Fundamentals of Wave Phenomena

387 Pages·2010·5.597 MB·English

by  Akira Hirose, Karl E. Lonngren

Fundamentals of
Wave Phenomena
Second Edition
Akira Hirose
UniversityofSaskatchewan
Karl E. Lonngren
UniversityofIowa
Raleigh,NC
scitechpub.com
PublishedbySciTechPublishing,Inc.
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Raleigh,NC27615
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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.