Fundamentals of Wave Phenomena Second Edition Akira Hirose UniversityofSaskatchewan Karl E. Lonngren UniversityofIowa Raleigh,NC scitechpub.com PublishedbySciTechPublishing,Inc. 911PaverstoneDrive,SuiteB Raleigh,NC27615 (919)847-2434,fax(919)847-2568 scitechpublishing.com Copyright©2010bySciTechPublishing,Raleigh,NC PrintedintheUnitedStatesofAmerica ISBN:978-1-891121-92-0 LibraryofCongressCataloging-in-PublicationData Hirose,Akira,1941- Fundamentalsofwavephenomena/AkiraHirose,KarlE.Lonngren.–2nded. p.cm.–(TheMarioBoellaseriesonelectromagnetismininformation&communication) Rev.ed.of:Introductiontowavephenomena.1985. Includesbibliographicalreferencesandindex. ISBN978-1-891121-92-0(hardcover:alk.paper)1.Waves.2.Wave-motion,Theoryof.I.Lonngren, KarlE.(KarlErik),1938-II.Hirose,Akira,1941-Introductiontowavephenomena.III.Title. QC157.H562010 531’.1133–dc22 2010009089 Contents Preface xi Publisher’sAcknowledgements xv Chapter1 ReviewofOscillations 1 1.1 Introduction 1 1.2 MassSpringSystem 2 1.3 EnergyTossinginMechanicalOscillations 7 1.4 OtherMechanicalOscillationSystems 11 1.5 ElectromagneticOscillation 17 1.6 DampedOscillation 19 1.7 ForcedOscillation 22 1.8 Problems 24 Chapter2 WaveMotion 29 2.1 Introduction 29 2.2 CreationofWavesonaString 30 2.3 Sinusoidal(Harmonic)Waves 33 2.4 WaveDifferentialEquation,PartialDifferentiation 37 2.5 NonsinusoidalWaves 42 2.6 PhaseandGroupVelocities,Dispersion 44 2.7 SuperpositionofTwoWaves,Beats 47 2.8 Problems 51 Chapter3 SomeMathematics 53 3.1 TaylorSeriesExpansion 53 3.2 ComplexNumbersandVariables 56 3.3 Problems 59 Chapter4 FundamentalsofMechanicalWaves 61 4.1 Introduction 61 4.2 Mass-SpringTransmissionLine 62 4.3 DerivationofaWaveEquation 64 4.4 EnergyCarriedbyWaves 68 4.5 MomentumCarriedbyWaves 71 4.6 TransverseWavesonaString 74 4.7 Problems 79 Chapter5 SoundWavesinSolids,Liquids,andGases 83 5.1 Introduction 83 5.2 SoundVelocityAlongaSolidRod 83 5.3 RigorousDerivationofVelocityofSoundinaSolidRod 87 5.4 SoundWavesinLiquids 90 5.5 SoundWavesinGases 93 5.6 IntensityofSoundWavesinGases 98 5.7 Problems 100 Chapter6 WaveReflectionandStandingWaves 103 6.1 Introduction 103 6.2 ReflectionataFixedBoundary,StandingWaves 103 6.3 ReflectionataFreeBoundary 108 6.4 TheoryofWaveReflection,MechanicalImpedance 112 6.5 Problems 119 Chapter7 SphericalWaves,WavesinaNonuniformMedia, andMultidimensionalWaves 121 7.1 Introduction 121 7.2 ConservationofEnergyFlowasApplied toSphericalWaves 122 7.3 NonuniformWaveMedium 125 7.4 MultidimensionalWaves 128 7.5 Problems 130 Chapter8 DopplerEffectofSoundWavesandShockWaves 133 8.1 Introduction 133 8.2 StationarySoundSourceandMovingObserver 134 8.3 MovingSoundSourceandStationaryObserver 137 8.4 GeneralExpressionforDoppler-ShiftedFrequency 139 8.5 ShockWaves 142 8.6 Problems 144 Chapter9 ElectromagneticWaves 145 9.1 Introduction 145 9.2 WaveEquationforanLCTransmissionLine 146 9.3 CoaxialCable 154 9.4 PoyntingVector 164 9.5 PlaneElectromagneticWavesinFreeSpace 167 9.6 ReflectionofElectromagneticWaves 172 9.7 ElectromagneticWavesinMatter 179 9.8 Problems 192 Chapter10 RadiationofElectromagneticWaves 197 10.1 Introduction 197 10.2 FieldsAssociatedwithaStationaryChargeandaCharge MovingwithaConstantVelocity 197 10.3 RadiationFieldsDuetoanAcceleratedor aDeceleratedCharge 201 10.4 RadiationfromanOscillatingDipoleCharge andaDipoleAntenna 207 10.5 CyclotronandSynchrotronRadiation 212 10.6 Problems 213 Chapter11 InterferenceandDiffraction 215 11.1 Introduction 215 11.2 InterferenceBetweenTwoHarmonicWaves 216 11.3 Young’sExperiment 219 11.4 Multi-slitStructure 224 11.5 OpticalInterferenceinThinFilms 231 11.6 DiffractionI(FraunhoferDiffraction) 236 11.7 ResolutionofOpticalDevices 239 11.8 DiffractionII(FresnelDiffraction) 241 11.9 Problems 246 Chapter12 GeometricalOptics 251 12.1 Introduction 251 12.2 ReflectionandRefraction 252 12.3 TotalReflection 260 12.4 ReflectionatSphericalSurfaces(Mirrors) 262 12.5 SphericalAberrationofMirrors 266 12.6 RefractionatSphericalSurfaces 268 12.7 Lenses 269 12.8 ChromaticAberration 272 12.9 OpticalInstruments 275 12.10 PhysicalMeaningofFocusing 280 12.11 MatrixMethodinGeometricalOptics 282 12.12 Problems 293 Chapter13 ParticleNatureofLight 299 13.1 Introduction 299 13.2 PhotoelectricEffectandEinstein’sPhotonTheory 299 13.3 HydrogenAtom 302 13.4 deBroglieWave 305 13.5 Problems 306 Chapter14 FourierAnalysesandLaplaceTransformation 307 14.1 Introduction 307 14.2 SumofSinusoidalFunctions 308 14.3 FourierSeries 311 14.4 FourierSpectrum 314 14.5 OperatorMethod 318 14.6 LaplaceTransform 323 14.7 Problems 328 Chapter15 NonlinearWaves,Solitons,Shocks,andChaos 331 15.1 Introduction 331 15.2 NonlinearOscillations 331 15.3 NonlinearWaveEquation 337 15.4 Fermi,Pasta,andUlamRecurrencePhenomena 343 15.5 KdVSolitonProperties 346 15.6 Shocks 352 15.7 Chaos 354 15.8 Conclusion 360 15.9 Problems 360 AppendixA ConstantsandUnits 363 A.1 FundamentalPhysicalConstants 363 A.2 DefinitionofStandards 363 A.3 DerivedUnitsofPhysicalQuantitiesandNotation 364 AppendixB TrigonometricIdentities,Calculus, andLaplaceTransforms 366 B.1 TrigonometricIdentities 366 B.2 Calculus 367 B.3 PowerSeries 368 B.4 LaplaceTransforms 369 AppendixC References 371 AppendixD AnswerstoSelectedProblems 372 Index 377 Preface Traditionally,thesubjectsdealingwithwavephenomenahavebeentaughtin differentdisciplines.Indeed,soundwavesandelectromagneticwavesareen- tirelydifferentphysicalphenomena.However,onceonerealizesthatallwaves, betheymechanicalorelectromagnetic,carrybothenergyandmomentumas they propagate and that the observable quantities are necessarily associated with energy propagation, then one can formulate a unified understanding of wave phenomena. To this end, we have produced a textbook that elucidates the general properties of wave phenomena, both linear and nonlinear, and illustratesthephysicalcontextsinwhichthevariouswavephenomenaoccur. The first edition was published twenty-five years ago. Somewhat to our surprise, the book is still in demand, which is why we decided to revise and update it. This update includes a significant revision of the presentation and numerousnewexamplesandproblems. Thiseditionconsistsoffifteenchapters.Webeginchapter1withareview of linear oscillations, both mechanical and electromagnetic. Oscillation sys- temscanbethesourceoftimeharmonicwavessincethemassinamechanical oscillationsystemandtheelectricalchargeinanelectromagneticoscillation system are all related to and can be used to excite various waves. The pen- dulum,orswing,isanoscillationsystemthatreadersmayhaveencountered before. Inchapter2westudygeneralpropertiesofwavemotion,thatis,without specifyingwhethertheyaremechanicalorelectromagneticwaves.Amathe- maticalexpressionforasinusoidalwaveandthewaveequationareintroduced. Phaseandgroupvelocitiesandbeatsaredescribed. Chapter 3 provides mathematical preparations that are needed to derive thewaveequationfromfirstprinciples, ∂2ξ ∂2ξ = c2 ∂t2 w∂x2 forvariouskindsofwaves.Itisusefulforthereadernotalreadyfamiliarwith Taylorseriesexpansiontechniques. Inchapters4and5,westudymechanicalwaves,includingthoseonsprings (longitudinal)andalongastretchedstring(transverse),andsoundwaves(lon- gitudinal) in solids, liquids, and gases. We show how Newton’s equation of motion can be converted into the wave equation for mechanical waves and canformulatetheenergyandmomentumthatisassociatedwiththesewaves. A detailed analysis to find the relationship is presented: [wave momentum transferrate=(waveenergytransferrate)/(wavevelocity)]. In chapter 6, we examine the reflection of mechanical waves at various boundaries. The wave equation allows two distinct solutions that propagate in opposite directions. When these two waves coexist, standing waves are created.Standingmechanicalwavesplayanimportantroleinmusicalinstru- ments.Theconceptofamechanicalimpedanceisintroducedasananalogyto thecharacteristicimpedancethatoneencountersinelectromagneticwavephe- nomena.Theconceptofwavereflectionisexplainedintermsofthereflection ofenergyandmomentum.Inwavereflectionatanimpedancediscontinuity, energyconservationisimposedtofindtheamplitudesofboththereflectedand transmittedwaves,assumingthatthewavemediaareinfinitelymassive.Mo- mentum conservation requires the consideration of momentum absorbed by themedia.Whenappliedtoreflectionofalightwaveincidentuponthesurface ofasemi-infiniteglassslab,theradiationpressureisnegative(forcedirected fromtheslabtoair)aspredictedbyPoyntingin1905.Usingthiswell-known phenomenon, the momentum of the wave transmitted into glass medium is uniquelydetermined,whichisconsistentwiththeclassicalformula. When a loudspeaker creates sound waves in air, the wave amplitude be- comessmallerasonemovesawayfromthespeaker.Thiseffectiscausedby a geometrical decrease of the amplitude of the wave as shown in chapter 7. Thevariationoftheamplitudecanalsooccurinone-dimensionalwavesifthe mediumisnotuniformorthewavevelocityvariesfrompointtopoint.This interesting(butdifficulttoanalyze)phenomenonisbrieflydiscussed. In chapter 8, the Doppler effect of sound waves is studied. Whenever a sound source or an observer are moving relative to the ambient air, the observer hears a frequency that is different from the true frequency emitted from the source. If the source is moving faster than the speed of sound, a shock wave can occur. A Doppler shift in electromagnetic waves is similar to that in sound waves, but there is a fundamental difference between the two. In electromagnetic waves, only the relative velocity between the wave sourceandtheobserverentersintothedescriptionoftheDopplershift.This is a consequence of one of Einstein’s postulates in relativity theory that the velocityofelectromagneticwavesinavacuumisindependentofthesource’s andtheobserver’svelocity. In chapter 9, we study the propagation and radiation of electromagnetic waves. We start with an LC transmission line that is an analogue to the mass-springmechanicaltransmissionlinedescribedinchapter3.Thereflec- tionofelectromagneticwavesisdiscussedintermsofthepossiblemismatch oftheelectricalcharacteristicimpedances,aswasdoneinchapter6forme- chanicalwaves.Electromagneticwavesinconductingmaterialssuchasmetals or plasmas require a special treatment. In such media, the wave equation is drasticallymodifiedandelectromagneticwavesbecomestronglydispersive, implying that the propagation properties depend upon the frequency of the waves. Inchapter10,welearnthattheradiationofelectromagneticwavesrequires the acceleration or deceleration of electric charges. The electric field lines associatedwithachargearecontinuous.Inthepresenceofchargeacceleration ordeceleration,theelectricfieldconsistsofanonradiationCoulombfieldand a transverse radiation field. The fields are linearly related and the radiation electricfieldcanthusbeobtainedfromtheknownCoulombelectricfield. In chapter 11, interference and diffraction, which are caused by the con- structive and destructive interference between more than one wave propa- gating in the same direction, are described. Depending on the phase of each wave,theresultingwaveamplitudeiseitherstrengthenedorweakened.Light doesnotalwaystravelalongastraightlinebecauseofthediffractivenatureof thewave.BothFraunhoferandFresneldiffractionsareanalyzed.Diffraction imposesalimitontheresolvingpowerofallopticalinstruments. Geometricalopticsisonebranchofopticsinwhichweareabletoneglect the wave nature of light, and this is discussed in chapter 12. Light can be assumedtopropagateinastraightlineinauniformmedium.Ofcourse,when lighthitsaboundarybetweentwomedia,lightchangesitspropagationdirec- tion.Afamiliarexampleisthereflectionandrefractionattheboundaryofair andglass.Opticaldevicessuchasmirrorsandlensesarediscussed,followed by a look at various optical instruments. The matrix method in geometrical opticsisapowerfultoolwhenanalyzingmultilenssystemsandthicklenses. Chapter13isanintroductiontomodernquantumphysics.Itisshownthat undercertaincircumstances,lightbehavesasacollectionofparticlesthatare calledphotons.Brieflydiscussedhereisthatenergeticparticlessuchasaccel- eratedelectronsalsohaveawavenature.Forexample,the(deBloglie)wave- lengthofenergeticelectronsinanelectronmicroscopeisordersofmagnitude shorterthananopticalwavelengthandthisiswhyanelectronmicroscopecan haveahigherresolvingpowerthananopticalmicroscope. Chapter 14 contains an introduction to Fourier series and Laplace trans- forms.Theconceptofthefrequencyspectrumisexplained.Also,itisshown howLaplacetransformscanconvertadifferentialequationintoanalgebraic equation. Thematerialcoveredinthepreviouschaptersmadethetacitassumption that the equations describing the physical phenomena could be linearized in that all perturbations of any physical quantity were small. In chapter 15, we introduce some mathematical techniques that are used to obtain analytical solutionstocertainrelevantnonlinearequations.Thesubjectsofsolitonsand chaosarealsodescribed. The MKS (meter, kilogram, second) unit system is used throughout the book.Sometraditionalandsomeconventionalunitsareemployedsuchasthe angstrom(1A◦ = 1×10−10 m)inopticsandtheelectronvolt(1eV = 1.6× 10−19J)fortheenergyofelementaryparticles.Inphysics,atravelingwaveis conventionallywrittenas Acos(kx−ωt)or(Aei(kx−ωt)),andin√engineering, it is written as Acos(ωt −kx) or (Aej(ωt−kx)) wherei = j = −1. We use the former representation but there are no fundamental differences between thetwo. The website for an undergraduate wave course at the University of Saskatchewan can be accessed at http://physics.usask.ca/∼hirose/ep225/, whichcontainsanimationofvariouswavemotions. Theauthorswishtoacknowledgethehelpfulandprofessionalassistance thatwereceivedfromourEditorDudleyKay,hisassistantKatieJanelle,and Production Manager Robert Lawless and the entire staff at SciTech in the preparationofthisbook. We also thank our colleagues and students for their comments and more importantly,ourwives,KimikoandVicki,fortheirtirelesssupport.