IntroductiontoSonarTransducerDesign
JohnC.Cochran RaytheonTechnologies,RI,USA(Retired)
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Contents
Preface xvii
1AcousticWavesandRadiation 1
1.1SmallSignals/LinearAcoustics 1
1.1.1Compressibility 2
1.1.2SmallSignals/LinearAcoustics 2
1.1.3RelationshipBetweenAcousticPressureandAcousticDensity 2
1.1.4Condensation 2
1.1.5TimeDerivativeUsingEulerianandLagrangianDescription 3
1.2TheEquationsofContinuity,Motion,andtheWaveEquationinaFluidMedia 3
1.2.1EquationofContinuityinaSingleDimension 3
1.2.2TheForceEquationinaSingleDimension 4
1.2.3TheWaveEquationinaSingleDimension 5
1.2.4GeneralizationoftheWaveEquationtoThreeDimensions 5
1.2.5HelmholtzWaveEquation 6
1.2.6VelocityPotential 6
1.3PlaneWaves 7
1.3.1HarmonicPlaneWaves 7
1.3.2PlaneWavesinanInfiniteMedia 7
1.3.3PlaneWaveAcousticIntensity 8
1.3.4PlaneWaveAcousticImpedance 8
1.4RadiationfromSpheres 8
1.4.1GeneralSolutiontoRadiationfromSpheres 9
1.4.2SphericalWaveAcousticImpedance 11
1.4.3Axis-SymmetricRadiationfromaSphere – theSphericalSource 11
1.4.4TheSimpleSphericalSource 12
1.4.5SourceStrength 12
1.4.6TheGeneralSimpleSource 13
1.4.7AcousticReciprocityandReciprocityFactor 13
1.5RadiationfromSourcesonaCylindricalSurface 14
1.5.1GeneralSolutiontoRadiationfromCylinders 15
1.5.2RadiationfromanInfinitelyLongCylinder 18
1.5.3TheSimple,InfinitelyLongCylindricalSource 19
1.5.4RadiationfromanInfinitelyLongStriponanInfinitelyLongCylinder 20
1.5.5RadiationfromaFiniteSourceonaCylinderwithaPeriodic z Dependence 21
1.5.6RadiationfromaFiniteSourceonaCylinderwithaUniform z Dependence 22
1.5.7TheSimpleCylindricalSource – RadiationfromaFiniteLengthCylinderinanInfinitely LongCylinderBaffle 25
1.6IntegralFormulations 26
1.6.1TheGreen’sFunction 27
1.6.2HelmholtzIntegralFormulations 28
1.6.3FarFieldApproximation 29
1.6.4AnApplicationoftheSimpleSourceIntegralFormulation – RadiationfromaFinite Cylinder 34
1.7LinearApertures 36
1.7.1FarFieldRadiation(Beam)PatternsasaFourierTransformoftheLinearAperture Function – theDirectivityFunction 36
1.7.2ASimpleRectangularApertureFunctionasanExampleofaLinearAperture 38
1.7.3TheTriangularWindowApertureFunctionasaLinearAperture 41
1.7.4TheCosineWindowApertureFunctionasaLinearAperture 43
1.7.5OtherLinearApertures 45
1.7.6TheFarFieldRadiationPatternofaLinearApertureonaCylindricalSurface 45
1.8PlanarApertures 49
1.8.1TheGreen’sFunctionforRadiationfromPlanarAperturesLocatedonaRigidPlane Baffle 49
1.8.2FarFieldRadiationPatternsasaFourierTransformofthePlanarApertureFunction 50
1.8.3TheRectangularPistoninanInfinitePlaneBaffle 52
1.8.4TheCircularPistoninanInfinitePlaneBaffle 54
1.8.5TheFarFieldRadiationPatternofaCircularAnnularRing 59
1.8.6TheEllipticalPistoninanInfinitePlaneBaffle 60
1.8.7ImpactofBoundaryImpedanceonRadiationPatternsfromPlanarApertures 60
1.9DirectivityandDirectivityIndex(DI) 63
1.9.1DefinitionofDirectivityandDirectivityIndex(DI) 65
1.9.2RelationshipBetweenSourceLevelandDirectivityIndex 67
1.9.3TheDirectivityofBaffledvs.UnbaffledSources 68
1.9.4TheDirectivityIndexofaBaffledCircularPiston 68
1.9.5TheDirectivityIndexofaBaffledRectangularPiston 70
1.9.6TheDirectivityIndexofaLineSource 70
1.10ScatteringandDiffraction 72
1.10.1ScatteringandDiffractionfromaRigidCylinder 72
1.10.1.1TheIncidentWave 72
1.10.1.2TheScatteredWave 73
1.10.1.3MatchingtheBoundaryConditionsfortheTotalField 73
1.10.1.4TheScatteredPressureFieldintheFarField 74
1.10.1.5TheTotalPressureField 74
1.10.1.6TheAveragePressureExertedontheCylinderbytheTotalPressureField 74
1.10.2TheDiffractionConstantforaRigidCylinder 76
1.10.3DiffractionConstantforaStriponaRigidCylinder 76
1.10.4DiffractionofaCylinderwithVariableBoundaryAdmittance 77
1.10.4.1TheIncidentWave 77
1.10.4.2TheBoundaryAdmittance 78
1.10.4.3TheScatteredWave 78
1.10.4.4MatchingtheBoundaryConditions 81
1.10.4.5TheBoundaryReflectionCoefficientandtheScatteredField 81
1.10.4.6TheTotalField 82
1.10.4.7TheAveragePressureExertedontheCylinderWithaVariableBoundary Admittance 82
1.10.4.8TheDiffractionConstantforaCylinderwithVariableBoundaryAdmittance 82
1.10.4.9TheTotalDiffractedFieldintheFarField 83
1.10.4.10TheTotalDiffractedFieldattheSurfaceoftheCylinder 83
1.10.5ScatteringandDiffractionfromaRigidSphere 84
1.10.5.1TheIncidentWave 84
1.10.5.2TheScatteredWave 85
1.10.5.3MatchingtheBoundaryConditionsfortheTotalField 85
1.10.5.4TheTotalPressureField 85
1.10.5.5TheScatteredPressureFieldintheFarField 86
1.10.5.6TheAveragePressureExertedontheSpherebythePressureField 86
1.10.6TheDiffractionConstantforaRigidSphere 87
1.10.7ScatteringandDiffractionfromaThinCylindricalRing 87
1.11RadiationImpedance 89
1.11.1IntroductiontoRadiationImpedance 89
1.11.2UnitsofAcousticRadiationImpedance 90
1.11.3WhatitMeanstobe ρc Loaded 90
1.11.4TheRelationshipBetweenResistanceandReactance – TheHilbertTransform 90
1.11.5TheRelationshipBetweenRadiationResistance,Directivity,andDiffraction Constant 92
1.11.6TheRadiationImpedanceofaSphericalRadiator 94
1.11.7TheRadiationImpedanceofaSimpleSourceRadiator 95
1.11.8TheRadiationImpedanceofaCircularPistonRadiatorinaPlaneBaffle 95
1.11.9TheRadiationImpedanceofaCircularPistonRadiatorattheEndofaTube 97
1.11.10TheRadiationImpedanceofaRectangularPistonRadiatorinaPlaneBaffle 98
1.11.11TheRadiationImpedanceofanInfinitelyLongStripRadiatorinaPlaneBaffle 100
1.11.12TheRadiationImpedanceofaCircularAnnularPistonRadiatorinaPlaneBaffle 101
1.11.13TheRadiationImpedanceofanEllipticalPistonRadiatorinaPlaneBaffle 103
1.11.14TheRadiationImpedanceofanInfinitelyLongCylindricalRadiator 103
1.11.15TheRadiationImpedanceofaFiniteCylindricalRadiator 104
1.11.16MutualRadiationImpedance 105
1.11.17TheMutualRadiationImpedanceBetweenSphericalRadiators 106
1.11.18TheMutualRadiationImpedanceBetweenTwoCircularPistonRadiatorsinaPlane Baffle 108
1.11.19TheMutualRadiationImpedanceBetweenTwoSquarePistonRadiatorsinaPlane Baffle 114
1.11.20TheMutualRadiationImpedanceBetweenaCircularPistonandanOuterAnnular Ring 116
1.11.21TheMutualRadiationImpedanceBetweenRectangularorSquarePistonsLocatedona CylindricalBaffle 118
1.11.22TheMutualRadiationImpedanceBetweenBandsonaCylindricalBaffle 124
1.12TransmissionPhenomena 125
1.12.1ReflectionandTransmissionofPlaneWaveswithNormalIncidenceataBoundary 126
1.12.2ReflectionandTransmissionofPlaneWavesObliquelyIncidentataPlane Boundary 129
1.12.2.1Snell’sLaw 130
1.12.2.2ReflectionandTransmissionFactorsforObliquelyIncidentPlaneWaves 131
1.12.2.3Brewster’sAngleortheAngleofZeroReflection 131
1.12.2.4TheCriticalAngleortheAngleofCompleteReflection 132
1.12.2.5EvanescentWaves 132
1.13AbsorptionandAttenuationofSound 133
1.13.1AbsorptionPhenomena 133
1.13.2AbsorptioninSeawater 134 References 135
2Mechanical/AcousticalEquivalentCircuits 137
2.1DifferentFormsofImpedance 138
2.2MechanicalEquivalentCircuits 139
2.2.1TheSimpleMechanicalSystem 139
2.2.1.1ASimpleMechanicalOscillator 139
2.2.1.2PhasorFormoftheSolutionstotheEquationsofMotion 139
2.2.1.3DampedOscillations 140
2.2.1.4ForcedOscillations 141
2.2.1.5CompleteSolutionforaSimpleOscillator 142
2.2.1.6AnalogytoElectricalCircuits 142
2.2.1.7BehavioroftheSteadyState,Forced,MechanicalOscillator 143
2.2.1.8EquivalentCircuitforaSimpleResonatorSystem 144
2.2.2IntroductiontoMobility 145
2.2.2.1MechanicalGenerators 145
2.2.2.2CombiningImpedanceandMobilityElements 145
2.2.2.3ElementsofMobilityandImpedanceAnalogs 147
2.2.2.4ExamplesofMechanicalSystemsDescribedbyMobilityAnalogs 149
2.2.2.5AnExampleofaGyratorConversion 150
2.2.2.6ConvertingfromMobilitytoImpedanceandViceVersa 151
2.3AcousticalEquivalentCircuits 153
2.3.1AcousticCircuitElements 153
2.3.1.1AcousticCompliance – theClosed-EndTube 153
2.3.1.2AcousticMass – theOpen-EndedTube 154
2.3.1.3AcousticResistance 156
2.3.1.4AcousticGenerators 156
2.3.1.5PressureEqualizationOrifices 156
2.3.1.6TheThinAcousticOrifice 159
2.3.1.7TheNarrowSlit 160
2.3.1.8TheAcousticMeshorPerforatedSheet 160
2.3.2AcousticEquivalentCircuits 161
2.3.2.1ExampleofanAcousticSystemDescribedbyanEquivalentCircuit 161
2.3.2.2AnotherExampleofanAcousticEquivalentCircuit – theHelmholtzResonator 161
2.4CombiningMechanicalandAcousticalEquivalentCircuits 163
2.5IntroductiontoTransduction 165
2.5.1TheTransducerasaTwo-PortEquivalentCircuit 165
2.5.2ReciprocalandAnti-ReciprocalTransducers 166
2.5.3TheElectromechanicalCouplingFactor 166
2.5.4ElectromechanicalTransformation 167
2.5.5Transmitters 167
2.5.6Receivers 169
2.5.7RelationshipBetweenTransmitandReceiveCharacteristics 170 References 171
3WavesinSolidMedia 173
3.1WavesinHomogeneous,Isotropic,Elastic,SolidMedia 173
3.1.1TheComponentsofStress 173
3.1.2TheEquationsofMotion 174
3.1.3TheComponentsofStrain 175
3.1.4TheRelationshipBetweenStressandStrain – TheConstitutiveEquations 177
3.1.4.1Hooke’sLaw – TensorForm 177
3.1.4.2Hooke’sLaw – MatrixForm 179
3.1.4.3TheDifferencesBetweenTensorandMatrixFormsoftheConstitutiveEquations 180
3.1.4.4Lame’sConstants 182
3.1.4.5Stiffnessvs.ComplianceMatrices 183
3.1.4.6ModifiedConstitutiveEquations 184
3.1.5AcousticWavesinIsotropicSolids 184
3.1.5.1TheAcousticWaveEquationforIsotropicSolids 184
3.1.5.2WavesofDilatationandDistortion 184
3.1.5.3AcousticPlaneWavesinIsotropicSolids 186
3.1.6LongitudinalWavesinBars 186
3.1.6.1VibrationsinaBarwithClampedBoundaryConditions 188
3.1.6.2VibrationsinaBarwithFreeBoundaryConditions 189
3.1.6.3EquivalentCircuitRepresentationforLongitudinalVibrationsinaBarwithArbitrary BoundaryConditions 190
3.1.6.4ATwo-PortRepresentationofLongitudinalVibrationsWithinaBar 192
3.1.6.5ImpactofDifferentLoadImpedancesontheLongitudinalVibrationsWithinaBar 193
3.1.6.6EquivalentCircuitRepresentationforaMass-LoadedBarwithOneFreeEnd 194
3.1.6.7EquivalentCircuitRepresentationforaMass-loadedBarwithOneEndClamped 196
3.1.6.8LumpedParameterEquivalentCircuitforaLongitudinalResonator 198
3.1.6.9TheEffectiveMassofaSpring 200
3.1.7EquivalentCircuitRepresentationsforSolidElements 202
3.1.7.1LongitudinalVibrationsWithinaHollowCylinder 202
3.1.7.2LongitudinalVibrationsWithinaConicalSection 204
3.1.7.3LongitudinalVibrationsWithinanExponentialSection 206
3.2Piezo-electricityandPiezo-electricCeramicMaterials 208
3.2.1TheNatureofPiezo-electricity 208
3.2.2Piezo-electricCeramicMaterials 211
3.2.3ThePiezo-electricCeramicConstitutiveEquations 212
3.2.4TheMeaningofthePiezo-electricCoefficients 214
3.2.5Piezo-electric,Elastic,andDielectricCoefficientNomenclature 215
3.2.6Piezo-electricCeramicMaterialProperties 216
3.2.7TheElectromechanicalCouplingCoefficient 219
3.2.8FurtherObservationsonthePiezo-electricConstitutiveEquations 220
3.3WavesinNon-Homogenous,Piezo-electricMedia 222
3.3.1VibrationsinRodsandDisks 223
3.3.1.1ConstitutiveEquations 223
3.3.1.2EquationsofMotionandStraininCylindricalCoordinates 224
3.3.1.3RadialModeVibrationsinThinDisks 224
3.3.1.4ThicknessModeVibrationsinThinDisks 228
3.3.1.5TheRelationshipBetweenDielectricConstantandCouplingFactorforVibrationsin ThinDisks 233
3.3.1.6LengthLongitudinalModeVibrationsinLong,ThinRodsorBars 235
3.3.1.7RadialModeVibrationsinLong,ThinRodsorBars 237
3.3.1.8TheRelationshipBetweenDielectricConstantandCouplingFactorforVibrationsin Long,ThinRods 240
3.3.1.9FrequencyConstantsforVibrationsinRodsandDisks 241
3.3.2VibrationsinPiezo-electricPlatesandParallelepipeds 242
3.3.2.1EquationsofMotionandStraininRectangularCoordinates 243
3.3.2.2LengthExpanderBarwithElectricFieldPerpendiculartoWidth – The31Mode Bar 244
3.3.2.3LengthExpanderBarwithElectricFieldParalleltoWidth – The33ModeBar 248
3.3.2.4ThicknessModeVibrationsinThinPiezo-electricPlateswiththeElectricFieldParallelto theThickness 250
3.3.2.5CoupledModeVibrationsinParallelepipedswithOneLargeDimension 254
3.3.2.6CoupledModeVibrationsinThinPiezo-electricPlateswiththeElectricField PerpendiculartotheThickness 256
3.3.2.7CoupledModeVibrationsinThinPiezo-electricPlateswiththeElectricFieldParallelto theWidth 258
3.3.2.8CoupledModeVibrationsinParallelepipedswithArbitraryDimensions 259
3.3.3VibrationsinPiezo-electricCeramicCylinders 261
3.3.3.1LongitudinalVibrationsinAxiallyPolarized,Piezo-ceramicCylinders 263
3.3.3.2LongitudinalVibrationsinRadiallyPolarized,Piezo-ceramicCylinders 272
3.3.3.3RadialVibrationsinRadiallyPolarized,Piezo-ceramicCylinders 281
3.3.3.4LongitudinalVibrationsinCircumferentiallyPolarized,Segmented,Piezo-ceramic Cylinders 285
3.3.3.5RadialVibrationsinCircumferentiallyPolarized,Segmented,Piezo-ceramic Cylinders 294
3.3.4VibrationsinRadiallyPolarizedSphericalShells 297
3.3.4.1BoundaryConditions 297
3.3.4.2ConstitutiveEquations 298
3.3.4.3TheEquationsofMotionandStrain 298
3.3.4.4KineticEnergyandEquivalentMass 299
3.3.4.5InternalEnergy 299
3.3.4.6ElectromechanicalCouplingCoefficient 300
3.3.4.7In-AirResonanceFrequencyofaSphericalShell 300
3.3.4.8EquivalentCircuitModelforaRadiallyPolarizedSphericalShell 300 References 303
4SonarProjectors 305
4.1ToolsforUnderwaterSonarProjectorDesign 305
4.1.1AssemblingCircuitElements 305
4.1.1.1Two-PortRepresentationsforNon-PiezoelectricComponents 305
4.1.1.2SeriesCombinationofTwo-PortNetworksforNon-PiezoelectricComponents 307
4.1.1.3ParallelCombinationsofTwo-PortNetworksforNon-PiezoelectricComponents 307
4.1.1.4Two-PortRepresentationsofPiezoelectricComponents 308
4.1.1.5CascadedCombinationsofTwo-PortNetworksforPiezoelectricComponents 309
4.1.1.6LadderNetworkAnalysis 311
4.1.2HowtoSpecifyaProjector 312
4.2SpecificApplicationsinUnderwaterSonarProjectorDesign 313
4.2.1FrequencyRangesforDifferentTypesofProjectors 313
4.2.2SphericalProjectors 314
4.2.2.1TheLossless,Air-BackedSphericalProjector 314
4.2.2.2TheLossy,Air-BackedSphericalProjector 320
4.2.2.3Fluid-FilledSphericalProjectors 321
4.2.3TheRadiallyPolarizedCylindricalProjector 323
4.2.3.1TheRadiallyPolarized,Air-BackedCylindricalProjector 323
4.2.3.2PrestressingforIncreasedPower-HandlingCapability 328
4.2.3.3TheRadiallyPolarized,Fluid-FilledCylindricalProjector 330
4.2.3.4TheRadiallyPolarized,SquirterProjector 332
4.2.3.5TheRadiallyPolarized,Free-FloodedCylindricalProjector 339
4.2.3.6TheFree-FloodedCylindricalProjectorwithaReflectorPlate 342
4.2.4CircumferentiallyPolarizedCylindricalProjectors – TheBarrelStaveProjector 343
4.2.4.1TheCircumferentiallyPolarized,Air-BackedCylindricalProjector 343
4.2.4.2TheCircumferentiallyPolarized,Free-FloodedCylindricalProjector 346
4.2.4.3TheCircumferentiallyPolarizedStripedCylindricalProjector 348
4.2.5TheTonpilzTransducer 352
4.2.5.1TheEndMass-LoadedTonpilzTransducer 353
4.2.5.2TheNodallyMountedTonpilzTransducer 355
4.2.6TheFlexuralDiskTransducer 355
4.2.6.1TheTrilaminarFlexuralDiskTransducer 357
4.2.6.2TheBilaminarFlexuralDiskTransducer 375
4.2.7FlatOvalFlextensionalProjectors 385
4.2.8SlottedCylinderProjectors 387
4.2.8.1GeometryandDescription 388
4.2.8.2WallThickness,Radii,andTaperFactors 390
4.2.8.3NeutralAxis 391
4.2.8.4DisplacementProfiles 392
4.2.8.5StressandStrainintheSCP 397
4.2.8.6KineticEnergyandEquivalentMass 398
4.2.8.7ConstitutiveEquationsforthePiezoceramicComponent 398
4.2.8.8VoltageAcrossElectrodesandDielectricDisplacement 398
4.2.8.9InternalEnergy 399
4.2.8.10FlexuralStiffness 400
4.2.8.11In-AirResonanceFrequency 400
4.2.8.12EffectiveElectromechanicalCouplingFactor, keff 401
4.2.8.13In-waterPerformance 401
4.2.8.14AnSCPExample 407
4.2.9MovingCoilTransducers 407
4.2.10TheLine-in-ConeTransducer 412
4.2.11Quarter-WavelengthResonators 415
4.2.12DiskProjectors 418
4.2.13TheHigh-FrequencyLineProjector 420
4.3SpecialTopicsinUnderwaterSonarProjectorDesign 422
4.3.1TechniquesforIncreasingBandwidth 422
4.3.1.1BandwidthIncreaseswithCoupling 422
4.3.1.2MechanicalTuningwithMatchingLayers 423
4.3.2PowerLimitationsinSonarProjectors 424
4.3.2.1ElectricFieldLimitations 424
4.3.2.2LossTangentLimitations 425
4.3.2.3StressLimitations 426
4.3.2.4ThermalLimitations 427
4.3.2.5CavitationLimitations 433 References 436
5SonarHydrophones 439
5.1ElementsofSonarHydrophoneDesign 439
5.1.1AnEquivalentCircuitforaSonarHydrophone 440
5.1.2TheImportanceofthePiezo-Ceramic g Constant 442
5.1.3AnEquivalentCircuitforaDielectricallyLossySonarHydrophone 442
5.1.4TheEffectofCableCapacitance 443
5.1.5TypicalResponseofaSonarHydrophone 444
5.2AnalysisofNoiseinHydrophone/PreamplifierSystems 445
5.2.1AmbientNoise 445
5.2.2TypesofEquivalentNoiseSources 446
5.2.3AmbientNoiseCouplingintoaSensor 447
5.2.4SensorSelf-Noise 448
5.2.5SensorSignaltoNoiseRatio 450
5.2.6PreamplifierNoise 450
5.2.7CombinedSensorandPreampSystemNoise,theEquivalentNoisePressure 452
5.2.8TheEquivalentNoisePressureatLowFrequencies 453
5.2.9ComparisonofSensorNoisewithAmbientNoiseExample 455
5.2.10HydrophoneFigureofMerit 456
5.2.11TheEffectofCableCapacitance – InsertionLoss 457
5.3SpecificApplicationsinUnderwaterSonarHydrophoneDesign 458
5.3.1UnidirectionalHydrophone 459
5.3.1.1BoundaryConditions 460
5.3.1.2EquationofMotionandStrain 460
5.3.1.3ConstitutiveEquations 460
5.3.1.4OpenCircuitVoltageSensitivity 460
5.3.2HydrostaticHydrophone 461
5.3.3SphericalHydrophone 462
5.3.3.1BoundaryConditions 463
5.3.3.2ConstitutiveEquations 464
5.3.3.3TheEquationsofMotionandStrain 464
5.3.3.4StressProfileinaSphericalHydrophone 464
5.3.3.5TheOpenCircuitSensitivityoftheSphericalHydrophone 465
5.3.3.6SphericalHydrophoneDepthLimitations 466
5.3.3.7TheEffectofaFillFluidonHydrophonePerformance 467
5.3.4CylindricalHydrophones 468
5.3.4.1TheRadiallyPolarizedCylindricalHydrophone 470
5.3.4.2TheCircumferentiallyPolarizedCylindricalHydrophone 488
5.3.4.3TheAxiallyPolarizedCylindricalHydrophone 493
5.3.5PVDFPolymerHydrophones 496
References 497
Appendix 499 Index 509
Preface
Thistextistheresultofmyeffortsovertheyearstounderstandthebroadsubjectofunderwater electroacoustictransducerdesign.Tofullyunderstandunderwatertransducers,onemustunderstandmanydifferentaspectsofphysics,electrical,andmechanicalengineering.Itisthebroad natureofacoustictransductionthatIenjoyverymuch.
Istartedworkingintheunderwatertransducerbusinessin1983.Ijoinedasmallcompany whichspecializedinthedesignandmanufactureo funderwateracoustictransducers.Atthetime, Ihadnoideawhatatransducerwas.Iquicklyfoundthatyouhadtobeajack-of-all-tradesin ordertobesuccessfulatmakingtransducers.The widerangeofexpertiserequiredtofullyunderstandunderwatertransducerdesignwasvery attractivetome.Iwasnotonetobepigeonholed intoanyoneparticularareaofengineeringdiscipline.Iembracedtheopportunitytobewhat Iconsideratrueengineertobe – anindividualwhoemploysthebasicprinciplesofphysics andchemistrytosolvepracticalproblemsandwhocanimplementthesesolutionsintoamanufacturableproduct.
Unfortunately,theprocessof “cominguptospeed” intransducerdesignismadeverydifficultby thelackofspecifictextsthataddressthesubject.Therearetextsthataddressthemechanicsof piezo-electricmaterialsandtextsonacoustictheory,butthereareveryfewtextsthataddress thebroadrangeoftopicsassociatedwithunderwatertransducerdesign.Asaresult,Iwasconstantlybombardedwith “rulesofthumb” thathadtheirbasisinsoundphysicalprinciplesbut forwhichnoonecouldaccount.Notbeingonetoacceptthe “rulesofthumb” onfaithalone, Ihaveendeavoredtounderstandthebasisforthetheoriesthatareapplicabletounderwatertransducerdesign.ItisthesebasicprinciplesthatIhopetodocumentinthistext.
Iconsiderthetexttobeanintroductorytextinunderwateracoustictransducerdesign.Thetext walksthroughthedevelopmentofvarioustheoriesstartingfromthefirstprinciples.Insomecases, themathematicaldevelopmentmayleadsomeonetothinkthatIshouldhavejumpedtotheanswer sooner.However,itisforthebeginnerinthisfieldthatIhavewrittenthisbook.Thebeginner shouldfeelthathecanfollowmathematicaldevelopmentcompletely.Ihaveputmanyoftheintermediatemathematicalstepsintothistext.
Thoughwrittenforthebeginnerinthisfield,thetextisalsofortheadvancedstudentorpracticing engineer.Themathematicaldevelopmentisfairlythoroughandrequiressomeexperiencewith advancedmathematicalfunctions(suchasBessel’sfunctions)inordertogetthemostoutofit.
Thetextisdividedintofivechapters.Thefirstchapterexploresthephysicsoftheacoustic mediumoutsideofthetransducer.Sincethepurposeofatransduceristogeneratesoundinthe water,wemustunderstandtheparametersthatimpactthedesignofthetransduceranditsability toproduceacousticpower.
Chaptertwobeginsthedevelopmentoftransducertheorybydevelopingequivalentcircuitsfor simplemechanicalandacousticalsystems.Theseprinciplescansometimesbeappliedtotransducer designbutmoregenerallyleadtoaphysicalunderstandingofhowamechanical/acousticaltransducerworks.
Chapterthreeaddressesacousticwavesinsolids.Inthischapter,wespecificallydevelopthetheoryforsoundpropagationinsolidsthatwillultimatelyimpactthedesignofthetransducer. Atransducerisasoliddevicethroughwhichacousticenergyflows.Wemustunderstandthepropagationofacousticenergyinsolidsinordertodesigntransducers.Thechapterstartsoffwithacousticwavesinnon-piezo-electricsolidsandthenmovesintoathoroughdiscussionofwavesinpiezoelectricsolids.Piezo-electricityisreviewedtothepointthatitisapplicabletotransducerdesign.
Chapterfourdiscussesprojectors.Thischapterbringstogetherelementsfromthefirstthree chapters.Toolsandlimitationstoprojectordesignarereviewed.
Chapterfivediscusseshydrophones.Sensorself-noiseanditsimpactonsystemdesignarethoroughlydiscussedinthischapter.
Enjoy. JohnC.Cochran,PhD
AcousticWavesandRadiation
Thischapterisdevotedtoissuesaffectingtheacousticenvironmentinwhichsonartransducersare expectedtoperform.Wewillfirstreviewthelineartheoryofacoustics.WewillthenformtheequationsofforceandmotionandderivetheHelmholtzwaveequation.TheHelmholtzwaveequation willformthebasisformostoftheanalysesconductedinthischapter.Thewaveequationissolvedin theseparablerectangular,spherical,andcylindricalcoordinatesystems.Fromthesesolutions,we derivetheradiationpatternsforsphericalradiatorsandradiatorslocatedoncylindricalbodies.The Helmholtzintegralformulationswillthenbereviewedasapreludetoourdiscussionintothefar fieldbeampatterns(BPs)ofplanarandlinearapertures.Wewillalsodefinetheunderlyingassumptionsinmakingthenearfield/farfielddistinctions.Theconceptofdirectivityanddirectivityindex (DI)isreviewednext.Inthenextsection,wewillreviewissuesaffectingscatteringanddiffraction. Understandingdiffractionisimportantinsystemdesignandespeciallyhydrophonedesign.Radiationimpedance,bothselfandmutual,isthenreviewed.Thediscussionofthemutualradiation impedancebetweenradiatorsisascloseaswewillgettoadiscussiononarrays.Finally,wewill reviewtransmissionphenomenaandtheabsorptionandattenuationofsound.Itiswiththecontentsofthischapterthatwehavealwaysstartedtransducerdesign.Wemustunderstandhowtransducersprojectand/orreceiveacousticenergybeforewecanstartdesigningtheinternalpartsofthe transducer.
1.1SmallSignals/LinearAcoustics
Inthissection,wewillbrieflyreviewthefoundationsoflinearacoustics.Theprocessoflinearizing theequationsofmotionandcontinuityiscentraltoourbeingabletodeveloptheHelmholtzwave equation.Thisequationwillformthebasicstartingpointformanyoftheacousticproblemsdealt withinthischapter.
Inordertolinearizetheequationsofmotionandcontinuity,wemustmakethebasicassumption thatacousticsignalsaresmallsignals.Theacousticpressureanddensityaresmallquantitiesrelativetothemacroscopicproperties.Intheprocessoflinearization,weassumethattheproductof twosmallquantitiescanbeignored.Wedon’tmakeanyclaimsastothevalidityofthisapproach. Theoverwhelmingamountofdatacollectedovermanyyearsbymanyengineersandresearchers substantiatesthelinearizationprocess.
1.1.1Compressibility
Thestartingpointforourdiscussionisaninherentpropertycalledcompressibility.Compressibility isthechangeinvolumeinfluidduetoachangeinpressureortemperature.Theprocessofcompressingafluidcanbeeitheradiabaticorisothermal.Itisnotessentialtoourdiscussionwhich processistakingplaceaslongasweknowthatfluidcompressiontakesplaceinacousticprocesses andtheabilitytocompressafluidisameasurablepropertyofthefluid.Thevolume, V,occupiedby n molesoffluidwithmolecularweight M anddensity ρ is:
Twopartialderivativesareusedtodescribethestateofthefluid.Thesearetheisothermal compressibilityandthecoefficientofthermalexpansion:
1.1.2SmallSignals/LinearAcoustics
Anacousticwaveinfluidmediamanifestsitselfasasmallchangeinpressure,density,and temperature.Weassumethatwithanacousticsignal,thefollowingchangesoccur:
P, ρ, and T aremacroscopicambientproperties,whichareinvariantinthespatialandtemporal realms.Thesmallsignalproperties p, δ, and τ arelocallyandtemporallyvariant.
Notethelinearbehavioroftheterms;therearenopowerterms.Thisispartofthelinearacoustic assumption.Itisveryaccurateforsmallsignals.Mostofourworkiswithlinearacoustics.Nonlinearacousticsisnotdiscussedinthistext.
1.1.3RelationshipBetweenAcousticPressureandAcousticDensity
Considerasmallchange(acoustic)inpressureofafluidmedia.Therewillbeacorrespondingly smallchangeinlocalizedfluiddensity.UsingEquation1.1-3:
1.1.4Condensation
Manyacousticiansusethetermcondensationtodescribesmalloracousticchangesindensity:
1.2TheEquationsofContinuity,Motion,andtheWaveEquationinaFluidMedia 3
1.1.5TimeDerivativeUsingEulerianandLagrangianDescription
Therearetwodifferentnotationsusedforderivativeswithrespecttotime.TheyaretheEulerand Lagrangenotations[1].
Partialderivativeswithrespecttotimeareusedtoindicatethatwearetalkingaboutaregionfixed inspace.ThisistheEulernotation;thepartialderivativeoffluidproperty f withrespecttotime representsthechangein f atafixedpointinspace.Thepartialderivativeoffluidproperty f with respecttotimeincludesthetimerateofchangeof f withinthefluidandalsothechangein f at x becausethefluidismovingpast.Anexamplemightbeafluid,suchaswater,whichflowspasta point,butwhichisgraduallychangedtooil.So,forexample,thedensityischangingwithrespectto timebecauseofcompressionatpoint x,butitwillalsobechangingbecausethefluidgradually changesfromwatertooil.IntheEulernotation,theobserverhasblindersonandisonlylooking atafixedregionofspace.
Theothernotation,alsocalledtheLagrangenotation,usesthetotalderivativeof f withrespectto timetorepresentthechangeinfluidproperty f asitmovesthroughpoint x attime t.IntheLagrange notation,theobserverismovingwiththefluidandobservesthechangeinfluidpropertiesashe/she crossesthedesignatedregion.Citingourpreviousexample,ifwewereinthewaterphaseofthe fluid,wewouldneverknowthatitchangesovertooil.Wewouldsimplyobservehowthedensity ofthefluidchangesasweflowalongwiththefluid.Thetotalderivativehasitsadvantages.For example,thetotalderivativeofthefluidvelocitywithrespecttotimeisthetrueaccelerationof aportionofthefluidandisproportionaltothenetforceactingonthisfluid.Thetotalderivative ofthefluiddensitywithrespecttotimemeasureswhetherthefluidisexpandingornotasitmoves.
Now,todeterminetherelationshipbetweenLagrangian(totalderivative)andEulerian(partial derivatives)notation,wecomparethepropertiesoffluidproperty f at x,t withitsvalueat x+dx, t + dt:
u representsthevelocityofthefluidinthe x direction,atpoint x.Weshalllaterusethetotal derivativeinthreedimensions.Thepreviousequationcanbegeneralizedtothreedimensionsas:
where u representsthefluidvelocityvectorand ∇ istheDeloperator(AppendixA.1.5).
1.2TheEquationsofContinuity,Motion,andtheWaveEquation
1.2.1EquationofContinuityinaSingleDimension
Theequationofcontinuityrepresentsamassbalanceforaregionoffluid.Fluidflowsintothe region,outoftheregion,oriscreatedwithintheregion.Forthepurposesofthistext,weignore fluidcreation.Thenetincreaseinmasswithinaregionmanifestsitselfasanincreaseinfluiddensitywithtime.Considerasliceoffluidofarea A andwidth Δx.Themassfluxoffluidat x isgivenby Auρ.Themassbalancebecomes:
Divideby Δx andtakethelimitas Δx goesto0:
Thisistheequationofcontinuity.Notethatitisanonlinearequationinthatitinvolvestheproductoftheterms ux and ρ .Wecanlinearizeitbynotingthatthedensitycanbemadeupfroman ambientbulktermthatisspatiallyandtemporallyinvariantandasmallsignaloracousticterm. Hence,referringbacktoEquations1.1-3 and1.1-6,thedensitycanbedefinedas:
SubstitutingthisintoEquation1.2-2,weobtainthefollowing:
Thisisstillanonlinearequationthatinvolvestheproductofsmallterms.Ourassumptionhas beenthat u, ρ,and s aresmallterms.Theproductofsmalltermsisevensmaller.Hence,ifwedrop thesmaller,secondorderterms,weobtainthelinearizedequationofcontinuity:
Anotherwaytoviewtheequationofcontinuity,whichincorporatestheacousticpressurerather thantheacousticdensity,andwhichwillbeusefulwhenweformulatethewaveequation,isas follows.CombiningEquations1.1-6and1.2-5,weobtain:
1.2.2TheForceEquationinaSingleDimension
Wehavejustfinisheddescribingamassbalanceonaregionoffluid.Wecalledthismassbalancethe equationofcontinuity.Inthissection,wedescribeasimpleforcebalanceonaregionoffluid. Influidmechanics,wenormallywritetheequationdescribingthebalanceofmomentumflux. Thisequationresultsintheequationofmotion.However,inthiscase,wewillwriteanequation describingthebalanceofforcesonaregionoffluid.Thisapproachleadsdirectlytotheacoustic waveequation.Forthisdevelopment,wedonotincludetheeffectsofviscosityorotherloss mechanisms.Weincludetheseeffectsinadifferentwaylaterinthistext.
Consideragainasliceoffluidofcross-sectionalarea A andwidth Δx.Theforceactingonthefluid at x is AP.Thenetforceactingonthesliceoffluidisequaltotheproductofthemassofthefluidand theaccelerationofthefluid:
ux isthevelocityofthefluidinthe x direction.Takingthelimitas Δx goesto0:
Thisisourforcebalanceequation.Weusethetotalderivativeforthefluidaccelerationbecause thisisthetrueaccelerationofthefluidandisproportionaltothenetforceactingonthefluid.
ConvertingtotheEulernotationandbyusingthenotationofEquation1.1-7 andEquation1.1-3,we obtain:
Onceagain,weconsideronlythefirstordertermsandweareleftwithourlinearizedforce equation:
1.2.3TheWaveEquationinaSingleDimension
WecaneliminateanyreferencetothevelocityinEquations1.2-10 and1.2-6bytakingthederivative ofEquation1.2-10withrespectto x andthederivativeofEquation1.2-6withrespecttotime(t)and combining:
Thisthenisthefirstorderequationofacousticmotionorthewaveequation.Aswewilllatersee, wecandefinethespeedofwavepropagationorspeedofsoundas:
1.2.4GeneralizationoftheWaveEquationtoThreeDimensions
Theequationofcontinuitycanbegeneralizedtothreedimensions:
Similarly,theforcebalanceequationbecomes:
Retainingthefirstorderterms,wehave:
Combiningthefirstorderquantitiesinthetwoequations,weobtainthewaveequationinthree dimensions.WetakethetimederivativeofEquation1.2-13 andthedivergenceofEquation1.2-15. Bynotingthatthedivergenceofthetimederivativeof u isequaltothetimederivativeofthedivergenceof u,weobtainthewaveequation:
The ∇2f operationisdefinedastheLaplacianofthescalarfield f.Thisequationappliestoany separablecoordinatesystem.Thereareelevenseparablecoordinatesystemsforthewaveequation. Wewilldealwiththree:rectangular,cylindrical,andsphericalcoordinatesystems.TheLaplace operator, ∇2,isgivenasfollowsforthesecoordinatesystems:
InRectangularorCartesiancoordinates x , y, z
InCylindricalcoordinates ρ, ϕ, z
InSphericalcoordinates
Wenoteimmediatelythattheradialdependenceforthesphericalcaseisinverselyproportional to r2,whiletheradialdependenceforthecylindricalcaseisinverselyproportionalto ρ.Hence,we canexpect,andasweshallseelater,thespreadinglossforthesphericalcaseis SLs =20log(r/1m) andthespreadinglossforthecylindricalcaseis SLc =10log(ρ/1m).
1.2.5HelmholtzWaveEquation
Normally,inacoustics,wedealwithharmonicorsinusoidalwaveforms.Withthisinmind,wecan makeanenormoussimplificationinthewaveequation.Wefirstallowthepressure(p),velocity(u), density(ρ),andallotheracousticpropertiestohavethefollowinggeneralproperty:
IfwesubstitutethisequationintoEquation1.2-16,weobtain:
ThisistheHelmholtzwaveequation. k iscommonlyreferredtoasthewavenumberandis givenas:
1.2.6VelocityPotential
Byusingthepotentialfunctioncalledthevelocitypotential,wecanconvertthedifferentialoperationforthevector u toascalaroperation.Thevelocityvectorcanbedescribedasthegradientofa scalarvelocitypotential.
1.3PlaneWaves 7
Substitutingthisintotheequationsforvelocity,pressure,andthewaveequation,weobtain:
ThevelocitypotentialcanalsobewritteninthecontextoftheHelmholtzwaveequation.Assumingasinusoidaldependenceontime,wehave:
ThevelocitypotentialisalsoasolutiontotheHelmholtzwaveequation.
1.3PlaneWaves
1.3.1HarmonicPlaneWaves
WeseekasolutiontoEquation1.2-19 inrectangularcoordinatesandinonedimension.Wewill firstconsiderthat k isarealnumber.Itisthenclearthatasolutiontotheequationis:
c1 and c2 areconstantstobedetermined.FromEquation1.2-10,wecanobtainthevelocityforthe planewave:
a x representsaunitvectorinthe x direction.
Thefirsttermineachequationrepresentsthepressureandparticlevelocityforawavetraveling inthe+x direction.Conversely,thesecondtermrepresentsawavetravelinginthe x direction. Theconstants c1 and c2 dependontheboundaryconditionsfortheparticularproblem.Sinceboth waveseachformasolutiontothewaveequation,wehaveconsideredthemhere.However,inmany cases,weconsiderawavetravelinginonlyonedirection.Weshallconsiderthisnext.
1.3.2PlaneWavesinanInfiniteMedia
Ifweconsiderasinglewavetravelinginaninfinitemedium,thenwecanwriteforthepressureand velocityatanypoint x andtime t,
where P representsthemagnitudeofthepressure.Wecanthereforeseethatthemagnitudeofthe velocityisrelatedtothemagnitudeofthepressureas:
Forplanewavespropagatinginanarbitrarydirection r,theexpressionforthepressureofthe planewaveisgivenby:
wherethewavenumberhasbeenvectorizedtoreflectthedirectionalcharacteristicsofthewave propagation:
andwherethepositionvector r isdefinedas:
1.3.3PlaneWaveAcousticIntensity
Wewanttofindthepowerperunitareaforourtravelingplanewave.Itcanbeshownfromenergy balanceconsiderationsthattheintensityoftheacousticwaveisgivenby:
wheretheasterix(∗)denotesthecomplexconjugate.Notethattheintensityisavector.Themagnitudeoftheintensityisgivenas:
Notetheunitsoftheacoustic,planewaveintensity:
1.3.4PlaneWaveAcousticImpedance
Theratioofacousticpressuretoparticlespeedisdefinedasthe specificacousticimpedance.For planewaves,thisis:
Theproductoffluiddensityandspeedofsoundisalsoknownasthe characteristicacousticimpedanceofthemedium andisaphysicalpropertyofthemedium.Thisproductreoccurscommonlyin acoustics.
1.4RadiationfromSpheres
Ourgoalinthissectionistosolvethegeneralproblemofradiationfromsphericalsources.The waveequationwillbesolvedbyusingthemethodofseparationofvariables.Thepressurefield forasphericalwavewillbederivedandthespecificacousticimpedanceofasphericalwavewill
1.4RadiationfromSpheres 9
beintroduced.Thegeometryusedthroughoutthetextfor asphericalsourceisshowninFigure1.4-1.
Theradiationfieldsofmanytransducerscanbeapproximatelymodeledafterageneralsphericalsource.Inthe nextsection,wesolvethegeneralizedwaveequationin sphericalcoordinates.Thisprovidesuswithafoundation forthensimplifyingtheresultingsolutiontodescribethe radiatedpressureandvelocityfieldsforauniformly pulsedsphericalsourceinanunboundedmedium.We considerradiationinanunboundedmediumbecause wearegenerallyinterestedindescribingtheperformance oftransducersinanoceanenvironment.
Inthenextsection,wedevelopthespecificacoustic impedanceforawaveemanatingfromasphericalsource. Aswillbeshownlater,astudyofthisimpedancecangive usagreatdealofinsightintotherelationshipbetween transducersize,efficiency,andtheabilitytoradiatepower.
Figure1.4-1 Sphericalcoordinate systemforradiationfromspheres.
Lastly,wedevelopaveryusefulsolutionforradiationfromanaxis-symmetricsphericalsource andweintroducetheconceptofthesimplesource.Thesimplesourceisaconceptthatisuseda greatdealintransducerdesigncircles.Thesimplesourceisdefinedasageneralsphericalsource wherethesizeofthesphericalsourceissmallrelativetoawavelength.Thepressurefieldemanatingfrommanylow-frequencytransducerscanbepredictedusingthesimplesourceapproximation.
1.4.1GeneralSolutiontoRadiationfromSpheres
Letusconsiderthewaveequationinsphericalcoordinates:
Weseekasolutionoftheform:
InsertingEquation1.4-2 into1.4-1andbydividingby R
ejωt,weobtain:
Multiplyingby r2sin2θ :
Thelasttermisafunctionof ϕ onlyandthereforemustequalaconstant.Settingthistermequal to n2,wehave:
Thegeneralsolutionis:
where c1 and c2 areto-be-determinedconstantsandwhere n =0,1,2, sothatthesolutioniscontinuousat0and2π .Substituting n 2 intoEquation1.4-4 anddividingbysin2θ :
Thefirstandfourthtermsdependon r only.Therefore,wecansetthesumofthesetermsequalto aconstant.Wewillfinditconvenienttosettheseequalto m(m +1)where m =0,1,2…
Letting ξ = kr,weobtain:
ThisisthesphericalBesselequation;thesolutionsarethesphericalHankelfunctionsofthefirst andsecondkindsoforder m [2].However,theHankelfunctionofthesecondkindrepresentsan inwardtravelingwaveandthusanunrealisticwaveforradiationfromasphericalsourceinan unboundedmedium.Hence,wedropthissolutionanduseonlythesphericalHankelfunction ofthefirstkind(hm):
jm isthesphericalBesselfunctionofthefirstkindoforder m and nm isthesphericalBesselfunction ofthesecondkind,sometimesreferredtoasWeber’sfunctionorNeumann’sfunction.TheremainingtermsfromEquation1.4-7 dependon θ only.Substituting m(m +1)forthetermsthatdependon r only,weobtain:
Ifwesubstitute η =cos(θ )intoEquation1.4-11,weobtaintheLegendreequation:
Thisequationhasthesolution:
P n m istheLegendrefunctionoforder m anddegree n. m mustberealandpositiveinorderforthe Legendrefunctiontobefiniteeverywhere.Thecompletesolutionforageneralsphericalwavecan bewrittenas:
wherewehavereplacedtheconstants c1 and c2 withmoregeneralconstants Amn and Bmn.
