viscouslossesin thethin air layerbetweenthecover plateandtheperforatedplateareintroduced.Secondly, en-ergy maydissipatein thefoamsupport.Thirdly, viscouslossesat theedgeof theperforationsincreasebecausetheperforationscannotradiateinto openhalf-space;thevibratingair hastoenterthesmallgapbetweentheplates.Thelattereffect is frequency dependent,becausetherelativevelocityof theair andtheplatehasamaximumat resonancefrequencies.

Mechel [1994a]investigatedacomparableconceptby athoroughanalysis;aslittedpanelabsorbercoveredby a limp foil, asshown in Fig. 2.13.His conclusionis thatthefoil coveredresonatorsareinferior to theslottedneckplatesdiscussedin Sec.2.4.1,despitetheadvantageof a smoothoutersurface.Amongthedisadvantageslistedare:

– Fewerexcitableresonances

– Outercovervulnerableto mechanicalattacks

– Gapwidth difficult to control

Chapter 3

Micr operforated panelswithhorn-shapedorifices

This chapterdescribesthe investigationof a specialconceptfor panelabsorbers;Panelswith horn-shapedopenings,asshown in Fig. 3.1. The inner part of eachhorn hasdimen-sionsapproximatelyequivalentto traditionalmicroperforatedpanels(MPPs),asdescribedinSec.2.4.1.Theconcept,calledmicrohorn, wasthoughtto havethefollowing virtues:

– Theincomingwave is offeredabetterimpedancematchthanwith normalMPPs

– Thelargersurfaceareaof theopeningincreasesthesystemresistance

– Theincreasedparticlevelocityin theinnerpartof thehornalsoincreasestheresistance

The microhornconcepthasbeenmodelledby threedifferentmethods,noneof which werecompletelysatisfactory. Becauseof this, no reliableestimateof anoptimalgeometrycouldbemade.Only onegeometrywasexperimentallytested.

OPOPOPOPOPOPOPOPOPOPOPOPOPOPOOPOPOPOPOPOPOPOPOPOPOPOPOPOPOQPQPQPQPQPQPQPQPQPQPQPQPQPQPQQPQPQPQPQPQPQPQPQPQPQPQPQPQPQd

b

Figure3.1: Front view andcross-sectionof microhornpanelabsorber.

25

26 Micr operforated panelswith horn-shapedorifices

∆z

r0

rh

mm - 1

r(z)

z

l

h

Figure3.2: Thegeometryof a typical microhorn. For theouterpart of thehorn, i. e.z in therange0–h, theshapeis givenby thefunctionr z� . Theinner part of thehorn is an ordinarycylindrical tubeof lengthl . For theIntegration Method,Sec.3.1.1,theouterpart is dividedinto cylindrical shellsof length∆z. Theshellsare very short tubes,and the impedanceofeach tubeis calculatedbyEq.(2.36).Theimpedancesof thetubesareaddedto givethetotalimpedanceof theouterpart of thehorn.

3.1 Models

The impedanceof a cylindrical, circular openingor tubemay be calculatedby the theorypresentedin Sec.2.3. However, the additionof the horn complicatesthe calculation. Thefollowing sectionspresentthreedifferentapproachesto thecalculationof the impedanceofthehorn-shapedpanelopenings.Thefirst methodis a semi-analyticmodel,calledthe Inte-grationMethod(IM), whereastheothertwo arethenumericFinite ElementMethod(FEM)andFinite DifferenceMethod(FDM), respectively.

Figure3.2shows thecross-sectionof asinglemicrohorn.To makethemodellingflexiblewith regardto variationsin dimensions,thehorn is divided into two parts. The inner (rear)part is an ordinary, cylindrical tubeof length l andradiusrh. The outer (front) part hasashapedescribedby the function r z� , wherez is the distancefrom the front opening. Thefront openinghasradiusr0. The lengthof theouterpart is h. Thearbitraryfunction r z� isonly subjectto theconstraintsr 0� � r0 andr h� � rh. Themicrohornpanelabsorberconsistsof a rectangulargrid of microhorns,wherethecenterto centerdistancebetweenthehornsisb, asillustratedin Fig. 3.1.Thepanelis placeda distanced in front of ahardwall.

3.1Models 27

3.1.1 Integration Method

A simplemodelof themicrohornabsorberusesthesimplifiedKirchoff modelfor cylindricalperforations,Eq. (2.36). For the calculationof the acousticimpedanceof the outerpart ofthe horn (z � h in Fig. 3.2), this part of the horn is divided into M shortcylindrical shells,analogousto a methodusedby Pfretzschner et al. [1999]. Thus, eachshell is a circularcylindrical tubeof length∆z � h

�M. Theacousticimpedanceof eachof theseshorttubesis

thenadded,andtheresultis thetotalacousticimpedanceof theouterhorn:

Zouter� M

∑mR 1

jωρ0 ∆z

π r2m

/ 1 � 2xm� � j

J1

�xm� � j �

J0

�xm� � j � 0 � 1

(3.1)

Heretheradiusof shellm is rm� 1

2

�rm� 1 � rm� , whererm

� r m∆z� . Theperforateconstantxm is givenby Eq.(2.37)with r � rm. Theacousticimpedanceof theinnerpartof thehornis

Zinner� jωρ0l

πr2h

/ 1 � 2x� � j

J1

�x� � j �

J0

�x� � j �S0 � 1

(3.2)

Here r � rh is usedin the expressionfor x. At both endsof the microhorn,inductive andresistiveendcorrectionsareincluded.Equations(2.12)and(2.32)give

Zend outer� 1

πr20

+ jωρ08r0

3π � � 2µρ0ω - (3.3)

and

Zend inner� 1

πr2h

+ jωρ08rh

3π � � 2µρ0ω - (3.4)

Notethattheendcorrectionof Eq.(2.32)is thetotal resistiveendcorrection(bothapertures).This hasbeensplit in two partshere. The acousticimpedanceof the air layer betweenthepanelandthewall is [Allard, 1993,Eq.2.21]

Zair� � j

za

b2 cotkd � (3.5)

whereza� ρ0c0 is thecharacteristicimpedanceof air. For smallcavity depths,kd � 1, the

impedanceZair equalsthe capacitive part of Eq. (2.10),andis inverselyproportionalto thecavity volumeV � b2d. This is thereasonthatd in Fig. 3.1 is measuredfrom thebackof theplate,andnot from therearapertureof themicrohorns.Thetotal acousticimpedanceof themicrohornabsorberis thesumof theabove impedances,

Zmh� Zair � Zend inner � Zinner � Zouter � Zend outer� (3.6)

andtheabsorptioncoefficientof thepanelis [MorseandIngard, 1968,Ch.6.3]

α � 1 � )))) Zmhb2 � za

Zmhb2 � za

)))) 2 (3.7)


The modeldescribedabove introducestwo sourcesof error. Firstly, the discretizationof the curvedsurfaceinto a seriesof short,cylindrical tubesintroducesanerror. This errorcanbereducedby increasingM. Themodelrequireslittle computingpower comparedto anumericalmethod(e.g. FDM), thusM canbemadevery large. Theseconderror is relatedto the velocity field in the microhorn. For large M, the velocity field in eachshort tubemin the range2 �� M � 1 is approximatelyequivalentto the velocity field in the neighboringtubes,sothereis no significanterrorcausedby thediscontinuitiesof thehorncross-section.Also, Eq. (2.36)seemsvalid for infinitesimalshorttubes.However, thevelocity field in thehorn is not axial, asassumedin the developmentof Eq. (2.36),but hasa significantradialcomponent.Theerrorcausedby this cannotbeeliminatedby increasingM, andcanonly beignoredfor long,narrow horns.Consequently, theIntegrationMethodcanonly beconsideredasa first approximation.

3.1.2 Finite ElementMethod

TheFinite ElementMethodis hereappliedto theentirelengthof themicrohorn.Thecom-puterprogramthat is used,FEMAK, doesnot have a specialelementtypefor viscoussoundpropagationin air [Johansenetal., 1996]. Instead,anequivalentfluid typeelement,designedfor soundpropagationin porousmaterialswith rigid frame,is used.Two versionsof this el-ementexist; onewhich usesa slightly correctedversionof a modelby Craggs [1978,1986],andanotherwhichusestheJohnson-Allardmodel[Allard, 1993,Ch.5]. Theformer, simplerversion,which is usedhere,is basedon theoryby Zwikker and Kosten[1949]. Adiabaticcompressionis assumed,so thermallossesareexcluded. Thecorrectionsto Craggs’modelconcerntheeffectivedensityandthecontinuityof thevolumevelocity. Theviscouslossesintheporousmaterialareincludedin thecomplex densityof theequivalentfluid.

Using the equivalentfluid modelto describeflow througha porousmaterial,Newton’sequationandthemassconservationequationcanbewritten [Johansenet al., 1996]� ∇p �,+ jωρ0ks

φ � σ - u (3.8)

∇ � u � � jωφρ0c2

0

p � (3.9)

whereφ is porosity, σ is flow resistivity, andks is thestructurefactor. Thesearecharacteristicparametersof theporousmaterial. Thevelocity u canbeeliminatedby thecombinationoftheequations.With theobviousassumptionof φ � 1 andks

� 1 for theair in themicrohorn,theresultis theHelmholtzequation,

∇2p � k21p � 0 � (3.10)

wherethecomplex, squaredwave numberis

k21� ω2

c20� jωσ

ρ0c20

� ω2

c20

+ 1 � jσρ0ω

- (3.11)

Eliminating velocity from the linearizedNavier-Stokesequation(Eq. (2.34)) andthe massconservation equation(Eq. (3.9) with φ � 1) also resultsin the Helmholtzequation. The

3.1Models 29

squaredwavenumberfor soundpropagationin viscousfluid is thenwrittenas

k22� ω2

c20

+ 1 � jωµρ0c2

0

- � 1

(3.12)

Becausek1 andk2 arecharacteristicparametersfor thesameviscousflow, describedby theHelmholtz equation,they canbe equated.Thus, the flow resistivity can be relatedto theviscosityby

σ � µω2

c20

� 1

1 � jωµρ0c2

0& µ

ω2

c20

(3.13)

For audiblefrequencies,theabsolutevalueof theratio in thedenominatoris equalto or lessthan1 � 5 � 10� 5. Hencethe approximationintroducesno significanterrors. For f � 200Hz,the equivalentflow resistivity σ & 2 � 44 � 10� 4Pas

�m2. Comparedwith the flow resistivity

of typical porousmaterialslike rock wool, which is in the orderof 10kPas�m2, the flow

resistivity by Eq.(3.13)is verysmall.The implementationof the equivalentfluid elementin FEMAK doesnot handleimpe-

dancesor sourcesat the edgeof the element. Therefore,it is requiredto add a layer ofordinary fluid elementsat the inner andouter endof the microhorn. The elementsat theouteropeningaredefinedto seeanincomingplanewave with amplitude1Pa parallelto thehorn axis. Theseelementsalsofacea specificimpedanceequalto za. The elementsat theinner openingfacethe specificimpedanceof the air layer betweenthe panelandthe wall,with theadditionof the innerendcorrectionπr2

h � Zend inner. Thethicknessof theair layer isadjustedto compensatefor the thicknessof the extra layer of elements.FEMAK supportsaxial symmetry, soonly half of thecross-sectionin Fig. 3.2 is includedin themodel.

Theoutputof theFEMAK simulationsis thecomplex pressureateachnode.To calculatethe specific impedanceat a nodefacing the incoming wave, the axial velocity acrossthecorrespondingelementm is needed.By Eq.(3.8),with φ � 0 � ks andσ & 0, thevelocity is

um� � 1

jωρ0� ∂ p

∂z & � 1jωρ0

� pm 2 � pm 1∆zm

� (3.14)

if the microhornaxis is in the z direction. Here,∆zm is the axial lengthof elementm, andpm 1 andpm 2 arethepressuresat thefront andrearsideof theelement.Theaveragespecificimpedancezmh at thefront microhornopeningis thencalculatedastheaverageof thespecificimpedanceat thefront nodeof eachelement,

zmh� 1

N

N

∑mR 1

pm 1um

� � jωρ01N

N

∑mR 1

pm 1pm 2 � pm 1 ∆zm � (3.15)

whereN is thenumberof elementsthatseestheincomingwave. Theacousticimpedanceofthemicrohornpanelabsorberis then,with theadditionof theouterendcorrection,

Zmh� zmh

πr20� Zend outer� (3.16)


andtheabsorptioncoefficient for thepanelis calculatedby Eq.(3.7)asbefore.Unfortunately, the resultsin Sec.3.3.1 indicatethat FEMAK cannothandlethe porous

materialelementtype whenusedlike this. The predictedabsorptioncoefficient of the res-onatorsystemis predictedto beapproximatelyzero.It seemsthattheequivalentfluid modelimplementedin FEMAK is not suitedfor “porousmaterials”with suchlow flow resistivitiesasusedhere.

3.1.3 Finite Differ enceMethod

Here,theFiniteDifferenceMethod[seee.g. ZienkiewiczandMorgan, 1983,Ch.1] is appliedto thelinearNavier-Stokesequation,Eq.(2.34),andthemassconservationequation,Eq.(3.9)(with φ � 1). With cylindrical coordinates,andassumingrotationalsymmetryof the horn,thevelocity vectorcanbewritten asu r � z� � u r � z� r � v r � z� z, andthepressureasp r � z� .Theequationsto besolvedfor p, u andv, arethen[LandauandLifshitz, 1959,§15]

jωρ0u � ∂ p∂ r � µ + ∂ 2u

∂ r2 � ∂ 2u∂z2 � 1

r∂u∂ r � u

r2 - � 0

jωρ0v � ∂ p∂z � µ + ∂ 2v

∂ r2 � ∂ 2v∂z2 � 1

r∂v∂ r- � 0

jω p � ρ0c20+ ∂u

∂ r � ∂v∂z � u

r- � 0

(3.17)

Theequationsaresolvedin theouterpartof thehorn,0 � z � h, asshown in Fig. 3.3. Thespecificimpedanceat z � h is an input parameterto the FDM model. Theouterpartof thehorn is divided into M axial stepsof length∆z � h

�M. Thus,thereareM � 1 rows of grid

points,in therangem � 0 �� M. Theshapeof thehornis givenby thearbitraryfunctionr z� ,asbefore.Dueto theshapeof thehorn,thetotal numberof grid pointsincreasessharplyasthenumberof radialstepsat theinnerendof thetubeincreases.Thenumberof radialstepsatm � M, denotedL M � , is thereforelimited by theavailablecomputingresources.To ensurethatno grid point is “wasted”at thesurfaceof thehorn,wheretheparticlevelocity is knownto beexactly zero,theradialsteplength∆r is setequalto rh

� ' L M � � 0 � 05( , sothatthegridpoint M � L M �� is closeto, but not exactly at, thesurface.Thenumberof radialgrid pointsfor row m, thatis L m� � 1, is thenearesthigherintegerto r m∆z� � ∆r.

With thevelocity at thegrid point l � m� written asul m � ul mr � vl mz andthepressureaspl m, theboundaryconditionsof theFDM modelcanbeexpressedasfollows:

– Symmetryat thecentralaxis(l � 0):

u0 m � 0 v� 1 m � v1 mu � 1 m � � u1 m p� 1 m � p1 m (3.18)

Notethesignchangeof theradialvelocitycomponentu.

3.1Models 31

rh

r0

∆z

∆r

h

z

r

m=M

m=0

l=0

Figure3.3: TheFinite DifferenceMethodwasusedto solveEqs.(3.17) for pressure p andparticle velocityu in the hatchedregion. Thegrid shownis not the actual grid used. Thespecificimpedanceat m � M is an input parameterto themodel.

– Constantsoundpressurep0 andaxial flow in front of outeropening(m � 0):

pl � 1� p0 vl � 1

� vl 0ul � 1

� 0(3.19)

Equations(3.17)arelinear, thustheimpedanceat any point is independentof p0. Thevalueof p0 is thereforearbitrary, andis setto p0 � 1.

– ConstantspecificimpedancezM at theboundaryof theinnerpartof thehorn(m � M),andaxial flow in theinnerpart(m $ M):

pl M � zMvl M vl M T 1� vl M

ul M T 1� 0 pl M T 1

� pl M (3.20)

HerezM� πr2

h Zair � Zend inner � Zinner! , by Eqs.(3.5),(3.4)and(3.2).


∆z

λ

∆r

∆z

∆rη

U UV V W WXY YZ Z[ [\ \ ] ]^ ^

_ _` ` a ab b

c cd d eeeeeeeeefffffffff

β

m+1

L-1L-2m-1

m

L

n

Figure3.4: Grid pointscloseto the curvedsurfaceof the microhorn, that is pointswhereη � 1 or λ � 1, are excludedfrom normal treatmentby theFinite DifferenceMethod. Thepressure and velocity at thesespecialpoints are determinedby boundaryconditionsandneighboringpoints,asdescribedbyEqs.(3.22)and(3.23).

– Thehornsurfaceis hard,andthereis no slip of theviscousflow:

∂ p∂n

)))) surface

� 0 u * surface� 0 � (3.21)

where∂ p�∂n is thederivative of thepressurein thedirectionof a vectorn normalto

thesurface.

Becauseof thecurvedsurface,pointscloseto thesurfacehave to be treatedspecially, witha proceduresummarizedby Crandall [1956]. Figure3.4 shows thegeometry. Points“closeto thesurface”aredefinedaspoints l � m� wherel $ L m � 1� or l � L m� , i. e. grid pointswhich lacksneighboringgrid pointseitherbelow or to the right. Thevelocity andpressureat suchpointsaredeterminedby theboundaryconditionsandthevaluesof theneighboringpoints.Thepressureat l � m� is givenby

pl m �hgi j pl m� 1 1 � tanβl m! � pl � 1 m� 1 tanβl m if βl m k arctan∆r∆z

pl � 1 m 1 � cotβl m! � pl � 1 m� 1cotβl m if βl m $ arctan∆r∆z � (3.22)

whereβl m is the anglebetweenthe z-axis andthe normalvectorn from the point l � m� tothesurface.Thevelocityat l � m� is

ul m � gi j λl lm1T λl lmul m� 1 if L m � 1�m� l � L

ηm1T ηm

ul � 1 m if l � L(3.23)

3.2Measurementsand simulations 33

For m � M, thevalueof λl M is undefined,soonly thelatterexpressionis used.Whenthepressureandvelocityatall grid pointshavebeendetermined,theimpedanceat

thefront openingof themicrohornis givenby theaverageof thespecificimpedanceat gridpointsat z � 0, with theadditionof theouterendcorrections(seeEq.(3.3)):

Zmh� 1

πr20 ' L 0� � 1( L n 0o

∑l R 0

pl 0vl 0 � Zend outer (3.24)

Theabsorptioncoefficientof themicrohornpanelabsorberis givenby Eq.(3.7).The FDM model is probablythe most reliable of the threepresentedmodels,but the

usefulnessandaccuracy is limited for thetypicalmicrohorngeometriesin question.To modelthe viscousflow in the inner part of the horn correctly, the grid sizehasto be quite small.Due to the shapeof the horns, the computationtime and memoryrequirementsmakes itinconvenientto investigateawiderangeof geometries,andpracticallyimpossibleto increasethenumberof grid pointsto anoptimallevel.

The threemodelspresentedabove wereprimarily meantto give a qualitative descrip-tion of differentmicrohorngeometries,andwerenot expectedto be quantitatively correct.Therefore,theendcorrectionsZinner andZouterwerenotcorrectedby Eq.(2.15)or Eq.(2.44).

3.2 Measurementsand simulations

3.2.1 Measured dimensionsof samples

Micr ohorn paneldimensions

Dueto thelimitationsof themodelspresentedabove,only onemicrohornpanelwasproducedfor impedancemeasurements.Thepanel,sized200 p 200mm, and1mm thick, weremadeof aluminium.Hornswerepunchedb � 15mm apartin a rectangulargrid. At thebottomofeachhorn,aholewith diameterapproximately0 � 5mmwasdrilled.

The modelsrequireaccuratevaluesof the dimensionsof the perforationsin the panel,includingtheradiusvs.depthfunctionr z� . To determinethis functionfor theperforationsinthesamplepanel,gluewasusedto makemoldsfrom threeof theperforations.TheglueusedwasSuperEpoxy120. Themoldswerethenmeasuredon a Hilger & Wattsprojectorscreen,type 601.301,with 50 p magnification.The measureddimensionsof the threeperforationswerequitesimilar, thereforetheaverageof thedimensionswasused.The innerperforationdiameterwasfoundto be2rh & 0 � 55mm wide. Theouterdiameterwasmeasuredto 2r0 &3 � 9mm. Thedepthof thehornwasapproximatelyh & 2 � 3mm. Thevalueof innerperforationlengthl would seemto bezero,but theexactvaluecouldnot bedeterminedfrom themolds.Figure3.5shows theaverageof themeasuredradiusof thehornasfunctionof depth.Giventheconstraintsr 0� � r0 andr h� � rh for thegivenh, a curve fitting to themeasuredr z�wasattempted.Several functionsweretried, e.g. a quadraticdependency anda polynomialfit. Theclosestfit to themeasuredshapeof thehornwasthelogarithmicfunction

r log z� � rh � 1K

ln � zh � 1 � z

h ! e� K r0 � rh � " � (3.25)


0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

z [mm]

r(z)

[mm

]

Measured Logarithmic

Figure3.5: Radiusr z� of horn asfunctionof depth.Measureddatacomparedto the loga-rithmic functionEq.(3.25)

with thearbitraryconstantK setto 1795m� 1 by themethodof leastsquares.This functionisalsoshown in Fig. 3.5.To evaluatetheinfluenceof theshapeof thehorn,thelinearfunction

r lin z� � r0 � �rh � r0 � z

h � (3.26)

correspondingto a conicalhornshape,wasalsousedin thesimulations.To supplementthemeasurementsof the molds,someof the microhornswerealsoexaminedin microscope.Itwasfoundthattheperimeterof theinnerperforationwasjaggedandonly remotelycircular.

Themethodusedto measurethemicrohornpanelgeometryintroducedseveralpossibleerrors,all of which maydecreasetheaccuracy of thepredictionsby themodels:

– Thepropertiesof theglueduringhardeningandchangesin temperaturearenotknown.If theglueshrunkduringhardening,themeasuredvaluesof r z� andh aretoo small.

– Theinnerpartof themoldmayhavebroken,or thegluemaynothavecompletelyfilledthehorn.Theresultwill bea toohigh measuredvalueof rh anda too low valueof h.

– Theprojectionsof themoldsweremeasuredmanuallywith a ruler. Therandomerrorsthusintroducedareonly partly compensatedfor by takingtheaverageof themolds.

– Becauseof theimportanceof thevalueof rh, asmallerrorin theprojectionmagnifica-tion mayalsointroducesignificanterrors.By useof a known sample,theerror in theprojectionsystemwasfoundto belessthan0 � 02mm.


After the initial measurements,it wasfoundthata panelvibrationmodeinterferedwiththeHelmholtzresonance.To eliminatethis problem,steelbarsof dimension5 p 5mm weregluedto therearsideof thepanel.Thestiffnessthusintroducedcausedthefrequency of thevibrationmodeto shift to a higherfrequency.

Micr operforated panel dimensions

An ordinarymicroperforatedpanelwasalsomeasuredto comparewith themicrohornpanel.The MPP thickness,andperforationlength,was l � 1mm. Like the microhornpanel,theperforationswere drilled in a rectangulargrid, with b � 15mm betweenthe perforations.Thediameterof theperforationsin thispanelwasmeasuredin microscope.It wasfoundthat2r & 0 � 52mm. As for themicrohornpanel,steelbarshadto begluedto therearsideto avoidinterferencebetweenplatevibrationmodesandtheHelmholtzresonance.

3.2.2 Impedancemeasurementsin Kundt’s tube

Thespecificimpedanceat normalsoundincidenceof themicrohornpanelandtheordinaryMPPwasmeasuredin a Kundt’s tubewith squarecross-section.Thestandardizedtransfer-functionmethodwasused[ChungandBlaser, 1980;ISO10534-2, 1996;JonesandStiede,1997].Figures3.6,3.7and3.8show anoverview of themeasurementsetup,themountingofthesample,andtherelevantdimensions,respectively. Thespecificimpedanceis givenby

z � jρ0c0H12sinkt � sink t � s�cosk t � s� � H12coskt � (3.27)

whereH12 is themeasuredtransferfunctionH12 betweenthemicrophones,s is thedistancebetweenthemicrophones,andt is thedistancebetweenthefront microphoneandthesample.Theinnerdimensionof thetubeis 200 p 200mm, with thelowestcut-off frequency at850Hz[Morseand Ingard, 1968,Ch. 9.2]. The measurementsweredonewith Brüel & Kjær con-densermicrophones,type 4165. The distancet was310mm, andthe microphonedistancewass � 150mm. According to Bodenand Åbom[1986] andNordtestACOU 095 [1996],this correspondsto a recommendedfrequency interval 113–907Hz. Outsidethis frequencyrange,themaximumerror in H12 increasessharply. Thesampleswerefastenedwith screwsto a squaresampleholder. The innerwidth andheightof thesampleholderis 190mm, andthe thickness(i. e. lengthalongthe tubeaxis) is 10mm. The outeredgehasa rubberbandplacedin a groove, to decreaseleakagebetweenthesampleholderandthe tubewall. Fourvaluesof the backcavity depthd wereused;44, 79, 149and219mm. Only the resultsford � 44mm arepresentedandcomparedwith simulationsin Sec.3.3below.

The transferfunction H12 wasmeasuredby the computerprogramWinMLS developedby MorsetSoundDevelopment[1999]. This programimplementstheMaximumLengthSe-quence(MLS) method[seee.g. Chu, 1990].ThePChadaHohnerARC 44 soundcardwithfour channels,threeof which wereusedhere. For all measurementsandcalibrations,thesamplingratewas11025Hz, thesequenceorder14,andthenumberof averageswas16.


Figure3.6: TheKundt’s tube, with microphoneamplifiers to the left. Two of thetubewalls,andalsothemovable“piston” which constitutetheback wall, aremadeof glass.Thisallowsfor visualcontrol of themountedsample, andalsovibration measurementsby laserinterfer-ometry, as describedin Ch. 4. In the backgroundcan be seenthe anechoic endpieceusedduring thecalibration.

Figure3.7:Themicrohornpanelandsampleholder, with tubetopcoverremoved.Thesampleholderwasheldin placebypinsnearthecorners. Thereflectingendpiececanbeput in anyposition,only limited by thepinsandthelengthof thetube.


w/ samplesample holder

ts d

mic. 1 end piecemic. 2

MLS-signal

WinMLS

Figure3.8: Thesetupusedfor normal incidenceimpedancemeasurementsof themicrohornpaneland the MPP. Thetransferfunctionbetweenthe microphoneswascalculatedby thecomputerprogramWinMLS [MorsetSoundDevelopment, 1999]. During all measurements,t � 310mmands � 150mm. Four valuesof d were used;44,79,149and219mm.

FEMAKI-DEAS I-DEAS to FEMAK MATLAB

FEMAK to I-DEAS

Figure3.9: Flowchart for theFinite ElementMethodsimulationsprocess.Thegeometryisdefinedin theCAD-programI-DEAS, andis translatedinto a formappropriatefor FEMAKbya conversionprogram.Theoutputis processedbyMATLAB , but mayalsobereturnedtoI-DEAS for graphicalrepresentation.

3.2.3 Simulations

Thesimulationsby theFinite ElementMethodweredoneon two differentgeometries.Thefirst was a microhornwith dimensionslike the measuredmicrohornpanel. The FEMAKinputfile wasgeneratedby theCAD-programI-DEAS[Lawry, 1998;SDRCSolutions, 2000],with an appropriateconversionprogram[Verdeille, 1998a,b],asindicatedin Fig. 3.9. Theelementswere approximately0 � 05 p 0 � 05mm. The simulationswere doneat 1

�3 octave

spacedpoints in the frequency range31.5–1600Hz. The othergeometrywasan ordinaryMPP, with perforationdiameter0 � 5mm, thickness1mm, andwith 4mm spacingbetweenperforations.Thedistanceto thehardwall was200mm. Theperforationswereassumedtobefilled with a porousmaterial,andtheflow resistivity of this materialwasvaried.For thisgeometry, the FEMAK input file waswritten manually. The elementswere0 � 125mm widein theradialdirection.In theaxial direction,thelengthwas0 � 25mm for theporousmaterial


Table3.1: Valuesof geometryparameters usedin Finite DifferenceandIntegration Methodsimulationsof microhorn,first series.Theback cavitydepthd was44mm.

Parameter Values used

mm

h 1 2

l 0 0.2 0.5

r0 1 2 5

rh 0.25 0.50

Table3.2: Valuesof geometryparametersusedin Finite Differencesimulationsof microhorn,secondseries.Outerhorn radiusr0 q 1 r 95mm and lengthh q 2 r 3mm for all simulations.Only hornshapefunctionr log wasused.Theback cavitydepthd was44mm.

Parameter Values used

mm

l 0 0.1 0.2 0.3

rh 0.250 0.260 0.265 0.275

elements,and1mmfor theordinaryair elementsateachperforationaperture.Thesimulationfrequency rangewas224–800Hz, with 1 s 6 octavespacedpoints.

Two seriesof Finite DifferenceMethod simulationswere carriedout. The first serieswasintendedto illustratethe influenceof largevariationsin themicrohorngeometryon theabsorptioncharacteristics.Thedimensionparametersandthevaluesusedin thesimulationsare given by Table 3.1. Both shapefunctions, r log given by Eq. (3.25) and r lin given byEq. (3.26), wereusedin thesesimulations,except for r0 q 5mm. This large outer radiusin combinationwith the conichornshape(r lin) couldnot be simulateddueto theextensivecomputermemoryrequirements.Simulationswith h q 5mm wereoriginally planned,butcouldnot be completed.This wasalsodueto the largememoryrequirements.The secondseriesof FDM simulationswasintendedto simulatethe measuredmicrohornpanelsampleandinvestigatethesensitivity to smallvariationsin thehorndimensions.Theparametersofthesecondseriesaregivenin Table3.2. For all theFDM simulations,thenumberof z-stepswereM q 70,andthenumberof r-stepsat z q h wereLM q 9. Thesevalueswerechosentogive a relatively squaregrid for dimensionscloseto themeasuredsample.With h q 2 r 3mmandrh q 0 r 275mm, thegrid lengthsare∆r t ∆z t 0 r 03mm. At f q 200Hz this is equivalentto 5 radialgrid pointsinsidetheviscousboundarylayerat z q h. Thesimulationsweredoneat 1s 3 octavespacedpointsin thefrequency range31.5–800Hz.

Thesimulationsby theIntegrationMethodwereassumedto givequick,approximativelycorrectpredictionsof theabsorptioncoefficient of variousmicrohorngeometries.This was

3.3Results 39

200 500 8000

0.5

1

Figure3.10:Finite Elementsimulationsof micro-perforationfilled with porousmaterial(seepage 37). Absorptioncoefficientasfunctionof frequencyfor differentflow resistivities. ,σ q 1kPass m2; , σ q 5kPass m2; , σ q 10kPass m2; , σ q 20kPass m2.

not the case(seeSec.3.3.1),so simulationsby the IM wereonly donefor the geometriesin Table3.1, with the samefrequency rangeasusedin the FDM simulations,andwith 1 s 6octavespacingbetweenthepoints.

The absorptioncoefficient of the microperforatedpanelsamplewas calculatedby thecomputerprogramFLAG, developedat NTNU [Vigran et al., 1991]. This programimple-mentsthe transfermatrix method[seee.g. Dunn and Davern, 1986;Brouard et al., 1995]for a numberof materials,includingMPPs.TheMPPimplementationin FLAG is basedonEq.(2.46).Unlike Melling [1973], theMPPimplementationin FLAG usesonly thestandardvaluefor theviscosity. StinsonandShaw[1985]statedthattheuseof an“effective” viscosity,which includestheeffect of thermallosses,is only appropriatefor long tubes.Additionally,the polynomialcoefficientsusedin FLAG differs from thosegiven in Eq. (2.45). The geo-metry parametersusedin the FLAG simulationof the MPP werethe sameasthe physicaldimensionsgiven in Sec.3.2.1. The FLAG programwasalsousedto simulateMPPswithresonancefrequenciesapproximatelyequalto thatof themeasuredmicrohornpanelsample.For this,thedimensionsof theMPPswere(a)r q 0 r 250mm, A q 120mm2, (b) r q 0 r 275mm,A q 140mm2 and(c) r q 0 r 325mm, A q 190mm2. For all FLAG simulations,d was44mmand l was1mm. Thesimulationsweredoneat 1 s 30 octave spacedpointsin the frequencyrange35–1094Hz.

3.3 Results

3.3.1 Comparison of simulation methods

Thepredictedabsorptioncoefficientof theFEM simulationof themicrohornwaspracticallyzerofor all frequencies.To verify that this resultwasnot anerror in thegeometrydefinitionor conversationprocess,a simplergeometrywasdefined. Figure3.10 shows the resultofdecreasingthe flow resistivity of a porousmaterialin a micro-perforation.The procedure


rh q 0 r 25mm, h q 2mm

50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1

Figure 3.11: Simulationsof microhorn geometries. Absorptioncoefficient as function offrequencyascalculatedby the Integration Methodand theFinite DifferenceMethod. Innerhorn lengthl q 0mm, outerhorn radiusr0 q 2mm for all graphs. , IntegrationMethod;

, Finite DifferenceMethod.

in Sec.3.1.2obviously fails to predicttheabsorptioncharacteristicsof a simpleMPP, whentheflow resistivity is givenby Eq. (3.13). It seemsthat,to simulateperforatescorrectly, theprogramFEMAK requiressomeresistive materialto supplytherequiredresistance,asweredoneby KristiansenandVigran [1994].

As mentionedin Sec.3.1.1,theIntegrationMethodcanonly beviewedasanapproximatemodel,if only dueto theinferior modellingof thevelocityfield in thehorn.This is confirmedin Fig. 3.11,which comparessimulationsby theIM andtheFinite DifferenceMethod.Onlythegeometrieswith l , r0 andr u zv approximatelyequalto theexperimentallytestedsample,i. e. l q 0mm, r0 q 2mmandr u zv q r log, arepresentedin this figure. In contrastto theFDMresults,andin contrastto whatcouldbeexpected,theIM resultsseemsrelatively insensitiveto variation of h and rh. The samemismatchbetweenthe modelsis also evident for the

3.3Results 41

rh q 0 r 25mm, r0 q 2mm

50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1

Figure3.12: Finite Differencesimulationsof microhorn geometries.Absorptioncoefficientasfunctionof frequencyfor hornshapesr log andr lin . Innerhorn lengthl q 0mm, outerhornlengthh q 1mm for all graphs. , r u zv q r lin u zv givenby Eq. (3.26); , r u zv q r log u zvgivenbyEq.(3.25).

geometriesnot shown, wherel q 0 r 2 and0 r 5mmandr0 q 1 and5mm. Consequently, in thefollowing sectionsonly theFDM resultsarepresented.

3.3.2 Variation of horn geometry

Horn shape

Theeffectof theshapeof thehornis illustratedin Figs.3.12and3.13,wherethehornshapesr log and r lin arecompared. The two figurescorrespondto h q 1 and2mm, respectively.For simplicity, l is 0mm for all graphs. Simulationsof geometrieswith r0 q 5mm werealsoattempted,but could not be completedbecauseFDM simulationswith horn shaper linrequiresevenmorememorythanr log. As couldbe expected,the resultsshow that thehorn



50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1

Figure3.13: Finite Differencesimulationsof microhorn geometries.Absorptioncoefficientasfunctionof frequencyfor hornshapesr log andr lin . Innerhorn lengthl q 0mm, outerhornlengthh q 2mm for all graphs. , r u zv q r lin u zv givenby Eq. (3.26); , r u zv q r log u zvgivenbyEq.(3.25).

shapeis of greaterimportancefor the longerhorns(Fig. 3.13),andevenmoreso for hornswith large r0 or small rh. It is lessintuitive that the resonancefrequenciesgenerallyarehigher for conic hornsthanfor logarithmichorns. This is despitethe fact that the volume(andhencethe nominalmass)of the conic horn is significantly larger than the volumeofthe logarithmichorn,giventhesamevaluesof r0, rh andh. Thereasonfor thedifferenceinresonancefrequency is probablythata logarithmichornhasa longer narrow portionof thehorn lengththana conic horn. As discussedby CraggsandHildebrandt(seepage14), theeffectivedensityof air in narrow tubescanbeupto ρe q 1 r 33ρ0 for smallvaluesof x. Hence,in somecases,the effective massof the logarithmichorn may be larger than the effectivemassof theconichorn.

3.3Results 43


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1

Figure3.14:Finite Differencesimulationsof microhorngeometries.Absorptioncoefficientasfunctionof frequencyfor differentouterradii r0. Inner horn lengthl q 0mm for all graphs.

, r0 q 1mm; , r0 q 2mm; , r0 q 5mm.

Horn dimension

Thesensitivity to variationin thehorndimensionsr0, rh, h andl , is visualizedby Figs.3.14,3.15and3.16. Thesefigurescorrespondto l q 0,0.2and0 r 5mmrespectively. By comparingthe correspondinggraphsin the threefigures, it can be seenthat the inner horn length linfluencesthe absorptioncharacteristicsas expected: an increasein l will in most casesincreasetheresonanceabsorptioncoefficientα0 dueto theincreasedresistance,andshift theresonancefrequency f0 towardslower frequenciesdueto theincreased(effective)mass.Thisseemsto bethemostsignificanteffectof anincreasein l . Exceptfor rh, theeffectof theothergeometryparametersarerelatively independentof l .

An increasein the outerhorn radiusr0 will generallydecreasethe Q-valueof the res-onance(aswasthe intendedpurposeof the horn-shapedorifices). While this is the domi-nanteffect for rh q 0 r 25mm, thedecreasein Q is alsoaccompaniedby a decreasein α0 for



50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1

Figure3.15: Finite Differencesimulationsof microhorn geometries.Absorptioncoefficientas functionof frequencyfor different outer radii r0. Inner horn length l q 0 r 2mm for allgraphs. , r0 q 1mm; , r0 q 2mm; , r0 q 5mm.

rh q 0 r 50mm, andespeciallyfor shortouterhorns,h q 1mm. It seemsthatalthoughwiderouteropeningsyield a largerflow ratethroughthe horn, this cannotbe fully utilized to in-creasethe viscouslossesif rh is large, andespeciallyif h is small. The reducedf0 that isexpectedfor increasedr0, dueto the increasedmassin thehorn, is alsomostnoticeableforshorthornswith largeinneropenings.

Theeffect of variationin theouterhorn lengthh is dependenton thevalueof r0 andrh.For thesmallestvalueof r0, i. e. 1mm, theouterpartof thehornmaybeconsideredasmainlyanextensionof thecylindrical innerpart. As wasthecasewith variationin l , anincreaseintheouterhornlengthh thereforeresultsin adecreasein f0 in thiscase.Ontheotherhand,forthe large r0 q 5mm, thereis alsoa dependency on thevalueof rh. For smallinneropenings,themaineffectof anincreasein h is a reductionin f0, asbefore.However, for rh q 0 r 50mmtheenergy dissipationin theinnerpartof thehornis muchless,andtheincreasedresistance

3.3Results 45


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1


50 100 5000

0.5

1

Figure3.16: Finite Differencesimulationsof microhorn geometries.Absorptioncoefficientas functionof frequencyfor different outer radii r0. Inner horn length l is 0 r 5mm for allgraphs. , r0 q 1mm; , r0 q 2mm; , r0 q 5mm.

causedby an increasein h will becomemoresignificant.Hence,α0 increases.In this case,theresonancefrequency is lesssensitive to changesin h.

Themostcritical parameter, rh, alsohasthemostcomplex dependency on theotherpa-rameters.The only generalobservation is that the resonancefrequency increaseswith rh.This is probablybecausethereductionin theeffectivemassdueto theincreasein theperfo-rateconstantx is largerthantheincreasein thenominalmassdueto theincreasedvolumeofthehorn. For larger0 q 5mm, themaximumabsorptioncoefficientdecreasesfor increasingrh, asexpected.For smallr0 q 1mm, theeffectof rh dependsonh andl . For largeh q 2mm,or large l q 0 r 5mm in combinationwith theshorth q 1mm, an increasein rh will increaseα0. On theotherhand,for theshortesttotal hornlength,h q 1mm, l q 0mm, themaximumabsorptiondecreaseswhenrh increases.It mayseemthatfor thelong tubes,theresistanceishigherthantheoptimum,sothatanincreasein rh will reducetheresistanceto amoreoptimal


l q 0mm

50 100 5000

0.5

1l q 0 r 1mm

50 100 5000

0.5

1

l q 0 r 2mm

50 100 5000

0.5

1l q 0 r 3mm

50 100 5000

0.5

1

Figure 3.17: Finite Differencesimulationscompared to measurementof microhorn panel.Absorptioncoefficientasfunctionof frequencyfor smallvariationsin inner radiusrh. Outerhornradiusr0 q 1 r 95mmandlengthh q 2 r 3mmfor all graphs.Thecavitydepthd q 44mm.

, rh q 0 r 250mm; , rh q 0 r 260mm; , rh q 0 r 265mm; , rh q 0 r 275mm.

level. For the shortertubes,the increasein rh resultsin lessoptimal resistance,andhencereducedα0.

3.3.3 Comparison with measurements

Thesecondseriesof FDM simulationswasintendedto illustratetheeffectof smallvariationsof thegeometryparameters,closeto thegeometryof thesamplepanel.Theresultsfor differ-entvaluesof l andrh areshown in Fig. 3.17.With regardto limitation setby themicrophoneseparation,themeasureddatawascut at 50Hz. Thespikesat f q 500Hz is dueto theplateresonance.Thefigureshows theexpectedshifting towardslower frequency for increasingl .More interesting,however, is theeffectof variationsin rh. An increasein rh doesnot mono-tonically reducethemaximumabsorptioncoefficient, but resultsin analternationin α0 and

3.4Summary 47

bandwidth.Thealternationpatternis thesamefor all valuesof l . It seemsthattheFDM im-plementationusedherecannotbetrustedto give quantitatively correctpredictionsfor smallvariationsof thecritical parameterrh. This is mostprobablydueto therelatively low numberof radialgrid points. As describedin Sec.3.2.3,for f q 200Hz thereareonly 5 radialgridpoints insidethe viscousboundarylayer at z q h. The placementof grid points insidetheviscousboundarylayer, wherethevelocityprofile is steep,will besignificantlyaffectedby asmallvariationof rh.

For all valuesof l exceptzero, the predictedresonancefrequenciesare lower than themeasuredresonancefrequency. The simulationsalso predict absorptionpeakswhich aregenerallytoo high andtoo broadcomparedto the measurement.Thereareseveralpossibleexplanationsof thediscrepancy:

– The dimensionsof the microhornpanelsamplemay have beeninaccuratelydeter-mined,asmentionedin Sec.3.2.1.Additionally, theshapeof theinnerhornopeningswereobservedto benot quitecircular. Thus,the real,averageinner radiusmayhavebeenlargerthanthevalueof rh thatwasused.

– The numberof grid points may be too small, so that the energy dissipationin theviscousboundarylayeris not modelledcorrectly.

– Theendcorrectionsof theouterhornopeningmaybe incorrect. Due to theshapeofthehorn,it is difficult to estimatehow muchof theopeningsareashouldbeassociatedwith theopening,andhow muchshouldbeconsideredaspartof thepanelsurface.Atoo high r0 will result in a too high associatedmass,leadingto a too low resonancefrequency.

3.3.4 Comparison with microperforated panels

Theleft chartin Fig. 3.18shows themeasuredabsorptioncoefficient of themicroperforatedpanelsamplecomparedwith FLAG simulationsof the samegeometry. The agreementisquitegood.Also, giventheconstraintsd q 44mm, r q 0 r 26mmand f0 t 180Hz, thisseemsto beanalmostoptimalconfiguration.Severalothercombinationsof A andt weresimulatedby FLAG, but noneresultedin significantlyhigherabsorptionat the samefrequency. Theright chart in Fig. 3.18 comparesthe absorptioncoefficient of threesimulatedMPPswiththemeasuredmicrohornpanelsample.Heretoo, thedimensionsof thesimulatedMPPsarelimited by d q 44mmandl q 1mm, andarechosensothat f0 t 250Hz.

3.4 Summary

The microhorn concepthas beeninvestigatedby analytical, numericaland experimentalmethods.The microhornconceptwasthoughtto increasethe absorptionbandwidth,com-paredto the ordinaryMPPs. However, the microhornsturnedout to be difficult to modelwith theaccuracy requiredto designreasonableoptimalpanelsfor experimentaltesting.Theexperimentson thesamplethatwasproduceddid neitherconfirmnor rejectthefeasibility ofthemicrohornconcept.


50 100 5000

0.5

1

i) MPP

50 100 5000

0.5

1

ii) Microhorn

Figure3.18: Comparisonof simulatedand measured absorptioncoefficient as functionoffrequencyfor the panel absorbersamples. i) FLAG simulationand measurementof mi-croperforatedpanelsample. For thesimulation,r q 0 r 26mm, l q 1mm, A q 245mm2 andd q 44mm. ii) FLAG simulationsof microperforatedpanelsand measurementof micro-horn panel sample. For the simulations,l q 1mm and d q 44mm. , r q 0 r 250mm,A q 120mm2; , r q 0 r 275mm, A q 140mm2; , r q 0 r 325mm, A q 190mm2.

Threemodelsweretried to simulatethemicrohorns.Of these,only theFiniteDifferenceMethodcould, to somedegree,be trustedto give qualitatively correctpredictions. Due tolimitationsin thenumberof grid points,theFDM resultscannotbetrustedto bequantitativelycorrect. Accordingto the FDM simulations,a high maximumabsorptioncoefficient andahigh absorptionbandwidthcanbe obtainedfor geometrieswherethe outer horn radiusislargeandthe innerradiusis small. Thelengthsof theouterandinnerpartsof thehorn,andtheshapeof thehorn,arealsosignificant,but subordinate,factors.Consideringthepossiblesourcesof errorin theFDM simulations,thesequalitativestatementsshouldneverthelessbecorrect.

Chapter 4

Doublepanelabsorbers

Thesubjectof studyin this chapteris a new typeof panelabsorber. It is a distributedHelm-holtz resonatorwith doubleplates,wherethe platesareperforatedwith holesor slits. Thekey ideais to elongatetheresonatorneckslaterally, andmake thewidth of theselateralslit-shapedneckssosmall thatviscouslossesbecomesignificant.Unlike theconceptdiscussedby Mechel [1994c], the narrow, lateral slits arenot “side branches”to the main resonatorneck,but ratheran integral part of the neck. Like the microperforatedpanels[Maa, 1987,1998], this new absorberconceptutilizes the viscouslossesinherentin the resonatoropen-ings. Hence,it doesnot requireextra resistive materialsto achieve a relatively broadbandedabsorptioncomparedto thesimplepanelHelmholtzresonators.

The conceptis implementedby mountingtwo parallel,smooth,perforatedmetalplatesclosetogether, separatedby a small distance.The perforationsform a regulargrid, andtheperforationsin the front plateareat a maximumdistancefrom thosein the rearplate. Theresultis thattheresonatornecksconsistof threesegments;front opening,rearopeningandthelateralslit formedby thegapbetweentheplates.Figure4.1showsasampleof adoublepanelabsorberwith circular perforations,as mountedfor impedancemeasurementin a Kundt’stube.Slit-shapedopeningsarealsoinvestigated.In this case,thelengthof thelateralslit canbevariedby lateraldisplacementof oneplaterelative to theotherplate.Theeffectsof suchdisplacementsarealsopresented.

4.1 Cir cular perforations

This sectiondescribestheinitial investigationson thedoublepanelabsorberconcept,wheretheperforationswerecircular, shorttubes.

4.1.1 Model

Thedoublepanelabsorberwith circularperforationsis heremodelledby a simpleanalyticalmodel. If normalsoundincidenceis assumed,symmetryallows animaginarytubeof cross-sectionareaA q b2 to be associatedwith eachperforation[Allard, 1993,Ch. 10]. Hereb

49

50 Doublepanel absorbers

Figure4.1: Theperforateddoublepanelabsorber, asmountedfor impedancemeasurementsin a Kundt’s tube. Theplatesare separatedbya numberof thin metalrings at theperimeter.Theopeningsin thetwo platesare placedsymmetricallywith respectto each other to max-imizethe lengthof the lateral slit formedby the gapbetweenthe plates. Thecenter-centerdistancebetweentheperforationsis thesamefor bothplates.

is the center-centerdistancebetweenthe perforations.The modelledgeometryis shown inFig.4.2,wherethecross-sectionof thetubeis indicatedwith asquare.Asshown in Sec.2.3.3,the influenceof the perforationsoutsidethe imaginarytube “walls” can be neglectedforperforationcross-sectionareasSwhicharesignificantlysmallerthanA. For the“worstcase”sampleusedin the measurements,the ratio Ss A is 0.01 (seeTable 4.1). Fok’s function,by Eq. (2.44), is then1.17. Thereforethe effect of interactionbetweenthe perforationsisexcludedfrom themodel.

As shown in Fig. 4.2,themodelincludesoneholefrom eachplate.Thefour cornerholesin therearplatecontribute1s 4 of aholeeach.Thegapwidth betweentheplatesis g, andtheplatesareti thick; i q 1 for front plateandi q 2 for rearplate.Thecalculationof impedanceis trivial for all partsof thesystem,excepttheimpedanceof thegapbetweentheplates;TheimpedanceZair of theair layerbetweentherearplateandthewall is givenby Eq. (3.5). Thespecificimpedanceof theperforationin platei is givenby Eq.(2.36).With theadditionof theclassicinductiveendcorrection(Eq.(2.12))andhalf theresistiveendcorrection(Eq.(2.32)),theimpedancecanbewritten

Zp w i q jωρ0l iπr2

i xy{z 1 | 2xi } | j

J1 ~ xi } | j �J0 ~ xi } | j �� 1 �

8r i

3π l i| j

dv

l i �� (4.1)

wherer i is theradiusof theperforationsin platei. Thelengthof theperforations,l i , equalsthe plate thicknessti . The perforateconstantxi is given by Eq. (2.37) with r q r i . Notethat Eq. (4.1) doesnot includethe endcorrectionsfor the perforationaperturesfacingthegapbetweentheplates.Theseendcorrectionsarenot needed,becausethecalculationof theimpedancebetweentheplatesincludestheimpedanceassociatedwith theseapertures.

To calculatethe impedanceof thegapbetweentheplates,thevelocity field betweentheplatesis required. For g � r i , the field is approximatelyequivalent to the field set up by

4.1Cir cular perforations 51

2r2

2r1

��

� � � � � � � � � � � � � � � ��

� ��

bd

g

i) Front ii) Side

Figure4.2: Geometryof perforateddoublepanelmodel. i) Only the cross-sectionA q b2

insidethesquare is includedin themodel. A unit volumeflow is assumedto flow from onehole in the front plate (filled circle) to 4 � 1s 4 q onehole in the rear plate. ii) Thedoublepanel is assumedto be placeda distanced in front of a hard back wall. The distancegbetweentheplatesis smallenoughto let viscouslossesbecomesignificant.

cylindrical radiatingsourcesat thepositionsof theperforations.Thesourceshaveradii equalto theperforationsradii r i . Volumeflow of unit sizeis assumedto flow parallelto theplatesfrom the sourceassociatedwith the front plateandinto the rearplatesource(i. e. sink). A32 � 32grid of pointsis definedacrosstheimaginarytubecross-section.Eachgrid pointhasanassociatedarea∆S q u bs 32v 2. For eachgrid point, thevelocityfieldsassociatedwith eachsourceis superposed.Thegrid pointsinsidetheperforationsareof courseexcludedfrom thecalculation.Whenthevelocityfield is calculated,theinductanceof thegapis

Lg q 2U2KE q 2

U2 z 12

ρ0

32

∑l � 1

32

∑m� 1 �� ul wm �� 2 ∆Sg� � (4.2)

whereKE is the kinetic energy of the air in the gap. Unit volumeflow, U q 1, is assumed.Thereactanceof thegapis calculatedby

Cg q 2p2PE q Ag

ρ0c20 � (4.3)

wherePEis thepotentialenergy of theair in thegap.Theviscousresistanceof theair in the


Lg21

Rg21

Rg21

Lg21

Cg Zair

Zp,2Zp,1U

p

Figure4.3: Equivalentcircuit for the perforateddoublepanelmodel. The impedanceZp w iof a perforation in plate i includesthe resistanceand inductanceof the perforation, andalsotheendcorrectionsof theperforationaperturesnot facingthegap.Theinductanceandresistancein the gap is divided in two. Thetwo parts are associatedwith the perforationapertureswhich facesthegap.

gapis calculatedby (seeEq.(2.29))

Rg q 2PU2 q ρ0ωdv∆S

U2

32

∑l � 1

32

∑m� 1 �� ul wm �� 2 � (4.4)

whereP is the energy perseconddissipatedby viscouslosses.Equations(4.2) to (4.4) arebasedonEqs.(9.1.10),(9.1.11)and(6.4.39)in thebookby MorseandIngard [1968]. Equa-tions(4.2)and(4.3)arevalid only whenthewavelengthis muchgreaterthanthedimensionoftheimaginarytube,b. This is satisfiedfor thefrequenciesandgeometriesusedhere.Theve-locity ul wm in Eq.(4.4)is,accordingto Morse& Ingard,thevelocity justoutsidetheboundarylayer. For geometrieswheretheplateseparationg is in theorderof theboundarylayerthick-ness,Rg mustbeconsideredto bea crudeapproximationonly. Expressionsfor the thermalresistances,assumingconstantpressuresover the surfaces,were includedin a preliminarymodel. As statedby Ingard [1953], thethermalresistanceswerefoundto benegligible. Tosimplify thecalculations,theseresistancesareexcludedfrom thecurrentmodel.

Figure4.3shows theequivalentcircuit for thedoublepanelabsorber. With Cg in parallelto theimpedanceof therearperforationandtheair layerbehindtheplates,thetotal acousticimpedanceof theimaginarytubeis calculatedasfollows:

Zrear q 12 ~ Rg

�jωLg � � Zp w 1 � Zair (4.5)

Zgap q ZrearZC

Zrear

�ZC

, with ZC q 1jωCg

(4.6)

Zdp q Zgap

�12 ~ Rg

�jωLg � � Zp w 2 (4.7)

Finally, theabsorptioncoefficient is calculatedby

α q 1 | �� ZdpA | za

ZdpA

�za ��

2

(4.8)


Table4.1: Perforation radii andgrid typesfor theplatesusedfor measurements.Two typesof grid were used;“X”, where the centerof the plate wasperforated,and “O”, where thecenterof the plate wasnot perforated. Thedistancebetweenperforationswasb q 17mm,andtheplateswere ti q 2mm thick.

Plate Perforation radius [mm] Grid type

A 1 O

B 0 r 75 X

C 0 r 75 O

D 0 r 5 X

E 0 r 5 O

F 0 r 25 X

4.1.2 Measurementsand simulations

Dimensionsof samples

For the experiments,six differentcircular aluminiumplateswereused. By combiningtwoand two plates,differentconfigurationsof doublepanelabsorberswereobtained. All theplateswereti q 2mm thick andhada diameterof 103mm. Theplateshadperforationswithcircularcross-section.Two typesof regulargrids,with b q 17mm betweentheholes,wereused.Table4.1shows thedimensionsof theplatesused.To keeptheplatesseparatedduringmeasurements,severalthin ringswerecutoutof coppersheets0.11and0 r 30mmthick. Theseringshadinnerdiameter97mm andouterdiameter103mm. It wasassumedthat theseringsdid not significantlyobstructtheflow of air throughthedoublepanelabsorber.

Impedancemeasurementsin Kundt’ s tube

The impedanceof the panelsweremeasuredin a standardKundt’s tube,shown in Fig. 4.4.Thediameterof the tubeis 100mm andthesampleholderdiameteris 103mm. Two plateswith differentgrid typesweremountedin thetubesothattheperforationsin thefront andrearplatewereat maximumdistancefrom eachother. SeeFigs.4.1 and4.2. The standardizedtransfer-functionmethod[ChungandBlaser, 1980;ISO10534-2, 1996]wasusedto measurethe impedanceof theabsorber. Thesetupwasbasicallythesameasin Fig. 3.8,exceptthatthe loudspeaker wasnot drivenby a MLS-signal,but a white noisesignal. The impedancewascalculatedby Eq. (3.27) after the transferfunction H12 betweenthe microphoneshadbeenmeasured.The microphoneswereBrüel & Kjær condensermicrophones,type 4165.The transferfunction wasmeasuredwith a ONO SOKKI dual channelFFT-analyzer, typeCF-940. Thedistancebetweenthe microphoneswass q 80mm. Thedistancebetweenthefront microphoneandthesamplewast q 180mm. Thelowestcut-off frequency of thetubeis approximately2kHz [Morseand Ingard, 1968,Ch. 9.2]. For the microphoneseparationusedhere,thearticleby BodenandÅbom[1986] suggeststhat thebestaccuracy is obtainedin thefrequency range212–1700Hz.


Figure4.4:ThecircularKundt’stubeusedfor theimpedancemeasurements.Notethatthefig-urehasbeenrotatedcounterclockwise. Thesamplesandthereflectingendpiecearemountedfrombelowandareheldin placeby the“clamp” at thebottom.There is no visualcontrol ofthemountedsamples.

Threeseriesof experimentsweredone. Tables4.2 and4.3 show the geometriesof theconfigurationsin the first andthird measurementseries. After measurement119,a small,possibleleakagein the reflectingendpiecewassealed.The secondseriesof experiments(measurements016–018) wasdonewith singleplatesonly. This wasdoneto observe theabsorptionof the perforations,without the effect of the lateralslit betweenthe plates.Thetree platesusedwere platesF, D and A in Table 4.1, with perforationdiameters0.5, 1.0and2.0, respectively. The measuringfrequency rangeswere0–500Hz for measurements001–018,and50–1050Hz for measurements101–123.

Simulations

The paneldimensionslisted in Tables4.2 and4.3 werealsousedin the simulationsby themodelpresentedin Sec.4.1.1.Thesimulationsweredoneat 1s 6 octave spacedpointsin thefrequency range39.7–2016Hz. Becausethe predictionsof the modelagreedpoorly withthe measurements(seeSec.4.1.3), simulationswere not carriedout for other geometries.The absorptioncoefficient of the single panelabsorberswere simulatedby the computerprogramFLAG, describedonpage39. FLAG hastwo modelsfor perforatedpanels;theMPPmodeldescribedearlier, anda simplermodelwhich is basedon the massof the air in theperforations,the classicmassendcorrection,and Ingard’s expressionfor the resistanceinandaroundthe perforations(Eqs.(2.6), (2.12) and(2.31), respectively). The latter modelfailedcompletelyin predictingtheresistanceof thesmallperforationsin plateF. Therefore,theFLAG MPPmodelusedin Sec.3.2.3is alsousedhere.


Table4.2: Dimensionsof themeasuredandsimulatedperforateddoublepanels,first series.SeeFig. 4.2 for definitionof symbols.For all geometries,d q 95r 5mm.

Measurement r1 r2 gno. mm mm mm

001 (Calibration)

002 1 0 r 75 0 r 3003 1 0 r 25 0 r 3004 0 r 25 1 0 r 3005 0 r 75 0 r 5 0 r 11006 0 r 75 0 r 5 0 r 3007 0 r 75 0 r 5 0 r 6008 1 0 r 5 0 r 6009 0 r 25 0 r 75 0 r 6010 0 r 75 0 r 75 0 r 11011 0 r 75 0 r 75 0 r 22012 0 r 75 0 r 75 0 r 3013 0 r 75 0 r 75 0 r 6014 0 r 75 0 r 75 0 r 9015 0 r 75 0 r 75 1 r 2

Table4.3: Dimensionsof themeasuredandsimulatedperforateddoublepanels,third series.SeeFig. 4.2 for definitionof symbols.For theseconfigurations,d q 29r 1mm.

Measurement r1 r2 gno. mm mm mm

101 (Calibration)

119 0 r 75 0 r 75 0 r 6120 0 r 75 0 r 75 0 r 6121 0 r 75 0 r 75 1 r 2122 1 0 r 75 1 r 2123 1 0 r 5 1 r 2


003

50 100 500 10000

0.5

1004

50 100 500 10000

0.5

1009

50 100 500 10000

0.5

1

Figure4.5: Measured and simulatedabsorptioncoefficient of perforateddoublepanelsasfunctionof frequency. Perforation radiusr q 0 r 25mm for oneplate. SeeTable4.2 for geo-metrydefinitions.

005

50 100 500 10000

0.5

1010

50 100 500 10000

0.5

1011

50 100 500 10000

0.5

1

006

50 100 500 10000

0.5

1012

50 100 500 10000

0.5

1002

50 100 500 10000

0.5

1

Figure4.6: Measured and simulatedabsorptioncoefficient of perforateddoublepanelsasfunctionof frequency. Perforation radius r i � 0 r 5mm for both plates,and gap width g �0 r 3mm. SeeTable4.2for geometrydefinitions.

4.1.3 Results

Theexperimentalwork indicatedthat theplateseparationinfluencedtheabsorptioncharac-teristicsquitestrongly, aswasexpected.Figures4.5to 4.8comparetheabsorptioncoefficientpredictedby theanalyticalmodelto themeasuredabsorptioncoefficient. For comparison,Fig. 4.9 shows the measuredabsorptioncoefficient of perforatedsinglepanels.The resultshave two implications:

– The modeldescribedin Sec.4.1.1,althoughapproximatelyadequatefor someof thegeometries,is clearlynotableto giveaccuratepredictionsof theabsorptioncoefficient.

– Thegeometriesof Tables4.2and4.3areclearlynot suitedasbroadbandabsorbers.


007

50 100 500 10000

0.5

1008

50 100 500 10000

0.5

1013

50 100 500 10000

0.5

1

014

50 100 500 10000

0.5

1015

50 100 500 10000

0.5

1

Figure4.7: Measured and simulatedabsorptioncoefficient of perforateddoublepanelsasfunctionof frequency. Perforation radius r i � 0 r 5mm for both plates,and gap width g �0 r 6mm. SeeTable4.2 for geometrydefinitions.

119

50 100 500 10000

0.5

1120

50 100 500 10000

0.5

1

123

50 100 500 10000

0.5

1121

50 100 500 10000

0.5

1122

50 100 500 10000

0.5

1

Figure4.8: Measured and simulatedabsorptioncoefficient of perforateddoublepanelsasfunctionof frequency. Perforation radius r i � 0 r 5mm for both plates,and gap width g �0 r 3mm. SeeTable4.3 for geometrydefinitions.


016

50 100 500 10000

0.5

1017

50 100 500 10000

0.5

1018

50 100 500 10000

0.5

1

Figure 4.9: Measured and simulatedabsorptioncoefficient of perforatedsinglepanelsasfunctionof frequency. Seepage54 for geometrydefinitions.

Themodelhandlesthegeometries003,004and009verywell (seeFig. 4.5). In thesege-ometries,theholesin thefront or rearplatearequitesmall,r q 0 r 25mm, andthegapwidth isnot very small,g � 0 r 3mm. Theagreementbetweenthemodelandthemeasurementsis dueto thefact that in thesecasesthe impedanceof thesmallperforation,Eq. (4.1), is of greaterimportancethanthe impedanceof the gap(seegeometry016 in Fig. 4.9). For the geome-trieswith largerperforations,r � 0 r 5mm, theagreementbetweenmodelandmeasurementsis clearlydependenton thevalueof g. For largevaluesof g (geometries115and121–123),the agreementis quite good,althougha little shiftedin frequency. For smallg (geometries005,010and011),themodelfails completely. Thecritical partof themodelis Eq.(4.4),theresistanceof thegap. As mentionedin Sec.4.1.1,this equationis not valid, andclearlyun-derestimatetheresistance,whenthedistancebetweentheplatesis comparableto (or smallerthan)theviscousboundarylayerthickness.

The geometriestestedareclearly not optimal. The main part of the energy dissipationis supposedto take placein thegapbetweentheplates.Therefore,analogousto theMPPs,thegapwidth betweenplatesshouldbein theorderof theviscousboundarylayerthickness.However, for the geometrieswherethis is the case,it seemsthat the distancebetweenper-forationsb is too large. In thesecases,the relative resistanceis significantlygreaterthanunity. For the geometrieswith larger g, the resistanceof the gapis low, andthe total rel-ative resistanceis lessthanunity. To achieve a reasonablebroadbandabsorption,optimumcombinationsof b andg, andto a lesserextent,r i , mustbefound.

The accuracy of themeasurements,especially002–018,waspoor for low frequencies.This is probablydueto thesmallmicrophoneseparation.Becauseof this, themeasurementdatawerecutat56Hz in all thefiguresabove.

4.2 Slit-shapedperforations with constantseparation

Theinitial investigationof theperforateddoublepanelabsorberwasunsatisfactory. A bettermodelwasrequiredto predict the impedanceof the gapbetweenthe plates,andmoreex-perimentswererequiredto validatethemodel.To simplify thegeometryof theslit betweentheplates,theperforationswerenow assumedslit-shaped.With this changein geometry, theflow in the narrow gapbetweenthe platesbecomesalmostunidirectional. The impedanceof flow in a thin, infinitely long gapis givenby Eq. (2.38). However, theeffect of thesharp

4.2Slit-shapedperforations with constantseparation 59

w1

w2

t1

t2

� � � � � � � � � � � � � � � � � � � � � � � � � ��

��

��

g

d q

b

Figure4.10: Thegeometryof the slitted doublepanelresonatorconcept(sideview, not atscale).Thecenter-centerdistancebetweenslits,b, is thesamefor thetwo plates.Theoffsetbetweenthecenters of theslits in thetwo platesis q. Thegapwidth g is in theorder of oneor twoviscousboundarylayers to allow viscouseffectsto becomesignificant

edgesof the perforationsis not easily incorporatedin a theoreticalmodel. Therefore,theFinite DifferenceMethodwasusedto simulatetheimpedanceof thegapbetweentheplates.Themodelpresentedbelow is a revisedversionof aninitial versionwhich did not allow lat-eraldisplacementof oneplaterelative to theother. Theinitial modelandsomeof theinitialmeasurementshave beendescribedearlier[Randeberg et al., 1999;Randeberg, 2000]. Be-sidesthepossibilityof lateraldisplacementandsomeminor changes,the initial andrevisedversionsarepracticallyequivalent.Therefore,only thelatteris presentedbelow.

4.2.1 Model

The geometryof the slitted doublepanelabsorberis shown in Fig. 4.10. As in Sec.4.1.1,the modelconsidersan imaginarytube. The cross-sectionareaof the tube,andhencethecross-sectionof thecavity volumeassociatedwith asetof slits, is A q b � 1. TheimpedanceZair of thecavity behindthedoublepanelis givenby Eq.(3.5),with b2 substitutedby A:

Zair q | jza

Acotkd q | j

za

bcotkd (4.9)

Thedistancebetweenthefront andrearslits is q. In thissection,wherethedistancebetweenthe slits is constant,q q b s 2. In general,q can take valuesin the range0–b, but due tosymmetry, only therange0–bs 2 is relevant.

Thespecificimpedanceof aslit in platei q 1 or2 isgivenbyEq.(2.38),thetotal inductiveendcorrectionof theslit is givenby Eq.(2.22),andthetotal resistiveendcorrectionis givenby Eq. (2.33). As explainedbelow, the full length l i q ti is not usedin the first of theseequations,but rathera length l �i � l i . Usingonly the inductive andresistive endcorrections


S1+Mm=S1m=

B2l=X+

B1l=L-

S1+Mm= S2+ =T

l=X

m=0

l=Ll=0

b

Rear slit

Front slit Front slit

z , m

x , l

Figure4.11:Thegeometryof theFDM modelfor a slitteddoublepanelresonator. Thefigureis at scalefor a panelwith typical dimensions.TheFDM is usedto solveEq. (4.11) in theshadedregion. Thesymbolsareexplainedin thetext. Notethecyclicsymmetry;Pointsto theright of l q L areequivalentto pointsto theright of l q 0.

of theaperturewhich doesnot facethegap,theimpedanceof aslit in platei canbewritten

Zsw i q jωρ0l �iwi �� ¡ 1 | tanh ~ xs } j �

xs } j ¢ � 1 | wi

π l �i log £ sin ¤ πwi

2b ¥{¦ | jdv

l �i § ¨© � (4.10)

wherexs is givenby Eq.(2.39)with w q wi , anddv is givenby Eq.(2.25).As mentionedabove, the impedanceof the gapbetweenthe platesis calculatedby the

FDM. AssumingCartesiancoordinates,the linear Navier-Stokesequation,Eq. (2.34), andthemassconservationequation,Eq. (3.9)with φ q 1, arewrittenas

jωρ0u

�∂ p∂x

| µ ª ∂ 2u∂x2

�∂ 2u∂z2 « q 0

jωρ0v

�∂ p∂z

| µ ª ∂ 2v∂x2

�∂ 2v∂z2 « q 0

jω p

�ρ0c

20 ª ∂u

∂x

�∂v∂z « q 0 � (4.11)

Theseequationsaresolved for pressurep u x � zv andvelocity u u x � zv q u u x � zv x � v u x � zv z intheregion shown shadedin Fig. 4.11. Thegapbetweentheplatesis dividedinto M verticalstepsof length∆z q g s M. The imaginarytubeis divided into L horizontalstepsof length∆x q bs L. L is determinedsothereareat leastL1 grid pointsin themostnarrow slit, andatleastL2 grid pointsin the tube. It is desirableto includethe edgeeffectsof the slits in theFDM simulations.Therefore,thesimulatedregionextendsa length∆zSi into theslit in plate


i. ∆zSi is setequalto g or ti s 2, whichever is smaller:

Si q¬ M if g � ti2

Mti2g if g � ti

2

(4.12)

Consequently, thereducedlengthin Eq. (4.10)is givenby

l �i q l i | ∆zSi (4.13)

Thevaluesof M, L1 andL2 aregivenasinputparametersto thesimulation.Thevaluesof WiandX in Fig. 4.11aredeterminedby therealslit widthswi , sothatwi areequalto or largerthanthewidth of thesimulatedslits,∆xWi . X is alsochosensothatX

�W1 s 2 � W2 s 2 is the

closestintegerto qs ∆x. In thosecaseswhereasimulatedslit is wider thanthecorrespondingrealslit, i. e. if grid pointsat l q L | W1, X or X

�W2 arenot exactly at therealslit surfaces,

thosepointsaretreatedspecially, asdescribedbelow.Theboundaryconditionsusedin thesimulationsare

– Repeatingsymmetry, i. e. pointsat l q 0 arethesameasl q L:

u0 wm q uL wm u� 1 wm q vL � 1 wmu1 wm q uL ® 1 wm (4.14)

The sameequationsare valid with v or p insteadof u. Any point at l q L

�n is

equivalentto l q n.

– Constantsoundpressurep0 andverticalflow in thefront slit (m � 0):

pl w � 1 q p0 vl w � 1 q vl w 0ul w � 1 q 0

(4.15)

Equations(4.11)arelinear, thustheimpedanceat any point is independentof p0. Thevalueof p0 is thereforearbitrary, andis setto p0 q 1Pa.

– ConstantspecificimpedancezT at the boundaryin the rearslit (m q T), andverticalflow in therearslit (m ¯ T):

pl w T q zTvl w T vl w T ® 1 q vl w Tul w T ® 1 q 0 pl w T ® 1 q pl w T (4.16)

Here,zT q w2 ¤ Zair

�Zsw 2 ¥ .

– All surfacesarehard,andthereis no slip of theviscousflow:

∂ p∂n �� surface

q 0 u ° surface q 0 � (4.17)


where∂ ps ∂n is thederivative of thepressurein thedirectionof a vectorn normaltoany surface. Oneexceptionwasmadeto the “no slip” condition,asthe velocity wasallowedto benon-zeroat thecornersof theslits.

The symmetrymakesit easyto handlesituationswherel q L (i. e. l q 0) is inside the rearslit: In suchcases,pointsoutsidethe interval l q 1–L arewrappedinside. Analogoustothemethodusedin Sec.3.1.3,pointsmorecloseto thesurfacethana normalhorizontalgridlength∆x aretreatedspecially. Thenumberηm is definedasthe ratio of the point–surfacedistanceto thegrid length,for grid line m. Thepressureat thespecialpoints,e. g. at l q X, iscalculatedby [Crandall, 1956]

pl wm q pl ® 1 wm � (4.18)

andtheparticlevelocity is calculatedby

uX wm q ηm

1

�ηm

uX ® 1 wm (4.19)

Analogousequationsareusedfor theothersurfaces,l q L | W1 andl q X

�W2.

The outputof the FDM is the pressureandvelocity distribution in the gapbetweentheplatesandinnerpartsof the slits. Theaverageacousticimpedanceat the entryof the frontslit is givenby thespecificimpedanceat m q 0 andtheacousticimpedanceof thefront slit,includingendcorrections(Eq. (4.10)):

Zdp q 1w1 u B1

�1v L

∑l � L � B1

pl w 0vl w 0 � Zsw 1 (4.20)

Theabsorptioncoefficientof theslittedpanelabsorberis givenby Eq.(4.8),with A q b � 1.As with theFDM modelusedto simulatethemicrohornsin Sec.3.1.3,theaccuracy of the

modelpresentedhereis limited by the availablecomputingresources.As the gapbetweenthe platesis the most importantparameterin determiningthe characteristicsof the doublepanelabsorber, the numberM of vertical stepscannotbe too small. However, the slits aretypically tentimeswider thang, thusit is not feasibleto requirethat∆x t ∆z. Thenumberofx-stepsin theinnerpartsof theslitsmaythereforenotbeoptimal.Additionally, theboundaryconditionsat the boundarybetweenthe FDM model and the analyticalexpressionsin theslits may inducesomeerrors. A constant,averagespecificimpedanceis usedin both slits.Theerrorsintroducedby theseassumptionsareprobablymuchlessthantheerrorthatwouldbe causedby not including the innerpartsof the slits (andhencethevelocity field nearthecorners)in theFDM model.



For the experiments,12 differentaluminiumplateswereused. The plateswerecombinedin pairs to form a numberof doublepanelconfigurations.The diameterof the plateswas103mm. The plateshad regularly spaced,slit-shapedperforations. Table 4.4 shows the


Table4.4: Slit width w, slit separation b, plate thicknesst and slit configuration for platesusedfor measurements.Two kindsof slit configurationswere used;“X”, where thediagonalof theplatewasslitted,and“O”, where thediagonalof theplatewasnot slitted.

Plate w t b Slit configuration

mm mm mm

A 1 1 15 O

B 1 r 5 1 15 O

C 1 r 5 2 15 O

D 1 r 5 2 15 X

E 3 2 15 O

F 3 2 15 X

G 2 3 15 X

H 3 3 15 X

I 1 1 20 X

J 1 r 5 2 20 X

K 1 r 5 2 20 O

L 2 3 20 O

dimensionsof theplatesusedfor measurements.Thesametypeof thin, annularcopperringsasdescribedin Sec.4.1.2wereusedto keeptheplatesseparatedduringmeasurements.Thedimensionsof theringswerethesameasbefore,i. e. 0.11and0 r 3mmthick.


The impedanceof the panelswere measuredin the samestandardKundt’s tube that wasusedto measurethe perforateddoublepanels. SeeFig. 4.4. Two plateswith differentslitconfigurations(centeredandnon-centered)weremountedin the tubeso that theslits of thefront andrearplateswereparallelto eachother. Thus,the valueof q in Fig. 4.10wasb s 2for thesemeasurements.Oneor moreof theannularringswereplacedbetweentheplatestokeepthemseparatedat a givendistanceg. Themeasurementmethodandsetupwasalmostidentical to what wasusedin Sec.4.1.2. However, the distancebetweenthe microphoneswasnow s q 200mm, andthedistancebetweenthefront microphoneandthesamplewast q300mm. For thismicrophoneseparation,thebestaccuracy is obtainedin thefrequency range85–680Hz [BodenandÅbom, 1986].Thedistanced to thehardbackwall was95r 5mm.

Two seriesof experimentswere conducted,the frequency rangeof which were 50–550Hz and100–600Hz, respectively. To testthesensitivity to plateconcavity, all thesecondseriesmeasurementswererepeatedwith thefront plateturnedbackto front. Threegapwidthsg weretested;0.11,0.22and0 r 52mm, usinganappropriatecombinationof thecopperrings.Tables4.5and4.6summarizesthedimensionsof themeasureddoublepanels.


Table4.5: Dimensionsof the measured slitteddoublepanels,first series.SeeFig. 4.10fordefinitionof symbols.For all geometries,b q 15mmandq q 7 r 5mm.

Measurement g t1 t2 w1 w2no. mm mm mm mm mm

000 (Calibration)

001 0.52 1.0 2.0 1.0 1.5

002 0.22 1.0 2.0 1.0 1.5

003 0.11 1.0 2.0 1.0 1.5

004 0.11 1.0 2.0 1.5 1.5

005 0.11 2.0 3.0 3.0 3.0

006 0.22 2.0 3.0 3.0 3.0

007 0.11 2.0 3.0 3.0 2.0

008 0.22 2.0 3.0 3.0 2.0

009 0.11 1.0 3.0 1.5 3.0

010 0.22 1.0 3.0 1.5 3.0

011 0.11 1.0 3.0 1.0 2.0

012 0.22 1.0 3.0 1.0 2.0

Table4.6: Dimensionsof themeasuredslitteddoublepanels,secondseries.SeeFig. 4.10fordefinitionof symbols.Measurements101–104hadb q 15mm andq q 7 r 5mm. Measure-ments105–107hadb q 20mmandq q 10mm.

Measurement g t1 t2 w1 w2no. mm mm mm mm mm

100 (Calibration)

101 0.11 2.0 3.0 3.0 3.0

102 0.22 2.0 3.0 3.0 3.0

103 0.11 2.0 3.0 1.5 3.0

104 0.22 2.0 3.0 1.5 3.0

105 0.22 1.0 3.0 1.0 2.0

106 0.11 2.0 2.0 1.5 1.5

107 0.22 3.0 2.0 2.0 1.5


Table4.7: Simulationparameters usedin the FDM modelof the slitteddoublepanels.Seepage60 for definitionof symbols.

M 10 Number of z-steps in gap between plates

L1 10 Minimum number of x-steps in the most narrow slit

L2 50 Minimum number of x-steps in the gap between the plates

Simulations

Becausethegapwidth is a very critical parameter, severalvaluesof g wereusedin thesim-ulationswhile theotherparametersfrom Tables4.5 and4.6 werekeptconstant.Thevaluesthat wereusedarenot tabulatedhere,but aregiven togetherwith the resultsin Sec.4.2.3.Thesimulationparametersusedareshown in Table4.7. Thesevalueswerefound to give agoodcompromisebetweensimulationtime andaccuracy. Thesimulationsweredoneat 1 s 3octavespacedpointsin thefrequency range39.7–2016Hz.

To comparethedoublepanelresonatorwith theMPPs,theabsorptionof two MPPswerecalculatedusingthe FLAG program(seepage39). For the FLAG calculations,the perfo-ration separationswerechosento give approximatelythe sameresonancefrequency astheresonancefrequenciesof thedoublepanelwith which theMPPsarecompared.MPPA hada0.65%perforation,andMPPB hada 0.39%perforation.Thepanelthicknesswas1mm, andtheperforationradiuswas0 r 5mm. Thesamebackcavity thicknessasusedduringmeasure-ments,d q 95r 5mm, wasassumed.Thesimulationsweredoneat 1 s 30octavespacedpointsin thefrequency range35–1094Hz.

4.2.3 Results

Both theexperimentsandthesimulationsshown in Figs.4.12to 4.14show that theabsorp-tion characteristicsarevery dependenton the gapwidth g betweenthe plates. The mostimportantobservationthatcanbedrawn from thesefiguresis thatanoptimumvalueof g ex-ists,for which thepanelabsorberhasahighabsorptionandarelatively largebandwidth.Thebandwidthof the resonanceincreaseswheng decreases,while theabsorptionat resonance,α0, hasa maximumvaluefor somevalueof g. The situationis analogousto the casewithMPPs,discussedby Maa [1987,1998] (seeSec.2.4.1). If the gapg betweenthe platesisconsideredanalogousto theperforationdiameterr, thefollowing observationscanbemade.For asmallvalueof g, therelativeresistanceθ is high,becauseθ increasesproportionallyto1s g4 (seeEq.(2.50)).Thus,adecreasein g increasesthebandwidth,asillustratedby Fig.2.9.On theotherhand,themaximumabsorptioncoefficientof Eq. (2.52),with θ t Y s g4, is

α0 t 4Y¤ g2

�Yg2 ¥ 2 � (4.21)

whereY is a constant.Thus,α0 hasa maximumfor g t 4} Y. Thesimulationresultsconfirmtheseeffects.Themaximumresonanceabsorptionseemsto occuratagapwidthalittle higher


001

50 100 500 10000

0.5

1

Figure4.12: Measurementsandsimulationsof slitteddoublepanels.Absorptioncoefficientas functionof frequencyfor different gap widthsg. SeeTable 4.5 for geometrydefinition.Measured gap width wasg q 0 r 52mm. Simulatedgap widths: � , g q 0 r 52mm; , g q0 r 58mm. Thesharppeaksat about500Hz are dueto plateresonances.

005

50 100 500 10000

0.5

1106

50 100 500 10000

0.5

1

Figure4.13: Measurementsandsimulationsof slitteddoublepanels.Absorptioncoefficientas function of frequencyfor different gap widths g. SeeTables4.5 and 4.6 for geometrydefinitions.Measuredgapwidthswereg q 0 r 11mm. Simulatedgapwidths:

�, g q 0 r 11mm;±

, g q 0 r 14mm.

than0 r 22mm. Also notethat someof the doublepanelmeasurementsshow sharppeaksatabout500Hz. Thesepeaksaredueto plateresonancesin the relatively thin plates(1mm)usedin someof thesedoublepanels.As this resonanceis narrow, andalsonot closeto theHelmholtzresonance,it shouldnot invalidatethemeasurements.

The simulationsby theFDM modeldescribedin Sec.4.2.1correspondreasonablewellwith themeasurements.Thediscrepancy in themaximumabsorptionis in somecasesmainlycausedby inaccuracy in thevalueof g. Theplatesarenever completelyplane,andmaybeslightly concaveor convex. As showedby thesimulations,a smalldeviation in g cansignifi-cantlyalter thevalueof α0. Thesensitivity to variationsin g, andthustheinfluenceof plate


002

50 100 500 10000

0.5

1006

50 100 500 10000

0.5

1

105

50 100 500 10000

0.5

1107

50 100 500 10000

0.5

1

Figure4.14: Measurementsandsimulationsof slitteddoublepanels.Absorptioncoefficientasfunctionof frequencyfor differentgapwidthsg. SeeTables4.5and4.6 for geometrydefi-nitions.Measuredgapwidthswereg q 0 r 22mm. Simulatedgapwidths:

�, g q 0 r 22mm;

±,

g q 0 r 24mm; ² , g q 0 r 26mm. Thesharppeaksat about500Hzaredueto plateresonances.

concavity appearto belessfor thelargeplateseparationg q 0 r 52mm. Thegapwidth alsoin-fluencestheresonancefrequency. Figure4.15showsall measurements,exceptmeasurement001whereg q 0 r 52mm. Theresults,especiallythosefor b q 15mm, show thatthemeasuredresonancefrequenciesdivideinto two groups.Thisgroupingis mainlydeterminedby thegapwidth. Within eachgroupof curves,the ratio of maximumandminimumair volumein theslitsandin thegapupto 3:1. Thisvariationin air volume(i. e. resonatormass)influencestheresonancefrequency relatively little. Ontheotherhand,for eachpairof configurationswhereonly g differs, the variationin total air volumeis about10%, while the differencein reso-nancefrequency is large.Consequently, theeffectiveresonatormassis mainlydeterminedbytheeffectivedensityof theair in thegap,andonly slightly dependentontheair volumein thegapandtheslits. Thismayalsobeconfirmedby approximatingtheeffectivemassof theslitsandthegapby theimaginarypartof Eq. (2.38).Table4.8shows theresultsfor sometypicaldimensions.Theresultshave two implications:Firstly, thetotal effectivemassof theslits ismuchlessthantheeffectivemassof thegapbetweentheplates.Therefore,theinfluenceon


Measurementswith b q 15mm

50 100 500 10000

0.5

1Measurementswith b q 20mm

50 100 500 10000

0.5

1

Figure4.15: Measurementsof slitted doublepanels. Absorptioncoefficient as functionoffrequencyfor different gap widths g. All measurementsin Tables4.5 and 4.6 are shown,exceptfor measurement001. , g q 0 r 22mm; , g q 0 r 11mm.

Table4.8: Effectivemassfor typical slit and gap widths. Calculatedby imaginary part ofEq. (2.38),with f q 200Hz andb q 15mm.

Length Width Effective mass

mm mm g/m2

l q t q 1 1 r 0 0 r 3163 r 0 0 r 097

l q b s 2 q 7 r 5 0 r 14 17r 80 r 26 9 r 6

theresonancefrequency is small. Secondly, thesquareroot of theratio of effective massofthegapis 1.36. This is closeto theratio of typical resonancefrequenciesof thetwo groupsof curvesin Fig. 4.15,i. e. 310and230Hz. Thepredictedresonancefrequenciesin Figs.4.12to 4.14aregenerallyabit highcomparedwith measurements.Likethediscrepancy in α0, thismay in somecasesbecausedby non-planarplates,leadingto a deviation in thevalueof g.However, becausethepredictedf0 is generallytoo high, it seemsthattheeffectiveresonatormassof themodelis a little too low for mostof theconfigurations.Onereasonfor this maybe that theFDM grid sizeis too large,so that theeffective massin thegap,especiallyneartheedgesof theslits, is not calculatedcorrectly.

To investigatethe effect of plateconcavity, the secondseriesof measurementswasre-peatedwith thefront plateturnedbackto front. Theeffect of this is shown in Fig. 4.16. Asshown, the turningof the front platemay leadto an increaseor a decreasein α0, or it mayhave no effect at all. The effect dependson the shapeof the plates,and the sensitivity islargestfor smallvaluesof g.

Figure4.17illustratestheeffect of theslit distanceb, whenall otherparameters(except


101(FrontplateE)

50 100 500 10000

0.5

1

104(FrontplateC)

50 100 500 10000

0.5

1

106(FrontplateJ)

50 100 500 10000

0.5

1

107(FrontplateL)

50 100 500 10000

0.5

1

Figure4.16: Measurementsof slitted doublepanels. Absorptioncoefficient as functionoffrequencywith andwithout reversion of front plate. SeeTable 4.6 for geometrydefinitions.

, front platenormal; , front platereversed.

q which equalsb s 2) arekeptconstant.Dueto thehighercavity volumeperslit for higherb,theshift towardslower frequenciesis expected.Thereductionof α0 for b q 20mmindicatesthat the resistancein this caseis too large. Thus,thereexists an optimumvalueof b for agivenplateseparationg.

Figure 4.18 comparesthe absorptionof someof the doublepanelmeasurementswithFLAG simulationsof sometypical MPPs. SeeSec.4.2.2for dimensionsof theMPPs. Thehigh correspondencebetweenthe simulationof MPPA andthemeasurementof configura-tion 008is a coincidence,but shows that thetwo conceptsmayhave comparableabsorptioncharacteristics.Note that the dimensionsof the measureddoublepanelspresentedin thissectionhavenot beenoptimized.


50 100 500 10000

0.5

1

Figure4.17: Measurementsof slitted doublepanels. Absorptioncoefficient as functionoffrequencyfor different valuesof b. SeeTables4.5 and 4.6 for geometrydefinitions. ,measurement012; , measurement105.

Measurementswith g q 0 r 22mm

50 100 500 10000

0.5

1

Measurementswith g q 0 r 11mm

50 100 500 10000

0.5

1

Figure 4.18: Measurementsof slitted doublepanelscompared with simulationsof MPPs.Absorptioncoefficient as function of frequency. SeeTable 4.5 and page 65 for geometrydefinitions. Left graph: , measurement008; , measurement002;

�, MPP A. Right

graph: , measurement009; , measurement003;

�, MPPB.

4.3 Slit-shapedperforations with adjustableseparation

Theresultsof theprecedingsectionindicatedthat thedoublepanelabsorbercouldhave ab-sorptioncharacteristicscomparableto or betterthan thoseof MPPs. Furtherinvestigationwasnecessary. Theavailability of a Kundt’s tubewith a squarecross-sectionmadeit possi-ble to experimentallyinvestigatetheeffect of relative lateraldisplacementof theslits in thetwo plates.This sectionpresentsmeasurementsandsimulationsof this effect. A completeinvestigationof theeffectof all eightgeometryparametersis alsopresented.

4.3Slit-shapedperforations with adjustableseparation 71

4.3.1 Model

Themodelusedin thissectionwaspresentedin Sec.4.2.1.Thevalueof theslit separationqis in therange0–bs 2. Thelimits of this interval correspondto whatwill becalledthe“open”and“closed”statesof thedoublepanelabsorber.



Four slitted doublepanelswereproducedfor measurements.Thedimensionsof the panelsarelistedin Table4.9.Thepanelsweremadeof steel.To allow theplatesto moverelativetoeachother, while keepinga constantdistancebetweentheplates,thin stripsof copperweregluedto the front plate. The rearplatewasfastenedwith screws to the front platethroughshort slits in the rearplate. Fig. 4.19 shows panelsampleC in the “open” position. Oneof the objectivesof the measurementson thesesampleswasto investigatethe influenceofpanelvibrationson the absorptioncharacteristics.Thus,unlike the microhornsamples,nosupportingbarswereused.


The specificimpedanceat normal soundincidenceof the adjustable,slitted doublepanelsampleswasmeasuredusingthesamesquareKundt’s tubeandequipmentthatwasusedformeasurementson microhorns. SeeSec.3.2.2. The transferfunction methodwas usedasbefore. In addition,to measurethe influenceof panelvibration, a PolytecOFV-2200laservibrometerwasused[Polytec, 1999] to measurethe vibration velocity at several positionson theplates.Thevibrometeroutputsa voltageproportionalto thevelocity amplitudein thedirectionof the laserbeam. During the measurementsdescribedhere,the laserbeamwasapproximatelynormalto thepanelsurface.Therefore,theoutputvoltagewasassumedto beapproximatelyproportionalto thenormalvelocity. Figures4.20and4.21show anoverviewof the measurementsetup,anda sketchshowing the relevant dimensions.The distancestands werethesameasin Sec.3.2.2,that is t q 310mm, ands q 150mm. Thesamplewas

Table4.9: Dimensionsof the measured adjustable, slitteddoublepanels.SeeFig. 4.10fordefinitionof symbols.All plateswere 0 r 7mm thick.

Panel g b w1 w2 qmm mm mm mm mm

A 0.25 14 3.0 3.0 0, 3, 5, 6 and 7

B 0.45 42 7.0 7.0 0, 4, 7, 10, 13, 16, 19 and 21

C 0.10 10 3.0 3.0 0, 2, 2.5, 3, 3.5, 4 and 5

D 0.12 10 2.0 2.0 0, 1, 1.5, 2, 2.5, 3, 4 and 5


Figure4.19:Sampleof adjustableslittedpanelabsorber, mountedto thesampleholder. Thepanelis in the“open” position,andis seenfromtherear side. Threeof thefivethin copperstripswhich keeptheplatesseparatedcanbeseen.Thesestripsweregluedto thefrontplate.

fastenedto thesampleholderwith screws, asshown in Fig. 4.19. Threevaluesof the backcavity depthd wereused;300,150and75mm.

For a giventransferfunctionbetweenthemicrophones,H12, thespecificimpedancewascalculatedby Eq. (3.27). ThetransferfunctionH12 wasmeasuredby thecomputerprogramWinMLS, describedin Sec.3.2.2.All four of thesoundcardchannelswereusedhere;onefortheoutputMLS-signal,two for themicrophonesignals,andonefor thevelocity signalfromthelaser. For all measurementsandcalibrations,themeasurementparameterswerethesameasin Sec.3.2.2:Thesamplingratewas11025Hz, thesequenceorder14,andthenumberofaverageswas16.

To relatethemeasuredplatevibrationquantitatively to thedipsandpeaksin theabsorp-tion coefficientasfunctionof frequency, themeasuredvelocitywasnormalizedwith a factorproportionalto theincidentsoundintensity. As givenby Vigran [1985],theincidentintensityis a functionof H12:

Ii ∝ ³ exp u jksv#| H122 � (4.22)

wherek q ω s c is the wavenumberands is the microphonedistance.Thenthe normalizedvelocity is givenby

vsf q Hlaser³ exp u jksv#| H122 � (4.23)


Figure4.20:TheKundt’stube, with microphoneamplifiers to theleft. Thepiston,which con-stitutetheback wall of thetube, is madeof thick glass,with a frameof aluminium.Thus,thelaserat therear endhasan almostcompleteview of themountedsample. In thebackgroundcanbeseentheanechoicendpieceusedduring thecalibration.

w/ samplesample holder

t

MLS-signal

WinMLS

s

end piecemic. 1 mic. 2

d

laser

Figure4.21: Thesetupusedfor normal incidenceimpedanceand vibration velocitymea-surementsof the slitted doublepanel. Thetransferfunctionbetweenthe microphonesandthevibrationnormalizedvelocity, Eq.(4.23),wascalculatedby thecomputerprogramsWin-MLS [MorsetSoundDevelopment, 1999] and Qintv (Sec.A.4). During all measurements,t q 310mmands q 150mm. Threevaluesof d were used;300,150and75mm.


whereHlaser is theFouriertransformof themeasuredvelocity impulseresponse.

Simulations

Thegeometriesin Table4.9weresimulatedby themodelpresentedin Sec.4.2.1.Alternativevaluesof the gapwidth g were also simulated. Thesevaluesare given togetherwith theresults.A comprehensivesetof simulationswasalsocarriedout to investigatetheinfluenceof all the eight geometryparameters(w1, w2, t1, t2, b, q, g andd in Fig. 4.10). For all theFDM simulations,thesimulationparameterswerethesameasin theprecedingsection,givenin Table4.7. Thesimulationsweredoneat 1s 3 octave spacedpointsin thefrequency range39.7–2016Hz, andalsofor thefrequency f q 2194Hz, correspondingto λ q d.

To comparethe slitted doublepanelresonatorwith MPPs,the absorptionof two MPPswerecalculatedusing the FLAG program(seepage39). As before,the perforationsepa-rationswerechosento give approximatelythe sameresonancefrequency asthe resonancefrequenciesof thedoublepanels.MPPA hada 0.94%perforation,andMPPB hada 0.76%perforation.Thepanelthicknesswas1mm, andtheperforationradiuswas0 r 5mm. Thebackcavity thicknessd q 150mm. Thesimulationsweredoneat1s 30octavespacedpointsin thefrequency range35–1094Hz.

4.3.3 Results

Note: To increasethe readability, all lengthdimensionsin the restof this sectionaregivenwithout units. Unlessotherwisestated,all lengthsaregiven in millimeters. Also notethattheplatethicknesst or theslit width w aregivenwithout indiceswhenoneor bothof thesedimensionsarethesamefor bothplates.

Velocity profiles

The Finite Differencemodel that wasappliedto the inner partsof the slits andto the gapbetweentheplates,is a linearmodel. Theassumptionof linearity fails if thevelocity in theresonatorneckis above a certainlevel. As mentionedin relationto Eq. (4.15), the drivingpressureof the model wasarbitrarily set to 1Pa (or 94dB). Note that this is the assumedsoundpressureinsidethefront slit, atm q 0 in Fig. 4.11.Basedonthis,themaximumvaluesof thehorizontalandverticalvelocity u andv werecalculatedfor onetypical geometry. Theresultsarepresentedin Table4.10.Themaximumvelocity, 5cms s is thehorizontalvelocityin thegapbetweentheplateswhentheresonatoris in the “closed” state,andthe frequencyequalsthe resonancefrequency. This velocity wasfound to be a factor100 lower thanthevelocity which mark the onsetof nonlinearity, asdiscussedin Sec.2.3.2. Thus,assumingthatthesoundpressurein thefront slit is notsignificantlylower thanthedriving pressure,thelinearmodelshouldbevalid for soundpressurelevelsup to about134dB.

Figure4.22shows thecalculatedvelocityfieldsfor thegeometriesandfrequencieslistedin Table4.10. Thevelocity hasa horizontalcomponentsomedistanceinsidetheslits, espe-cially in therearslits. This justifiestheextensionof theFDM a distance∆zSi into theslits.In general,thevelocity profilesin theslits andin thegapbetweentheplateshave thequali-tative propertiesdescribedby CraggsandHildebrandt [1984] (seepage14). Thedimension


f q 50Hz, q q 6 f q 50Hz, q q 0

f0 q 300Hz, q q 6 f0 q 550Hz, q q 0

f q 2194Hz, q q 6 f q 2194Hz, q q 0

Figure 4.22: Velocity fields of adjustableslitted doublepanels. The rear slit is up in thefigures.Thegeometriesandfrequenciesare thesameasin Table4.10.Notethat to increaseclarity, onlyeverysecondgrid pointhasbeenincluded.Alsonotethatthescalesin thefiguresdiffersbya factor1000.Pleasereferto Table4.10to compare thevelocities.


Table4.10: Calculatedmaximumvelocitiesof a typical slitteddoublepanelin the “open”(q q 0) and“closed” (q q 6) states.Theothergeometryparameterswere: w q 2, t q 1, g q0 r 16, b q 12 andd q 155. Therowsin bold typefacerepresentscalculationsfor frequenciesnearresonance. Thefrequency2194Hz correspondsto d q λ .

f q u (max) v (max)

Hertz mm cm/s cm/s

50 0 0 r 007 0 r 005550 0 0 r 5 12194 0 0 r 009 0 r 001

50 6 0 r 7 0 r 1300 6 5 12194 6 0 r 01 0 r 002

of thegapin this exampleis of theorderof theviscousboundarylayerthickness,andsothevelocity profile in thegapis parabolic.Theslit widthsarelargecomparedto the boundarylayerthickness,sothevelocityprofilestherehavea flat centralpart.This is bestobservedinthefront slit, wherethehorizontalvelocitycomponentsarelessdominantthanin therearslit.

Theshapesof thevelocity fieldsin the“closed” statearesurprisinglysimilar for thelowfrequency, f q 50Hz andtheresonancefrequency, f q f0 q 300Hz, althoughthedifferencein magnitudeis about10, accordingto Table4.10. Thefrequency f q 2194Hz correspondsto λ t d, i. e. a standingwave in thebackcavity. For this frequency, thereis almostno netflow in theresonatorneck. Therelatively low velocitieslisted in thetablecorrespondto the“rotational” flow thatcanbeseenin thefigure.Theeffectof thestandingwaveis alsoevidentin the“open” statefor thesamefrequency. It is interestingto notethatat resonance,themax-imum verticalvelocity (correspondingto thevelocity of theflow in theslits) is equalfor the“closed” and“open” statesof thedoublepanelresonator. However, thehorizontalvelocitiesdiffers by a factorof 10. As the viscousenergy dissipationis proportionalto the squareofthevelocity to asurface,it is clearthattheslitscontributesvery little to thedissipationof thesoundenergy.

Effect of geometryparameters

Theresultsdiscussedin Sec.4.2.3gaveanindicationof therelative importanceof themanygeometryparametersusedin themodel.As themodelwasverifiedto bein reasonablyaccor-dancewith experiments,a largeseriesof simulationswasdoneto systematicallyevaluatetheeffect of all eightgeometryparameters.Themainresultsof thesesimulationsarepresentedin thefollowing paragraphs.

Symmetryof plates The velocity fields in the slits wereexpectedto be similar, so it wasassumedthattheplatesshouldbeinterchangeable.An exchangeof platesshouldnotalterthe


w q 2

50 100 500 1000 20000

0.5

1t q 1

50 100 500 1000 20000

0.5

1

Figure4.23: Effectof exchange of platesof different thicknessandwith differentslit widths.Absorptioncoefficientas functionof frequency. SeeFig. 4.10for definitionof symbols.Forbothgraphs,b q 12, q q 6, d q 75 andg q 0 r 16. Left graph: , t1 q 1 andt2 q 2; ,t1 q 2 andt2 q 1. Rightgraph: , w1 q 1 andw2 q 2; , w1 q 2 andw2 q 1.

absorptioncharacteristics.However, asdemonstratedabove, the velocity fields in the frontandtherearslitswerenotcompletelysimilar. Figure4.23showstheeffectof anexchangeofplatesof differentthicknessandwith differentslit widths.Thisfigureshowsthattheplatesofdifferentdimensionsarenot completelysymmetric,but thedifferenceis sosmall that thereis probablyno needto considerthis effect aspartof a designprocess.To excludethesmalleffectof asymmetry, all theothergeometriespresentedherehadt q t1 q t2 andw q w1 q w2.

Effectof slit widthandplatethickness Figure4.24shows theeffectof achangein theplatethicknesst andthe slit width w. Simulationswerealsodonewith t q 2 andw q 2. Theseresultsare excludedto simplify the figure. By comparingthe left column with the rightcolumnof Fig. 4.24, it is clear that the plate thicknessis practically insignificant,at leastfor the relatively normal thicknessesconsideredhere. This hasthe practicalconsequencethat the plate thicknesscan be chosensolely on the basisof physicalrequirements.Theslit width, on the other hand,may seemto have a significantinfluenceon the absorptioncharacteristics,accordingto thegraphsin Fig. 4.24. This is probablynot thecase.As seenfrom Fig. 4.10on page59, theeffectivelengthof thegapbetweenthefront slit andtherearslit is q | w1 s 2 | w2 s 2 q q | w. A changein w will thereforechangetheeffectivegaplengthequivalently. By comparingthe top left graphin Fig. 4.24 with the middle left graphinFig. 4.27,theeffectof w maybecomparedwith theeffectof q.

Effectof plateseparation Theplateseparationwasexpectedto be themostimportantpa-rameter. Thediscussionin thebeginningof Sec.2.2andtheresultsof Sec.4.2.3indicatethatfor a givenvalueof b (or q), thereexistsanoptimumvalueof g. Theexistenceof this opti-mumis confirmedby thegraphsin Fig.4.25.To increasetheclarity, eachgraphincludesonlyevery third of thegeometriesthatweresimulated.Thegraphsclearly indicatetheoptimumvalueof g for thethreevaluesof b simulated.As expected,theoptimumvalueof g increase


t q 1, b q 12,g q 0 r 16

50 100 500 1000 20000

0.5

1

t q 3, b q 12,g q 0 r 16

50 100 500 1000 20000

0.5

1

t q 1, b q 12,g q 0 r 12

50 100 500 1000 20000

0.5

1t q 3, b q 12,g q 0 r 12

50 100 500 1000 20000

0.5

1

t q 1, b q 24,g q 0 r 16

50 100 500 1000 20000

0.5

1

t q 3, b q 24,g q 0 r 16

50 100 500 1000 20000

0.5

1

Figure4.24: Effect of plate thicknessand slit width. Absorptioncoefficient as functionoffrequency. SeeFig. 4.10for definitionof symbols.For all graphs,d q 75 andq q bs 2. ,w q 1; , w q 3.


b q 12,d q 75

50 100 500 1000 20000

0.5

1b q 12,d q 155

50 100 500 1000 20000

0.5

1

b q 18,d q 75

50 100 500 1000 20000

0.5

1b q 18,d q 155

50 100 500 1000 20000

0.5

1

b q 24,d q 75

50 100 500 1000 20000

0.5

1b q 24,d q 155

50 100 500 1000 20000

0.5

1

Figure4.25: Effectof plateseparation. Absorptioncoefficientas functionof frequency. SeeFig. 4.10 for definitionof symbols.For all graphs,w q 2, t q 1 and q q b s 2. Thecurvescorrespondto plateseparationsg q 0 r 10, 0 r 16, 0 r 22, 0 r 28, 0 r 34, 0 r 40, 0 r 46, 0 r 52 and0 r 58.Thicker linescorrespondto larger valuesof g.


for largervaluesof b, andis independentof d. However, theabsorptionbandwidthdecreasesfor largervaluesof b. Thus,if high absorptionover a largefrequency rangeis required,theplateseparationis limited to a ratherlow value. Theresonancefrequency increaseswhengincreases,dueto thereductionof theeffectivemass.

Effect of center-centerdistancebetweenslits The effect of variation in the center-centerdistancebetweenslits is shown in Fig. 4.26. Theseresultsaresimilar to thosein Fig. 4.25,andshows thatfor agiveng thereexistsanoptimumvalueof b. As discussedabove,a largervalueof g correspondsto a largeroptimumb, at thecostof a reducedabsorptionbandwidth.The resonancefrequency decreaseswhenb increases.Simulationswith g q 0 r 40 are notincludedin Fig. 4.26. Thesesimulationsshowed the sametendency asg q 0 r 20, with themaximumabsorptioncoefficientoccurringat b q 22.

Effectof slit separation Figure4.27shows theeffect of variationin theslit separation,i. e.the lateraldistanceq betweenthe centersof the front andrearslits. As shown in Fig. 4.26,for g q 0 r 16 themaximumabsorptioncorrespondsto b q 8. Thus,for b q 12 theabsorptionis maximumfor a slit separationq � b s 2, while for b q 6, the slit separationbs 2 givesthelargestpossibleabsorption.If b is larger thanthe optimumvalue,the optimumvalueof qis lessthanb s 2. As with g andb, a changein q causesa frequency shift. As shown in themiddleleft graph,this canbeusedto makearesonatorwith shiftableresonancefrequency.

Summary It hasbeenfound that for the slitted doublepanelabsorber, the platesare in-terchangeable.For platesof normalthickness,the thicknessof the platesis of no practicalsignificance.A changein theslit width, or a changein thelateraldistancebetweenthefrontandrearslits bothcausea changein theeffectivelengthof lateralgap. For a givenapplica-tion, thereexistsanoptimumcombinationof this effective gaplengthandthegapwidth. Achangein plateseparationg, center-centerdistancebetweenslits b, or slit separationq maycausea significantchangein the maximumabsorption,absorptionbandwidthor resonancefrequency. As describedin Sec.4.3.2, the slit separationcanbe easilyby changed.Thus,the adjustableslittedpanelconceptoffers thepossibilityof resonatorswherethe maximumabsorptionor theresonancefrequency caneasilybechanged.

Panel vibrations

As statedearlier, oneof the objectivesof the measurementsdescribedherewasto evaluatethe influenceof panelvibrationson the absorptioncharacteristics.Figures4.28 and 4.29shows themodulusandphaseof themeasurednormalizedvibrationvelocity asfunctionoffrequency for several positionsandconfigurations. All thesemeasurementsweredoneonpanelA, definedin Table4.9. Thefirst four peaks,in thefrequency range20–55Hz, seemsto be practicallyequalfor all graphs,andareprobablyanartifactof the measuringsystem.The main vibration modeof the panel,at 77Hz, shows the expectedcharacteristics;Thevibrationvelocity is smallerneartheedgeof thepanelthanit is at thecenter. Thevelocity islargerfor a deeperbackcavity. Whenthepanelis in the“open” state,thevibrationvelocitybecomessignificantlysmaller. This is in accordancewith whatwasreportedby Tanakaand


g q 0 r 12,d q 75

50 100 500 1000 20000

0.5

1

g q 0 r 12,d q 155

50 100 500 1000 20000

0.5

1

g q 0 r 16,d q 75

50 100 500 1000 20000

0.5

1g q 0 r 16,d q 155

50 100 500 1000 20000

0.5

1

g q 0 r 20,d q 75

50 100 500 1000 20000

0.5

1g q 0 r 20,d q 155

50 100 500 1000 20000

0.5

1

Figure4.26:Effectof center-centerdistancebetweenslits. Absorptioncoefficientasfunctionof frequency. SeeFig. 4.10for definitionof symbols.For all graphs,w q 2, t q 1 andq q bs 2.Thecurvescorrespondto center-centerdistancebetweenslits, b q 6, 8, 10, 12, 14, 16, 18,20, 22and24. Thicker linescorrespondto larger valuesof b.


b q 6, g q 0 r 16

50 100 500 1000 20000

0.5

1b q 6, g q 0 r 40

50 100 500 1000 20000

0.5

1

b q 12,g q 0 r 16

50 100 500 1000 20000

0.5

1b q 12,g q 0 r 40

50 100 500 1000 20000

0.5

1

b q 24,g q 0 r 16

50 100 500 1000 20000

0.5

1

b q 24,g q 0 r 40

50 100 500 1000 20000

0.5

1

Figure4.27: Effect of slit separation. Absorptioncoefficient as functionof frequency. SeeFig. 4.10 for definitionof symbols.For all graphs,w q 2, t q 1 and d q 75. Thecurvescorrespondto slit separationsq q 0, b s 12, bs 6, b s 4, bs 3, 5b s 12 and b s 2. Thicker linescorrespondto larger valuesof q.


“Closed”panel,centerof panel,d q 75

50 100 500 10000

0.5

1


50 100 500 10000

0.5

1

“Closed”panel,nearedgeof panel,d q 75

50 100 500 10000

0.5

1


50 100 500 10000

0.5

1

“Open” panel,centerof panel,d q 150

50 100 500 10000

0.5

1

Figure4.28:Measuredpanelvibration for somemeasuringpositionsandpanelstates.Mod-ulusof normalizedvibration velocityasfunctionof frequency. Theordinatehasbeenscaledsothat themaximumvalueequals1. ThemeasurementsweredoneonpanelA (seeTable4.9).

, front plate; , rear plate.



50 100 200−pi

0

pi


50 100 200−pi

0

pi


50 100 200−pi

0

pi


50 100 200−pi

0

pi

“Open” panel,centerof panel,d q 150

50 100 200−pi

0

pi

Figure4.29:Measuredpanelvibrationfor somemeasuringpositionsandpanelstates.Phaseof normalizedvibration velocityas functionof frequency. Themeasurementswere doneonpanelA (seeTable4.9). , front plate; , rear plate.


PanelA

50 100 500 10000

0.5

1PanelB

50 100 500 10000

0.5

1

Figure4.30: Modulusof normalizedpanelvibration velocityand absorptioncoefficient asfunctionof frequency. SeeTable 4.9 for geometrydefinitions. Theback cavity depthwasd q 150 for bothpanels.Thevelocity, measuredat thecenterof thepanel,hasbeenscaledsothat themaximumvalueequals1. , vibrationvelocity; , absorptioncoefficient.

Takahashi[1999] for perforatedpanelswith largeopenings(seepage21). Thegraphsshowthatthevibrationvelocitiesof thetwo platesseemto havethesameamplitudeand,asshownin Fig. 4.29,practicallythesamephase.Consequently, theplateseparationis not influencedby panelvibrations. Figure4.30comparesthe vibration modeswith the dips andpeaksofthe absorptioncoefficient for two differentpanels. It is evident that the panelresonancesmay influencethe absorptioncoefficient significantly, especiallyfor panelresonanceswithfrequenciesneartheHelmholtzresonance.It is alsoworthnotingthatpanelresonancesbelowtheHelmholtzresonanceresultin dips in theabsorptioncoefficient,while panelresonancesabovetheHelmholtzresonanceresultin absorptionpeaks.Thisis probablydueto therelativephaseof the panelvibrationvelocity andtheparticlevelocity in the panelopenings.Whenthesevelocitiesare in phase,the particle velocity in the panelopenings(i. e. the velocityrelative to thepanel)is smaller. Theresultis adecreasein theenergy dissipationandadip intheabsorptioncoefficient.

Samplemeasurementsand simulations

Figures4.31and4.32comparesthe measuredandsimulatedabsorptioncoefficient for thepanelslisted in Table4.9. The panelswerein the “closed” state.The agreementbetweenmeasurementsandsimulationsis relatively goodfor panelsA andB. For panelsC andD,however, thereal,averagegapseparationseemto havebeensignificantlydifferentfrom whatwasassumedandlisted in Table4.9. In the caseof panelC, this discrepancy could be ob-servedvisually. In anattemptto estimatethereal,averagegapwidth, severaldifferentvaluesof g weresimulated.Theplottedsimulationsg q 0 r 22for panelC andg q 18for panelD hadthe bestagreementwith measurements.Subsequentsimulationsof panelD (seeFigs.4.33and4.34)haveusedg q 0 r 18.

The measuredand simulatedeffect of a variation in the slit separationq is shown inFig. 4.33. Notethat thecurveshave beencut at 100and850Hz to excludetheeffectsof the


PanelA, d q 75

50 100 500 10000

0.5

1PanelB, d q 75

50 100 500 10000

0.5

1

PanelA, d q 150

50 100 500 10000

0.5

1PanelB, d q 150

50 100 500 10000

0.5

1

PanelA, d q 300

50 100 500 10000

0.5

1PanelB, d q 300

50 100 500 10000

0.5

1

Figure4.31: Simulationsandmeasurementsof slitteddoublepanels.Absorptioncoefficientas function of frequency. SeeTable 4.9 for geometrydefinitions. All panelswere in the“closed” state. Left column: measured gap width g q 0 r 25, simulatedgap width:

�, g q

0 r 25. Rightcolumn:measuredgapwidthg q 0 r 45, simulatedgapwidth: ² , g q 0 r 45.


PanelC, d q 75

50 100 500 10000

0.5

1PanelD, d q 75

50 100 500 10000

0.5

1

PanelC, d q 150

50 100 500 10000

0.5

1PanelD, d q 150

50 100 500 10000

0.5

1

PanelC, d q 300

50 100 500 10000

0.5

1PanelD, d q 300

50 100 500 10000

0.5

1

Figure4.32: Simulationsandmeasurementsof slitteddoublepanels.Absorptioncoefficientas function of frequency. SeeTable 4.9 for geometrydefinitions. All panelswere in the“closed” state. Left column: measuredgapwidth g q 0 r 10, simulatedgapwidths: , g q0 r 10;

�, g q 0 r 22. Rightcolumn: measuredgapwidth g q 0 r 12, simulatedgapwidths:

±,

g q 0 r 12; ² , g q 0 r 18.


PanelA, Measurements

100 5000

0.5

1PanelD, Measurements

100 5000

0.5

1

PanelA, Simulations

100 5000

0.5

1PanelD, Simulations

100 5000

0.5

1

Figure4.33: Simulationsandmeasurementsof slitteddoublepanels.Absorptioncoefficientasfunctionof frequencyfor several valuesof q. SeeTable4.9 for geometrydefinitions.Theback cavity depthwasd q 150 for both panels. For panelA, the curvescorrespondto slitseparationsq q 0, 3, 5, 6 and7. For panelD, thecurvescorrespondto slit separationsq q 0,1, 2, 2 µ 5, 3, 4 and5. Thicker linescorrespondto larger valuesof q.

mainvibrationmodeof thepanelandhigherordermodesin theKundt’s tube. As shown inFig. 4.27,theabsorptioncoefficient canbevery sensitive to variationsin q for intermediatevaluesof q. Thus,someof the discrepancy betweenthe simulationsandmeasurementsforintermediatevaluesof q maybedueto inaccuraciesin themeasuredvalueof q. Thegeneralbehavior with varyingq is predictedby thesimulationswith a reasonabledegreeof accuracy.

Figure4.34comparestheabsorptioncoefficientsof the“best” slitteddoublepanels(panelA andD) with simulationsof typical MPPs.Thecomparisonwith MPPsconfirmsthepoten-tial of theslitteddoublepanelabsorber. Theresultspresentedin Figs.4.33and4.34werethemotivationfor thework to bedescribedin thenext section.

4.4Adjustable slitted panel absorber 89

PanelA andMPPA

100 5000

0.5

1PanelD andMPPB

100 5000

0.5

1

Figure4.34:FLAG simulationsof MPPsandmeasurementsof slitteddoublepanels.Absorp-tion coefficientasfunctionof frequency. SeeTable4.9andpage74 for geometrydefinitions.Theback cavitydepthwasd ¶ 150 for all panels. , measuredabsorptioncoefficient;

�,

MPP A; ² , MPPB.

4.4 Adjustable slitted panelabsorber

The doublepanelabsorberwith adjustableslit separationseemedto have absorptionchar-acteristicsequivalent to or betterthan typical microperforatedpanels. However, all mea-surementsandsimulationsdescribedin the precedingsectionsare valid for normalsoundincidenceonly. This sectiondescribesthe investigationson the doublepanelabsorberin adiffusesoundfield.

4.4.1 Model

Therelative,specificimpedanceof anair cavity in front of ahardwall is

ζair u β v·¶ | jcosβ

cot u kdcosβ v � (4.24)

whereβ is the angleof incidenceandd is the thicknessof the cavity. If a locally reactingperforatedpanel,i. e. apanelwith impedanceindependentof theangleof soundincidence,isplacedin front of theair cavity, theimpedancebecomes

ζ u β v¸¶ ³ u θ � jωχ v � ζair u β v ´ cosβ � (4.25)

whereθ andχ is therelative, specificresistanceandreactanceof theperforatedpanel.Theabsorptioncoefficientof thesystemis givenby

α u β v¸¶ 4ℜ u ζ v° ζ ° 2 � 2ℜ u ζ v � 1 � (4.26)


andthe absorptioncoefficient in a diffusesoundfield is givenby the integral of α u β v [seee. g. MorseandIngard, 1968,Ch.9.5]:

αst ¶º¹ 2π

0dβ � ¹ π » 2

0α u β v sinβ cosβ dβ (4.27)

This integralmaybeapproximatedby

αst ¶ 2πM

∑m� 1

α u βm v sinβmcosβm ∆β � (4.28)

where∆β ¶ π ¼ 2M, andβm ¶ m∆β , andM is aninteger.For simplicity, theslitteddoublepanelconsideredin thischapteris assumedto belocally

reactingso that the theoryabove is applicable. However, the panelresistanceθ andreac-tanceχ arenot explicitly givenby themodeldescribedin Sec.4.2.1. As shown in relationto Eq. (4.16), the impedancein front of the panelis dependenton the input parameterzT ,which is dependenton the impedanceof the backcavity. Therefore,the calculationof theimpedancein a diffusefield by Eq. (4.28)would requirethe time-consumingcalculationofthe impedancefor a large numberof incidenceangles. As a morefeasiblealternative, theimpedanceof thegapbetweenthepanelsis hereapproximatedby Eq.(2.38):

Zgap ½ 0 µ 5jωρ0b¼ 2g

1 | tanh ~ xs } j �

xs } j ¢ � 1

(4.29)

Thefactor0.5 is dueto thefactthat,asseenfrom thefront slit, therearetwo gaplengthsb¼ 2in parallel.Theperforateconstantof thegap,xs, is givenby Eq.(2.39),with w substitutedbyg. Thetotal, relative,specificimpedanceof thesystemis thengivenby

ζdp u β v·¶ £ ζair u β v za

b

�Zsw 2 � Zgap

�Zsw 1 ¦ bcosβ

za � (4.30)

with Zsw i givenby Eq. (4.10). Note that theslit length l �i ¶ l i ¶ t for i ¶ 1 � 2. Thestatisticalabsorptioncoefficient is calculatedby Eq.(4.28).

Themodeldescribedhereis intendedto beanquickapproximation,giventheinfeasibilityof the Finite DifferenceMethod. For simplicity, the modeldoesnot includecorrectionsforendeffects[seee. g. CremerandMüller, 1978b,pp.335-338].



Basedon theresultsof theprecedingsections,25 full scaleadjustableslittedpanelabsorberswereproduced.The basicdesignwasthe sameasfor the samplesdescribedin Sec.4.3.2.Figure4.35shows oneof the panels.Table4.11summarizesthe dimensionsof the panels.As shown in the figure, therearefour sectionsof slits. Eachsectionis 140mm wide, with8mm spacing.To keepthe front andrearplatesseparated,five stripsof adhesive tapewere


Figure 4.35: Sampleof full scaleadjustableslitted panel absorber. The panel is in the“closed” state, and is seenfrom the front side. Barely visible are the screws usedto fas-tenthefront plateto therear plateandthesupportingframe.

Table4.11: Dimensionsof full scaleadjustableslittedpanels.SeeFig. 4.10for definitionofsymbols.Therear platewas600 � 600mmandthefront platewas594 � 600mm.

Dimension Length (mm)

w 3

t 0.7

g 0.16

b 12

q 0, 3, 4, 5 and 6 for d ¶ 75mm0, 3, 3.5, 4 and 6 for d ¶ 155mm


fastenedto therearplatebetween(andoutside)thesectionsof slits. The tapewas0 µ 16mmthick. Visualinspectionshowedthattheplateswerenotcompletelyplane,sothereal,averagedistancebetweenthe plateswasprobablysignificantlylarger thanthis. The rearpanelwasfastenedwith screwsto asquareframeof steelbars.To supportthepanelmechanically, therewasalsoonesteelbar at the middle of the panel. The cross-sectionof the steelbarswas8 � 10mm. As shown in Fig. 4.35,thefront platewasfastenedto thebackplatewith screwsthroughshortslits in thefront plate.Theseslits wereplacedalongthestripsof tape.

Absorption coefficientmeasurementsin reverberation room

The absorptioncoefficient of the panelswas measuredaccordingto ISO 354 [1985] in astandardreverberationroom with volume 268m3. SeeFig. 4.36. To minimize the effectof inaccuraciesin the plate separation,the panelswere laid with the front down, so thatthe weightof the steelframewould help keepthe platestogether. This is equivalentto thesituationif thepanelsweremountedin asuspendedceiling. Thepanelswerelaid onT-profilesteelbars,which werefastenedto a rectangularframeof wood.Theheightof theframewas70mm. The total areaof the setupwas3 µ 024m � 3 µ 072m ¶ 9 µ 29m2, which is a little lowcomparedwith the10–12m2 that is suggestedby ISO354 [1985]. An extra frame,80mmhigh, wasalsomadeso that two differentbackcavity depthscouldbe tested.Including thethicknessof theT-profilesteelbars,thedistanced from thepanelto thefloor was75mmand155mm.

Two microphoneswereusedin six positionsin the room. The microphonesusedwereBrüel& Kjær type4145.Theabsorptioncoefficientwasmeasuredin 1¼ 3 octavebandsin therange100–5000Hz.

Simulations

Doublepanelswith dimensionsasgivenin Table4.11weresimulatedby themodeldescribedin Sec.4.4.1. The numberof discreteangleswereM ¶ 45. As an attemptto find the real,averagegapwidth, several othervaluesof the plateseparationg werealsosimulated.Thevalueg ¶ 0 µ 25mmresultedin a goodmatchwith themeasurements.

4.4.3 Results

Figure4.37shows thereverberationroommeasurementsof theadjustable,slittedpanelab-sorber. The effect of q is asexpected(comparewith Figs.4.27and4.33). The maximumabsorptionis lower thanwasexpected,however. As notedabove,this is probablydueto thatfactthattheaverageplateseparationwassignificantlylargerthan0 µ 16mm.

Figure4.38 shows the measurementsof the doublepanelabsorberin the “closed” and“open” states,comparedto simulationby the model describedin Sec.4.4.1. Simulationsfor theplateseparationg ¶ 0 µ 25mmcorrespondsreasonablywell with themeasurements,atleastfor d ¶ 75mm.


Figure4.36: Full scaleadjustableslittedpanelsin thereverberation room.Thepanelswerelaid facedownon supportingT-profiles.Thetotal areaof thesampleswas9 µ 29m2.


d ¶ 75mm

100 500 1000 50000

0.5

1d ¶ 155mm

100 500 1000 50000

0.5

1

Figure 4.37: Measurementsof slitted doublepanelsin a diffusesoundfield. Absorptioncoefficient as function of frequencyfor several valuesof q. SeeTable 4.11 for geometrydefinitions. For the left graph, the curvescorrespondto slit separationsq ¶ 0, 3, 4, 5 and6mm. For theright graph,thecurvescorrespondto q ¶ 0, 3, 3 µ 5, 4 and6mm. Thicker linescorrespondto larger valuesof q.

4.5 Summary

The doublepanelabsorberconcepthasbeeninvestigatedby analytical,numericalandex-perimentalmethods.Two variantshave beenexamined:onewherethe plateshave circularperforations,andonewith slitted plates. It hasbeenfound that the doublepanelabsorberis practically insensitive to the plate thickness.The most importantdesignparametersarethe thicknessof thegapbetweentheplatesandtheeffective lengthof this gap,i. e. thedis-tancebetweenthepanelopeningsof thetwo plates.With anoptimaldesign,thedoublepanelabsorbermayhaveanabsorptionbandwidthequalto or betterthanMPPs.

The slitted variantwassimpler to model thanthe perforatedvariant. Additionally, theslitted doublepanelofferedtwo distinct featuresthat arenot easily implementedwith theperforatedvariant,or with microperforatedpanels:

– Frequencyshift. The resonancefrequency f0 of the panelabsorbercanbe shiftedbyadjustingthelateraldistancebetweentheslits in thefront andtherearplate.Fromthesimulations,it seemsthatat frequency shift of at leastoneoctaveis possible(atnormalsoundincidence),while keepingtheabsorptioncoefficientat 0.7or larger.

– Variation in absorptioncoefficient. Also by adjustingthe lateral slit separation,theabsorptionat resonance,α0 canbevariedfrom almostunity to avaluecloseto zero.

To achieve a reasonableabsorptionbandwidth,thegapwidth g hasto bequitesmall, inthe range0.1–0 µ 3mm. In this range,the sensitivity to variationsin g is very large. Themethodthat hasbeenusedto keepg at a specifiedvalueis far from perfect. Thus, in thedesignprocessprecautionsmustbetakento ensurethat thevalueof g is keptasconstantaspossible.

4.5Summary 95

d ¶ 75mm, simulatedg ¶ 0 µ 16mm

100 500 1000 50000

0.5

1d ¶ 155mm, simulatedg ¶ 0 µ 16mm

100 500 1000 50000

0.5

1


100 500 1000 50000

0.5

1


100 500 1000 50000

0.5

1

Figure4.38: Simulationsandmeasurementsof slitteddoublepanelsin a diffusesoundfield.Absorptioncoefficientasfunctionof frequencyin “closed” and“open” states.SeeTable4.11for geometrydefinitions. , measurement,“closed”; , measurement,“open”; ² , simu-lation, “closed”;

�, simulation,“open”.

Chapter 5

Conclusions

This thesishaspresentedmy work with two new perforatedpaneldesigns.Themicrohornpanelabsorberis amicroperforatedpanelwherethetraditionalcylindrical

orificeshavebeenshapedassmallhorns.Thepurposesof thehorn-shapeareto increasethesurfaceareaassociatedwith eachopening,increasetheparticlevelocity in the innerpartofthe horn,andoffer a betterimpedancematchto the incomingsoundwave. The microhornconcepthasbeeninvestigatedby theuseof threemodels;theFinite DifferenceMethod,theFiniteElementMethod,andanintegrationmethodbasedon theanalyticalexpressionfor theimpedanceof a tube. Only the resultsof the FDM could, to somedegree,beconsideredasqualitatively correct.Accordingto theseresults,a relatively high absorptioncoefficient canbeobtainedoverarelatively largebandwidthfor microhornswith largeouterradiusandsmallinnerradius.Themicrohornabsorbermayhaveapotentialasan“enhanced”microperforatedpanel. However, the model needsto be refinedto give more accuratepredictionsand toevaluatethefeasibilityof themicrohornpanelconcept.

Thedoublepanelabsorberconsistsof two parallelperforatedor slittedplatesseparatedbya smalldistance.Themainpartof thesoundenergy dissipationtakesplacein thesmallgapbetweenthe plates.The slitted variantis moreeasilymodelledthanthe perforatedvariant,andoffers two specialfeatures. By moving the platesrelative to eachother in the lateraldirection,themaximumabsorptioncoefficientcanbeadjustedfrom unity to almostzero,andthe resonancefrequency canbeadjustedby anoctave (for normalsoundincidence).Thereare eight possibledesignparameters.FDM simulationsand experimentson a numberofsampleshaveshown thatthemostimportantparametersaretheeffectivelengthandthicknessof thegapbetweentheplates.Thelatterparameteris verysensitive. Theplatethicknesswasfoundto bepracticallyinsignificant.Themeasuredandsimulatedabsorptioncoefficientsofthedoublepanelabsorberhave a bandwidthequivalentto, or slightly betterthanMPPs.Tobeof practicaluse,however, thepaneldesignhasto beimprovedsothatthedistancebetweentheplatescanbekeptasconstantaspossible,andaccordingto specifications.

Both panelabsorberconceptsillustratethatthedistributedHelmholtzresonatorswithoutporousabsorbersmay obtainan acceptableabsorptionbandwidthby optimizing the designof theresonatororifices.

97

98 Conclusions

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Appendix A

Documentationof functions

Theheadersof all MATLAB-, Perl-andshellfunctionsusedin themodelling,measurementsanddataanalysisarepresentedhere.

A.1 General functions

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105

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