The Physics of Vibrations and Waves, 6th Edition

570 Pages·2005·2.345 MB·English

by  H. J. Pain

THE PHYSICS OF VIBRATIONS
AND WAVES
Sixth Edition
H. J. Pain
Formerly of Department of Physics,
Imperial College of Science and Technology, London, UK
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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