Lecture Notes on Acoustics and Noise Control

381 Pages·2022·12.09 MB·English

by  Hejie Lin, Turgay Bengisu, Zissimos P. Mourelatos

Hejie Lin
Turgay Bengisu
Zissimos P. Mourelatos
Lecture Notes
on Acoustics
and Noise
Control
Lecture Notes on Acoustics and Noise Control
(cid:129)
Hejie Lin Turgay Bengisu
Zissimos P. Mourelatos
Lecture Notes on Acoustics
and Noise Control
HejieLin TurgayBengisu
GeneralMotors OaklandUniversity
Warren,MI,USA Rochester,MI,USA
ZissimosP.Mourelatos
MechanicalEngineering
OaklandUniversity
Rochester,MI,USA
ISBN978-3-030-88212-9 ISBN978-3-030-88213-6 (eBook)
https://doi.org/10.1007/978-3-030-88213-6
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Preface
Objectives of the Book
Lecture Notes on Acoustics and Noise Control provides a mathematical backbone
for acoustics and noise control. This mathematical foundation is comprised of
straightforward mathematical derivations and formulations of sound waves. All
formulations are builtfrom the acousticwave equation based on thekinetictheory
of gases. The mathematical formulations are condensed into a minimalistic yet
complete acousticfoundation for a one-semestercourse. The formulations covered
inthisbookcanbeusedasreliablereferencesforresearchersandengineersworking
inthefieldofacousticsandnoisecontrol.
Style
Thisbookiswritteninacoursenotesformatforaone-semestercourseof“acoustics
and noise control.” This course is offered to senior undergraduates and beginning
graduate students without a prior background of acoustics or noise control. The
materialsinthisbookarecompiledfromaseriesofnoise,vibration,andharshness
(NVH)coursestaughtbytheauthorsofthisbookduringthepast14yearsatOakland
University.
Straightforward derivations of formulations are presented in the course notes.
Whenthederivationofformulasisthemainobjectiveofasection,thefinalformulas
areplacedatthebeginningofthatsectiontoserveasacompassofderivation.The
properties of sound can be observed and understood through the derivation of the
formulation. Class examples and homework are included to support and reinforce
the derivations. Through the derivations, examples, and homework, students can
understandthephysicsbehindtheformulas.
v
vi Preface
Twosignificantprojectsinnumericalanalysisofsoundpropagationareincluded
in this course. The first project revolves around numerically calculating the sound
pressure induced by a vibrating plate. This project requires students to understand
the formulations for acoustic point sources and apply them to the real-world situa-
tions.Thefirstprojectconcludesthefundamentalacousticstheoryinthefirsthalfof
the course (Chaps 1, 2, 3, 4, 5 and 6). The second project requires numerical
calculationofthesoundpowertransmissioninpipes.Thisprojectrequiresstudents
tounderstandtheformulaforpowertransmissioninpipesasfiltersandapplythemto
real-world problems. The second project concludes the filter design in the second
halfofthecourse(Chaps.7,8,9,10,11and12).
Prerequisites
Students should have a basic knowledge of practical experience with calculus and
differentialequations.
The Big Picture
The first half of this book covers the fundamentals of acoustic wave formulations.
Chapter 1 reviews complex numbers and introduces four equivalent forms of
complex numbers for harmonic waves. Understanding these four equivalent forms
of complex numbers is crucial for understanding the mathematical formulations of
acousticsandnoisecontrol.Chapters2,3,and4deriveandsolvetheplaneacoustic
wave equation. Chapters 5 and 6 derive and solve the spherical wave equation. At
theendofChap.6,Project#1isincluded toreinforceandconsolidatethelearning
from Chaps. 1 to 6 (the first half of the course). This project requires students to
numericallycalculatesoundpressureusingthepointsourceformulationobtainedin
Chap.6.
Thesecondhalfofthisbookcoversapplicationsofacousticsinthefieldofnoise
control.Chapters7and8explainandformulatethesoundresonanceinrectangular
cavitiesandsoundpropagationinwaveguides.Chapter9introducestheconceptof
weighted sound pressure levels. Chapter 10 introduces the basic formulations for
noisecontrolofroomacoustics.Chapters11and12coverthetheoryofthreebasic
acoustic filters: the high-pass filter, the low-pass filter, and the band-stop (Helm-
holtz)filters.
Preface vii
AttheendofChapter11,Project2Aisincludedtonumericallycalculatesound
powertransmissioninpipelinesusingtheformulasdevelopedinthischapter.Atthe
endofChapter12,Project2Bisincludedtomodellow-passfilters,high-passfilters,
and band-stop filters as pipes with side branches. In this project, sound power
transmissionoffilterswillbecalculated.
Warren,MI,USA HejieLin
Rochester,MI,USA TurgayBengisu
Rochester,MI,USA ZissimosP.Mourelatos
Contents
1 ComplexNumbersforHarmonicFunctions. . . . . . . . . . . . . . . . . . 1
1.1 ReviewofComplexNumbers. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 ComplexNumbersinPolarForm. . . . . . . . . . . . . . . . . . . . . . 3
1.3 FourEquivalentFormstoRepresentHarmonicWaves. . . . . . . 4
1.4 MathematicalIdentity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 DerivationofFourEquivalentForms. . . . . . . . . . . . . . . . . . . 8
1.5.1 ObtainForm2fromForm1. . . . . . . . . . . . . . . . . . . 8
1.5.2 ObtainForm3fromForm2. . . . . . . . . . . . . . . . . . . 9
1.5.3 ObtainForm4fromForm3. . . . . . . . . . . . . . . . . . . 9
1.6 VisualizationandNumericalValidationofForm1
andForm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Space-TimeHarmonicFunctionsExpressedinFour
EquivalentForms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.9 ReferencesofTrigonometricIdentities. . . . . . . . . . . . . . . . . . 21
1.9.1 TrigonometricIdentitiesofaSingleAngle. . . . . . . . . 21
1.9.2 TrigonometricIdentitiesofTwoAngles. . . . . . . . . . . 23
1.10 AMATLABCodeforVisualizationofForm1
andForm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 DerivationofAcousticWaveEquation. . . . . . . . . . . . . . . . . . . . . . 27
2.1 Euler’sForceEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 EquationofContinuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 EquationofState. .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . .. 34
2.3.1 EnergyIncreaseduetoWorkDone. . . . . . . . . . . . . . 35
2.3.2 PressureduetoCollidingofGases. . . . . . . . . . . . . . 37
2.3.3 DerivationofEquationofState. . . . . . . . . . . . . . . . . 38
2.4 DerivationofAcousticWaveEquation. . . . . . . . . . . . . . . . . . 39
2.5 FormulasfortheSpeedofSound. . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 FormulaUsingPressure. . . . . . . . . . . . . . . . . . . . . . 42
2.5.2 FormulaUsingBulkModulus. . . . . . . . . . . . . . . . . . 43
ix
x Contents
2.5.3 FormulaUsingTemperature. . . . . . . . . . . . . . . . . . . 45
2.5.4 FormulaUsingCollidingSpeed. . . . . . . . . . . . . . . . . 46
2.6 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 SolutionsofAcousticWaveEquation. . . . . . . . . . . . . . . . . . . . . . . 49
3.1 ReviewofPartialDifferentialEquations. . . . . . . . . . . . . . . . . 51
3.1.1 ComplexSolutionsofaPartialDifferential
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 TrigonometricSolutionsofaPartialDifferential
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 FourBasicComplexSolutions. . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 FourBasicTravelingWaves. . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 FourBasicStandingWaves. . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.5 ConversionBetweenTravelingandStandingWaves. . . . . . . . 63
3.6 Wavenumber,AngularFrequency,andWaveSpeed. . . . . . . . 68
3.7 VisualizationofAcousticWaves. . . . . . . . . . . . . . . . . . . . . . 71
3.7.1 PlottingTravelingWave. . . . . . . . . . . . . . . . . . . . . . 71
3.7.2 PlottingStandingWave. . . . . . . . . . . . . . . . . . . . . . 72
3.8 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 AcousticIntensityandSpecificAcousticImpedance. . . . . . . . . . . . 81
4.1 Pressure-VelocityRelationship. . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.1 Pressure-VelocityRelationshipsforBTW. . . . . . . . . 83
4.1.2 Pressure-VelocityRelationshipsforBSW. . . . . . . . . . 86
4.1.3 Pressure-VelocityRelationshipsinComplex
FunctionForm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 RMSPressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2.1 RMSPressureofBTW. . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2 RMSPressureofBSW. . . . . . . . . . . . . . . . . . . . . . . 91
4.3 AcousticIntensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3.1 AcousticIntensityofBTW. . . . . . . . . . . . . . . . . . . . 94
4.3.2 AcousticIntensityofBSW. . . . . . . . . . . . . . . . . . . . 95
4.4 SpecificAcousticImpedanceExpressedasRealNumbers. . . . 96
4.4.1 SpecificAcousticImpedanceofBTW. . . . . . . . . . . . 97
4.4.2 SpecificAcousticImpedanceofBSW. . . . . . . . . . . . 98
4.5 SpecificAcousticImpedanceExpressedasComplex
Numbers. . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . 103
4.5.1 IssueswithRealImpedance. . . . . . . . . . . . . . . . . . . 103
4.5.2 DefinitionofComplexImpedance. . . . . . . . . . . . . . . 103
4.6 ComputerProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.7 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.8 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.8.1 DerivativesofTrigonometricandComplex
ExponentialFunctions. . . . . . . . . . . . . . . . . . . . . . . 114
4.8.2 TrigonometricIntegrals. . . . . . . . . . . . . . . . . . . . . . . 115
Contents xi
5 SolutionsofSphericalWaveEquation. . . . . . . . . . . . . .. . . . . . . . . 117
5.1 SphericalCoordinateSystem. . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2 WaveEquationinSphericalCoordinateSystem. . . . . . . . . . . 119
5.3 PressureSolutionsofWaveEquationinSpherical
CoordinateSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4 FlowVelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.4.1 FlowVelocityinRealFormat. . . . . . . . . . . . . . . . . . 124
5.4.2 FlowVelocityinComplexFormat. . . . . . . . . . . . . . . 126
5.5 RMSPressureandAcousticIntensity. . . . . . . . . . . . . . . . . . . 128
5.6 SpecificAcousticImpedance. . . . . . . . . . . . . . . . . . . . . . . . . 132
5.7 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6 AcousticWavesfromSphericalSources. . . . . . . . . . . . . . . . . . . . . 137
6.1 ReviewofPressureandVelocityFormulas
forSphericalWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2 AcousticWavesfromaPulsatingSphere. . . . . . . . . . . . . . . . 139
6.3 AcousticWavesfromaSmallPulsatingSphere. . . . . . . . . . . . 141
6.3.1 Near-FieldSolutionsofaSmallSpherical
Source(kr(cid:1)1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.2 Far-FieldSolutionsofaSmallSpherical
Source(kr(cid:3)1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4 AcousticWavesfromaPointSource. . . . . . . . . . . . . . . . . . . 146
6.4.1 PointSourcesFormulatedwithSourceStrength. . . . . 146
6.4.2 FlowRateasSourceStrength. . . . . . . . . . . . . . . . . . 147
6.4.3 PointSourceinanInfiniteBaffle. . . . . . . . . . . . . . . . 149
6.5 AcousticIntensityandSoundPower. . . .. . . . . . . . . . . . . .. . 150
6.6 ComputerProgram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.7 Project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.8 Objective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.9 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 ResonantCavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.1 1DStandingWavesBetweenTwoWalls. . . . . . . . . . . . . . . . 168
7.2 NaturalFrequenciesandModeShapesinaPipe. . . . . . . . . . . 170
7.3 2DBoundaryConditionsBetweenFourWalls. . . . . . . . . . . . 177
7.3.1 2DStandingWaveSolutionsoftheWave
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.3.2 2DNatureFrequenciesBetweenFourWalls. . . . . . . 179
7.3.3 2DModeShapesBetweenFourWalls. . . . . . . . . . . . 181
7.4 3DBoundaryConditionsofRectangularCavities. . . . . . . . . . 184
7.4.1 3DStandingWaveSolutionsoftheWave
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.4.2 3DNaturalFrequenciesandModeShapes. . . . . . . . . 184
7.5 HomeworkExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192