The Physics of Vibrations and Waves, 6th Edition PDF
Preview The Physics of Vibrations and Waves, 6th Edition
THE PHYSICS OF VIBRATIONS AND WAVES Sixth Edition H. J. Pain Formerly of Department of Physics, Imperial College of Science and Technology, London, UK Copyright#2005 JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester, WestSussexPO198SQ,England Telephone (+44)1243779777 Email(forordersandcustomerserviceenquiries):[email protected] VisitourHomePageonwww.wileyeurope.comorwww.wiley.com AllRightsReserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystemor transmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanning orotherwise,exceptunderthetermsoftheCopyright,DesignsandPatentsAct1988orunderthe termsofalicenceissuedbytheCopyrightLicensingAgencyLtd,90TottenhamCourtRoad, LondonW1T4LP,UK,withoutthepermissioninwritingofthePublisher.Requeststothe PublishershouldbeaddressedtothePermissionsDepartment,JohnWiley&SonsLtd, TheAtrium,SouthernGate,Chichester,WestSussexPO198SQ,England,oremailedto [email protected],orfaxedto(+44)1243770620. Thispublicationisdesignedtoprovideaccurateandauthoritativeinformationinregardtothesubject mattercovered.ItissoldontheunderstandingthatthePublisherisnotengagedinrendering professionalservices.Ifprofessionaladviceorotherexpertassistanceisrequired,theservices ofacompetentprofessionalshouldbesought. OtherWileyEditorialOffices JohnWiley&SonsInc.,111RiverStreet,Hoboken,NJ07030,USA Jossey-Bass,989MarketStreet,SanFrancisco,CA94103-1741,USA Wiley-VCHVerlagGmbH,Boschstr.12,D-69469Weinheim,Germany JohnWiley&SonsAustraliaLtd,33ParkRoad,Milton,Queensland4064,Australia JohnWiley&Sons(Asia)PteLtd,2ClementiLoop#02-01,JinXingDistripark,Singapore129809 JohnWiley&SonsCanadaLtd,22WorcesterRoad,Etobicoke,Ontario,CanadaM9W1L1 Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsin printmaynotbeavailableinelectronicbooks. LibraryofCongressCataloging-in-PublicationData (tofollow) BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN0470012951hardback ISBN047001296Xpaperback Typesetin10.5/12.5ptTimesbyThomsonPress(India)Limited,NewDelhi,India. PrintedandboundinGreatBritainbyAntonyRoweLtd,Chippenham,Wiltshire. Thisbookisprintedonacid-freepaperresponsiblymanufacturedfromsustainableforestry inwhichatleasttwotreesareplantedforeachoneusedforpaperproduction. Contents Introduction to First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction to Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Introduction to Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Introduction to Fourth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Introduction to Fifth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Introduction to Sixth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 Simple Harmonic Motion 1 DisplacementinSimpleHarmonicMotion 4 Velocity andAccelerationinSimpleHarmonicMotion 6 EnergyofaSimpleHarmonic Oscillator 8 SimpleHarmonicOscillations inanElectricalSystem 10 SuperpositionofTwoSimpleHarmonic VibrationsinOneDimension 12 SuperpositionofTwoPerpendicular SimpleHarmonicVibrations 15 (cid:1) Polarization 17 SuperpositionofaLargeNumbernofSimpleHarmonic Vibrationsof EqualAmplitudeaandEqualSuccessivePhaseDifferenced 20 (cid:1)SuperpositionofnEqualSHMVectorsofLengthawithRandom Phase 22 SomeUsefulMathematics 25 2 Damped Simple Harmonic Motion 37 MethodsofDescribingtheDampingofanOscillator 43 3 The Forced Oscillator 53 TheOperation ofiuponaVector 53 Vectorform ofOhm’sLaw 54 TheImpedanceofaMechanical Circuit 56 BehaviourofaForcedOscillator 57 v vi Contents BehaviourofVelocity vvinMagnitudeand PhaseversusDrivingForceFrequencyx 60 BehaviourofDisplacementversusDrivingForceFrequencyx 62 ProblemonVibrationInsulation 64 SignificanceoftheTwoComponentsoftheDisplacementCurve 66 Power SuppliedtoOscillatorbytheDrivingForce 68 VariationofP withx.AbsorptionResonanceCurve 69 av TheQ-ValueinTerms oftheResonanceAbsorptionBandwidth 70 TheQ-ValueasanAmplificationFactor 71 TheEffectoftheTransientTerm 74 4 Coupled Oscillations 79 Stiffness(orCapacitance)Coupled Oscillators 79 Normal Coordinates,DegreesofFreedomandNormal ModesofVibration 81 TheGeneral MethodforFindingNormalMode Frequencies,Matrices, EigenvectorsandEigenvalues 86 MassorInductanceCoupling 87 CoupledOscillations ofaLoadedString 90 TheWaveEquation 95 5 Transverse Wave Motion 107 PartialDifferentiation 107 Waves 108 Velocities inWaveMotion 109 TheWaveEquation 110 SolutionoftheWaveEquation 112 Characteristic ImpedanceofaString(the stringasaforcedoscillator) 115 Reflection andTransmissionofWavesonaString ataBoundary 117 Reflection andTransmissionofEnergy 120 TheReflected andTransmittedIntensityCoefficients 120 TheMatchingofImpedances 121 StandingWavesonaStringofFixedLength 124 EnergyofaVibratingString 126 EnergyinEachNormal ModeofaVibrating String 127 StandingWaveRatio 128 WaveGroupsandGroupVelocity 128 WaveGroupofManyComponents. TheBandwidthTheorem 132 TransverseWavesinaPeriodicStructure 135 LinearArray ofTwoKindsofAtomsinanIonicCrystal 138 AbsorptionofInfraredRadiationbyIonicCrystals 140 DopplerEffect 141 6 Longitudinal Waves 151 SoundWavesinGases 151 Contents vii EnergyDistributioninSoundWaves 155 IntensityofSoundWaves 157 Longitudinal WavesinaSolid 159 Application toEarthquakes 161 Longitudinal WavesinaPeriodicStructure 162 Reflection andTransmissionofSoundWavesatBoundaries 163 Reflection andTransmissionofSoundIntensity 164 7 Waves on Transmission Lines 171 IdealorLossless TransmissionLine 173 CoaxialCables 174 Characteristic ImpedanceofaTransmission Line 175 Reflections fromtheEndofaTransmissionLine 177 ShortCircuitedTransmissionLineðZ ¼0Þ 178 L TheTransmissionLineasaFilter 179 EffectofResistance inaTransmissionLine 183 Characteristic ImpedanceofaTransmission LinewithResistance 186 TheDiffusionEquation andEnergyAbsorptioninWaves 187 WaveEquation withDiffusionEffects 190 Appendix 191 8 Electromagnetic Waves 199 Maxwell’sEquations 199 Electromagnetic WavesinaMediumhavingFinitePermeabilityland Permittivitye butwithConductivityr¼0 202 TheWaveEquation forElectromagnetic Waves 204 IllustrationofPoyntingVector 206 ImpedanceofaDielectric toElectromagneticWaves 207 Electromagnetic WavesinaMediumofPropertiesl,eand r(wherer6¼0) 208 SkinDepth 211 Electromagnetic WaveVelocity inaConductorandAnomalousDispersion 211 WhenisaMediumaConductororaDielectric? 212 Whywill anElectromagnetic WavenotPropagate intoaConductor? 214 ImpedanceofaConductingMediumtoElectromagnetic Waves 215 Reflection andTransmissionofElectromagnetic WavesataBoundary 217 Reflection fromaConductor(NormalIncidence) 222 Electromagnetic WavesinaPlasma 223 Electromagnetic WavesintheIonosphere 227 9 Waves in More than One Dimension 239 PlaneWaveRepresentationinTwoandThreeDimensions 239 WaveEquation inTwoDimensions 240 viii Contents WaveGuides 242 Normal ModesandtheMethodofSeparationofVariables 245 Two-DimensionalCase 246 Three-Dimensional Case 247 Normal ModesinTwoDimensionsona RectangularMembrane 247 Normal ModesinThreeDimensions 250 FrequencyDistributionofEnergyRadiatedfrom aHotBody.Planck’sLaw 251 DebyeTheoryofSpecificHeats 253 Reflection andTransmissionofaThree-Dimensional Waveata PlaneBoundary 254 TotalInternalReflection andEvanescentWaves 256 10 Fourier Methods 267 FourierSeries 267 Application ofFourierSineSeries toaTriangularFunction 274 Application totheEnergyintheNormalModesofaVibratingString 275 FourierSeriesAnalysis ofaRectangularVelocityPulseonaString 278 TheSpectrumofaFourierSeries 281 FourierIntegral 283 FourierTransforms 285 ExamplesofFourierTransforms 286 TheSlitFunction 286 TheFourierTransformAppliedtoOpticalDiffractionfrom aSingleSlit 287 TheGaussianCurve 289 TheDirac DeltaFunction, itsSiftingProperty anditsFourierTransform 292 Convolution 292 TheConvolutionTheorem 297 11 Waves in Optical Systems 305 Light.WavesorRays? 305 Fermat’sPrinciple 307 TheLawsofReflection 307 TheLawofRefraction 309 Rays andWavefronts 310 RayOpticsandOpticalSystems 313 Power ofaSphericalSurface 314 MagnificationbytheSpherical Surface 316 Power ofTwoOptically RefractingSurfaces 317 Power ofaThinLensinAir(Figure 11.12) 318 Principal PlanesandNewton’sEquation 320 OpticalHelmholtz EquationforaConjugatePlaneatInfinity 321 TheDeviationMethodfor(a)TwoLensesand(b)aThickLens 322 TheMatrixMethod 325 Contents ix 12 Interference and Diffraction 333 Interference 333 DivisionofAmplitude 334 Newton’sRings 337 Michelson’sSpectral Interferometer 338 TheStructure ofSpectral Lines 340 Fabry--PerotInterferometer 341 ResolvingPoweroftheFabry--PerotInterferometer 343 DivisionofWavefront 355 Interference fromTwoEqualSourcesofSeparationf 357 Interference fromLinearArray ofNEqualSources 363 Diffraction 366 Scale oftheIntensityDistribution 369 IntensityDistributionforInterferencewithDiffraction fromNIdenticalSlits 370 Fraunhofer Diffraction forTwoEqualSlitsðN ¼2Þ 372 Transmission DiffractionGrating (NLarge) 373 ResolvingPowerofDiffractionGrating 374 ResolvingPowerinTermsoftheBandwidthTheorem 376 Fraunhofer Diffraction fromaRectangular Aperture 377 Fraunhofer Diffraction fromaCircularAperture 379 Fraunhofer FarFieldDiffraction 383 TheMichelsonStellar Interferometer 386 TheConvolutionArrayTheorem 388 TheOpticalTransferFunction 391 FresnelDiffraction 395 Holography 403 13 Wave Mechanics 411 OriginsofModernQuantumTheory 411 Heisenberg’s UncertaintyPrinciple 414 Schro¨dinger’sWaveEquation 417 One-dimensionalInfinitePotentialWell 419 SignificanceoftheAmplitudew ðxÞoftheWaveFunction 422 n ParticleinaThree-dimensional Box 424 NumberofEnergyStatesinIntervalEtoEþdE 425 ThePotential Step 426 TheSquarePotential Well 434 TheHarmonic Oscillator 438 ElectronWavesinaSolid 441 Phonons 450 14 Non-linear Oscillations and Chaos 459 FreeVibrations ofanAnharmonicOscillator--LargeAmplitudeMotionof aSimplePendulum 459 x Contents ForcedOscillations – Non-linearRestoringForce 460 ThermalExpansionofaCrystal 463 Non-linearEffectsinElectrical Devices 465 ElectricalRelaxation Oscillators 467 Chaos inPopulationBiology 469 Chaos inaNon-linearElectrical Oscillator 477 PhaseSpace 481 RepellorandLimit Cycle 485 TheTorusinThree-dimensionalðxx_;x;t)PhaseSpace 485 ChaoticResponse ofaForcedNon-linearMechanical Oscillator 487 ABriefReview 488 Chaos inFluids 494 RecommendedFurther Reading 504 References 504 15 Non-linear Waves, Shocks and Solitons 505 Non-linearEffectsinAcoustic Waves 505 ShockFrontThickness 508 EquationsofConservation 509 MachNumber 510 Ratios ofGasProperties AcrossaShockFront 511 StrongShocks 512 Solitons 513 Bibliography 531 References 531 Appendix 1: Normal Modes, Phase Space and Statistical Physics 533 Mathematical DerivationoftheStatisticalDistributions 542 Appendix 2: Kirchhoff’s Integral Theorem 547 Appendix 3: Non-Linear Schro¨dinger Equation 551 Index 553 Introduction to First Edition The opening session of the physics degree course at Imperial College includes an introduction to vibrations and waves where the stress is laid on the underlying unity of concepts which are studied separately and in more detail at later stages. The origin of this shorttextbookliesinthatlecturecoursewhichtheauthorhasgivenforanumberofyears. Sections on Fourier transforms and non-linear oscillations have been added to extend the range of interest and application. At the beginning no more than school-leaving mathematics is assumed and more advanced techniques are outlined as they arise. This involves explaining the use of exponentialseries,thenotationofcomplexnumbersandpartialdifferentiationandputting trial solutions into differential equations. Only plane waves are considered and, with two exceptions, Cartesian coordinates are used throughout. Vector methodsare avoided except for the scalar product and, on one occasion, the vector product. Opinioncanvassedamongstmanyundergraduateshasarguedfora‘working’asmuchas for a ‘reading’ book; the result is a concise text amplified by many problems over a wide rangeofcontentandsophistication.Hintsforsolutionarefreelygivenontheprinciplethat an undergraduates gains more from being guided to a result of physical significance than from carrying out a limited arithmetical exercise. The main theme of the book is that a medium through which energy is transmitted via wave propagation behaves essentially as a continuum of coupled oscillators. A simple oscillator is characterized by three parameters, two of which are capable of storing and exchangingenergy,whilstthethirdisenergydissipating.Thisisequallytrueofanymedium. The product of the energy storing parameters determines the velocity of wave propagation through the medium and, in the absence of the third parameter, their ratio governs the impedance which the medium presents to the waves. The energy dissipating parameter introduces a loss term into the impedance; energy is absorbed from the wave system and it attenuates. This viewpoint allows a discussion of simple harmonic, damped, forced and coupled oscillators which leads naturally to the behaviour of transverse waves on a string, longitudinal waves in a gas and a solid, voltage and current waves on a transmission line and electromagnetic waves in a dielectric and a conductor. All are amenable to this common treatment, and it is the wide validity of relatively few physical principles which this book seeks to demonstrate. H. J. PAIN May 1968 xi Introduction to Second Edition The main theme of the book remains unchanged but an extra chapter on Wave Mechanics illustrates the application of classical principles to modern physics. Any revision has been towards a simpler approach especially in the early chapters and additional problems. Reference to a problem in the course of a chapter indicates its relevancetotheprecedingtext.Eachchapterendswithasummaryofitsimportantresults. Constructive criticism of the first edition has come from many quarters, not least from successive generations of physics and engineering students who have used the book; a second edition which incorporates so much of this advice is the best acknowledgement of its value. H. J. PAIN June 1976 xii
The list of books you might like
