WilliamC. ElmoreMark A. HealdDepartment of Physics SwarthmoreCollegePhysics of WavesMcGraw-Hill Book CompanyNewYork, St. Louis, SanFrancisco, London, Sydney, Toronto, Mexico, PanamaPhysics oj WavesCopyright 1969 by McGraw-Hili, Inc. All rights reserved.Printedinthe UnitedStates ofAmerica. No part ofthispublicationmay be reproduced, stored in aretrieval system,ortransmitted, inany formorby anymeans, electronic,mechanical, photocopying, recording, or otherwise, withoutthe priorwrittenpermission ofthe publisher.Library of CongressCatalogCardNumber68-58209192601234567890MAMM7654321069DedicatedtothememoryofLeigh PageProfessorof Mathematical PhysicsYale UniversityPrefaceClassical wave theorypervades muchof classical andcontemporaryphysics.Becauseoftheincreasingcurriculardemandsofatomic, quantum, solid-state,andnuclearphysics, theundergraduatecurriculumcannolongeraffordtimeforseparatecourses in many oftheolder disciplinesdevotedtosuchclasses ofwavephenomenaasoptics, acoustics, andelectromagneticradiation. Wehaveendeavoredtoselect significant material pertainingtowave motionfromalltheseareasofclassical physics. Our aimhasbeen to unify the study of wavesbydevelopingabstract andgeneralfeaturescommontoall wavemotion. Wehave done this by examining a sequence of concrete and specific examples(emphasizing the physics of wave motion)increasing in complexity and sophisti-cation as understanding progresses. Although we have assumed that the mathe-matical background of the student has included only a year's course in calculus,wehaveaimedat developingthestudent'sfacilitywithappliedmathematicsbygradually increasing the mathematical sophistication of analysis as thechaptersprogress.At Swarthmore College approximately two-thirds of the present material isoffered asasemester courseforsophomores orjuniors,followingasemester ofintermediatemechanics. Muchofthetext isanenlargement ofaset ofnotesdevelopedoveraperiod of yearstosupplementlecturesonvariousaspectsofwavemotion. Thechapteron electromagneticwavespresentsrelatedmaterialwhichour studentsencounter as part of asubsequent course. Both courses areaccompaniedbyalaboratory.A fewtopicsin classical wavemotion(forthemost partomitted fromourformal coursesfor lackoflecturetime) havebeenincludedtoroundout thetreatment of the subject. We hope that these additions, includingmuch ofChapters 6,7, and 12, will make the text more flexible for formulating courses tomeet particular needs. We especiallyhope that the inclusionof additionalmaterial tobecoveredina one-semester course will encourage the serious stu-dent ofphysicstoinvestigatefor himself topics not coveredinlecture. Starsidentify particular sections or whole chapters thatmay be omitted without lossof continuity. Generallythismaterial issomewhat moredemanding. Many oftheproblemswhichfolloweachsectionformanessential part of thetext. Intheseproblems the student is asked to supply mathematical details forcalcula-tionsoutlined inthesection, orheisaskedtodevelopthe theoryfor relatedvUlIm Prefacecases that extendthecoverageof the text. Afewproblems (indicatedbyanasterisk) go significantly beyondthe level ofthetext and areintendedtochal-lengeeventhebeststudent.Thefundamental ideasof wavemotionareset forthinthefirst chapter,usingthestretchedstringas aparticular model. InChapter 2 the two-dimen-sional membrane is usedtointroduce Bessel functions andthecharacteristicfeatures of waveguides. InChapters 3and4elementaryelasticitytheoryisdevelopedandappliedtofind thevarious classesof waves that canbesup-portedby arigidrod. Theimpedanceconcept isalsointroduced atthispoint.InChapter5acousticwavesinfluids arediscussed, and, amongotherthings,the number ofmodes in a box is counted. These firstfive chapters complete thebasictreatmentofwavesin one. two, andthreedimensions, withemphasisonthecentralidea of energy and momentum transport.Thenext threechaptersare optionsthatmay beusedto give a particularemphasis to a course. Hydrodynamic waves at a liquidsurface (e.g., waterwaves)are treatedin Chapter 6. In Chapter 7 generalwaves in isotropic elasticsolids are considered, after a development of theappropriate tensor algebra(with its future use in relativity theory kept in mind). Although electromagneticwaves are undeniablyof paramount importance inthe real worldof waves,wehavechosentoarrange theextensivetreatment of Chapter8as optionalmaterial becauseof thephysical subtlety and analytical complexity of electro-magnetism. ThusChapter 8might eitherbeignoredorbemadeamajor partofthecourse, depending ontheinstructor's aims.Chapter 9 is probably the most difficult and formal of the central core of thebook. In it approximate methods are considered for dealing with inhomogeneousand obstructedmedia, in particular theKirchhoff diffraction theory.The casesofFraunhoferandFresnel diffractionareworkedout inChapters 10and11,withsome care to showthat their relevance is not limitedtovisible light.Chapter 12 removes the idealizations of monochromatic waves and pointsources by consideringmodulation,wavepackets, and partialcoherence.Conspicuously absent from our catalog of waves is a discussion of the quan-tum-mechanical variety. Manyof ourchoicesof emphasisandexampleshavebeenmadewithwavemechanics inmind, butwehavepreferred to stay inthecontext of classical waves throughout. We hope, rather, that a student willapproachhis subsequent course in quantummechanics well-armedwiththephysical insight andanalytical skillsneededtoappreciatetheabstractionsofwave mechanics. Wehavealsorestrictedthediscussiontocontinuummodels,leavingthetreatment of discrete-mass and periodic systems to later courses.Wearegrateful toMrs. AnnDeRosefor her patience and skill in typingthemanuscript.WilliamC. ElmoreMark A. HealdContentsPreface vii1 Transverse Waves on a String 11.1 Thewave equation foran idealstretched string 21.2 A general solution of theone-dimensionalwave equation 51.3 Harmonic or sinusoidal waves 71.4 Standing sinusoidal waves 141.5 Solvingthewaveequationbythemethodofseparationof variables 161.6 The general motion of afinitestring segment 191. 7 Fourier series 231.8 Energy carried bywaves on astring 311.9 The reflection andtransmission ofwaves atadiscontinuity 39*1.10Another derivation ofthewave equation forstrings 42*1.11 Momentum carried by a wave 452 Waves on a Membrane 502.1 Thewave equation forastretched membrane 512.2 Standing waves on arectangularmembrane 572.3 Standing waves on acircular membrane 592.4 Interferencephenomenawithplanetraveling waves 653 Introduction to the Theory of Elasticity 713.1 The elongation of arod 723.2 Volume changes in anelastic medium 763.3 Shear distortion in aplane 783.4 Thetorsion of roundtubes and rods 813.5 The statics of asimplebeam 833.6 Thebending of a simplebeam 863.7 Helical springs 914 One-dimensional Elastic Waves 934.1 Longitudinalwaveson aslender rod 94(a) The waveequation 94" Content.r(b) Standing waves 95(C) Energy and power 96(d) Momentumtransport 974.2 The impedance concept 984.3 Rodswithvarying cross-sectional area 1044.4 The effect of smallperturbations on normal-modefrequencies 1074.5 Torsionalwaves on around rod 1124.6 Transverse waves on aslender rod 114(a) Thewaveequation 114(b) Solution ofthewave equation 116(c) Traveling waves 117(d) Normal-mode vibrations 1194.7 Phase and group velocity 1224.8 Waves on a helical spring 127*4.9 Perturbation calculations 1315 Acoustic Waves in Fluids5.1 Thewave equation forfluids*5.2 The velocity of sound in gases5.3 Plane acoustic waves(a) Traveling sinusoidal waves(b) Standing waves of sound5.4 The cavity(Helmholtz) resonator5.5 Spherical acoustic waves5.6 Reflection and refraction at aplane interface5.7 Standing wavesin a rectangular box5.8 The Doppler effect*5.9 Thevelocity potential*5.10Shock waves*6 Waves on a Liquid Surface6.1 Basic hydrodynamics(a) Kinematicalequations(b) The equation of continuity(c) TheBernoulli equation6.2 Gravity waves6.3 Effect of surface tension6.4 Tidalwaves andthetides(a) Tidalwaves(b) Tide-generating forces135135139142143145148152155160164167169176177177180181184190195195197Content.. :Ii(C) Equilibriumtheory of tides 199(d) The dynamical theory of tides 2006.5 Energy andpower relations 203*7 Elastic Waves in Solids 2067.1 Tensors and dyadics 2067.2 Strain as adyadic 2137.3 Stress as adyadic 2177.4 Hooke's law 2227.5 Waves in an isotropic medium 225(a) Irrotational waves 226(b) Solenoidal waves 2277.6 Energy relations 229*7.7 Momentum transportby a shear wave 234*8Electromagnetic Waves 2378.1 Two-conductor transmission line 238(a) Circuit equations 239(b) Wave equation 240(c) Characteristic impedance 241(d) Reflection fromterminal impedance 242(e) Impedancemeasurement 2438.2 Maxwell's equations 2478.3 Plane waves 2538.4 Electromagnetic energy and momentum 2568.5 Waves in a conductingmedium 2638.6 Reflection and refraction at aplane interface 268(a) Boundary conditions 268(b) Normal incidence on a conductor 269(c) Oblique incidence on a nonconductor 2718.7 Waveguides 281(a) The vectorwave equation 281(b) General solution for waveguides 283(c) Rectangular crosssection 287*(d) Circular cross section 2908.8 Propagation in ionized gases 2998.9 Sphericalwaves 3049 Wave Propagation in Inhomogeneous and ObstructedMedia3099.1 TheWKBapproximation 3109.2 Geometrical optics315"Ii Content,9.3 The Huygens-Fresnelprinciple904 Kirchhoff diffractiontheory(a) Green'stheorem(b) The Helmholtz-Kirchhoff theorem(c) Kirchhoff boundary conditions9.5 Diffractionof transversewaves*9.6 Young'sformulationofdiffraction32232632732832933433610 Fraunhofer Diffraction 34110.1 Theparaxial approximation 34210.2 The Fraunhofer limit 34310.3 The rectangular aperture 347lOA The single slit 35110.5 Thecircular aperture 36110.6 The double slit 36510.7 Multiple slits 369*10.8 Practical diffractiongratings for spectral analysis 375(a) Gratings of arbitrary periodic structure 375(b) The grating equation 377(c) Dispersion 378(d) Resolving power 379*10.9 Two-dimensional gratings 381*10.10Three-dimensional gratings 38511 Fresnel Diffraction .19211.1 Fresnel zones 392(a) Circular zones 392(b) Off-axis diffraction 396(c) Linear zones 39811.2 The rectangular aperture 401(a) Geometry andnotation 401(b) TheCornu spiral 40311.3 The linear slit 40611.4 The straight edge 41212 Spectrum Analysis of Waveforms 41612.1 Nonsinusoidal periodic waves 41712.2 Nonrecurrentwaves 42012.3 Amplitude-;modulatedwaves 4261204 Phase-modulatedwaves 42812.5 Themotion of awavepacket in adispersivemedium 43112.6 The Fourier transformmethod 43612.7 Properties oftransferfunctions12.8 Partial coherence in awavefieldAppendixesA. Vector calculusB. TheSmith calculatorC. Proof oftheuncertainty relationIndexContents siii441445451451460465469Physics ofWavesoneTransverseWaves on a StringWestart thestudyofwavephenomenabylooking at aspecial case, thetransversemotionof aflexiblestring undertension. Variousmethodsforsolv-ing the resulting wave equation are developed, andthe solutions found are thenusedto illustrate anumber of important properties ofwaves. Theemphasisinthepresent chapteris primarilyondevelopingmathematical techniques thatprovetobeextremelyuseful intreatingwavephenomenaofa morecomplexnature.Itwill be foundimpressivetoviewinretrospect therather formidabletheoretical structure that can be based on a study of the motion of such a simpleobject asaflexiblestring undertension.I1.1 The Wave Equation for an Ideal Stretched StringWesuppose thestringtohavea mass Aoper unit lengthandtobeunder aconstant tensionTOmaintainedbyequal andoppositely directedforcesappliedat its ends. In the absence of a wave, the string is straight, lying along the x axisof a right-handed cartesiancoordinatesystem. We further suppose that thestring is indefinitely long; later we shallconsider the effect of end conditions.Evidentlyif thestringislocallydisplacedsidewaysasmall amount andquickly released, i.e., if it is "plucked," the tension in the string will give rise toforcesthat tendtorestorethestringtothepositionofitsinitialstateofrest.However, the inertia of the displacedportion of the string delays an immediatereturntothis position, andthemomentumacquiredbythedisplacedportioncausesthestringtoovershoot itsrest position. Moreover, becauseof thecon-tinuityof thestring, thedisturbance, whichwasoriginallyalocal one, mustnecessarily spread, or propagate,alongthestring astimeprogresses.Tobecome quantitative,let us apply Newton's second lawto any elementdXof thedisplacedstringtofind the differential equationthat describes itsmotion. Tosimplifytheanalysis, supposethat themotionoccursonlyinthexy plane. We use the symbol,., for the displacement in the y direction(reservingthe symboly, along withx and z, for expressing position in athree-dimensionalframeof reference). Weassumethat ,." whichisafunctionof positionxandtimet, iseverywheresufficiently small, so that:(1) Themagnitude ofthetensionTO is aconstant, independent of position.(2) Theangleofinclinationofthedisplacedstringwithrespect tothexaxisat any pointis small.(3) Anelement dx ofthestring can beconsideredto havemoved only inthetransversedirection as aresult ofthewave disturbance.We also idealize the analysis by neglecting the effect of friction of thes u r r o u n d ~ing air in dampingthemotion, theeffect of stiffness that areal string(or wire)may have, andtheeffect of gravity.Asaresultof sidewaysdisplacement, anetforceacts on anelement dx ofthe string, sincethe small angles al anda2 defined inFig. 1.1.1are, in general,not quite equal. We see fromFig. 1.1.1 that this unbalanced force has theycomponent To(sina2 - sinal) andthex component TO(COSa2- COSal). Sinceal and a2 are assumed to be verysmall, we mayneglect the x componententirelytandalsoreplacesinabytana =a,.,/ax inthe y component. Accord-t In Sec. 1.11 it is foundthat the x component neglectedhere is responsible for the transportof linearmomentumby atransverse wave traveling onthe string.1.1 TheWalle Equation/or an Ideal Stretched Strinll 3IIIIII1112II~ r IrFig. 1.1.1 Portion of displaced string. (The magnitude of the sideways displacement is greatlyexaggerated.)ing to Newton's second law,the latter forcecomponent must equalthemass ofthe element AO dx times its acceleration in the y direction. Therefore at all times(a7/2 a7/1) a27/TO - - - = AO dx -, (1.1.1)ax ax at2where7/, themeandisplacement of theelement, becomes the actual displace-ment atapointofthestringwhendx ~ O. Thepartial-derivativenotationisneeded forboththetimederivative andthespace derivative since7/ is afunc-tionofthetwoindependent variablesx andt. Thepartial-derivativenotationmerely indicates that x is to be held constant in computing time derivatives of7/andt isto beheld constant in computing space derivatives of7/.Next we divide (1.1.1)throughby dX and pass to the limitdX ~ O.By thedefinitionof a second derivative,lim~ (a7/2_a7/1) = a27/,~ ..... o dx ax ax ax2(1.1.1)becomesa27/ a ~TO- = AO-'ax2at2We chooseto write(1.1.2)intheforma27/ 1 a27/-=--,ax2c2at2wherec== ( ~ Y / 2(1.1.2)(1.1.3)(1.1.4)4 TransllerseWalles on a Strin,will be shown tobethevelocityofsmall-amplitudetransversewavesonthestring (c after the Latinceleritas, speed). Wenowturnourattentiontode-velopingmethodsfor solvingthisone-dimensional scalarwaveequationandtodiscussing anumberofimportant properties ofthesolutions. Equation(1.1.3)is the simplest member ofalargefamilyofwaveequationsapplyingtoone-two-, andthree-dimensionalmedia. Whateverwecan learn about the solutionsof(1.1.3) willbe usefulin discussing morecomplicated wave equations.Problems1.1.1 An elementary derivation of the velocity of transverse waves on a flexible string undertension is based on viewing atraveling wave from areference framemoving in the x directionwith a velocity equal to that of the wave. In this moving frame the string itself appears to moveProb. 1.1.1 String seen frommoving frame.backward past the observer with a speed c, as indicated in the figure. Find c by requiring thatthe uniformtensionTOgiverisetoacentripetal forceonacurvedelement asofthestringthatjustmaintainsthemotionof theelement ina circular path. Does this derivation implythat atraveling wave keeps its shape?1.1.2 A circular loop of flexible rope is set spinning with a circumferential speed1>0. Findthetension if the linear density is>'0. What relation doesthis case have toProb. 1.1.1?1.1.3 The damping effect of air on a transverse wave can be approximated by assuming thata transverseforceb Of//ot per unitlengthactssoastoopposethetransversemotionofthestring. FindhowEq. (1.1.2)ismodifiedbythisviscous damping.1.1.4 Extendthetreatment inProb. 1.1.3toincludethepresence ofanexternally appliedtransverse driving force Fv(x,tl perunit length acting onthe string.1.1.5 Use theequationdevelopedin Prob. 1.1.4to find theequilibriumshapeunder theactionofgravityofahorizontal segmentof string of linear massdensity>'0 stretchedwith atensionTObetweenfixedsupports separated adistance I.Assume that the sag is small.(1.2.1)1.1 AGeneral Solution of the One-dimensionalWalle Equation 51.1.6 Thedensityofsteel pianowireisabout 8 g/cm3 Ifasafeworkingstressis 100,000lb/in.!, what is themaximum.velocitythat canbeobtainedfor transversewaves? Doesitdepend on wire diameter?1.2 A General Solution of the One-dimensional WaveEquationApartial differential equationstatesarelationshipamongpartial derivativesof a dependent variable that is a function of two or more independent variables.Suchanequation, ingeneral, hasa muchbroader classof solutions thananordinarydifferential equationrelatinga dependent variable toa singleinde-pendent variable, suchas theequationfor simpleharmonic motion. As withordinary differential equations, it is often possible to guess a solution of a partialdifferential equationthatmeetstheneedsof someparticular problem. For ex-ample, we mightguess thatthere exists a sinusoidal solution ofthewave equa-tion(1.1.3)ofthe form,., = Asin(o:x + (3t + -y).Indeed, substitution ofthis functionin(1.1.3) showsthatit satisfiestheequa-tionprovided ({3/0:)2= c2 Although this solutionrepresents a possible formthat waves on astretchedstring cantake, itis farfromrepresentingthemostgeneral sort of wave, asthefollowinganalysis shows.We rewrite(1.1.3) intheforma2,., 1 a2,., ( a a) (a a )ax2 - ~ at2 = ax - c at ax + c at ,., = 0,where thedifferential operator operating on,., hasbeensplitintotwofactors.Thisfactorizationispossiblewhenc isnotafunctionofx (or t). Theformofthese operators suggests changing to two new independent variables u= x- ctand'll = X + ct. It is easyto show that (Prob. 1.2.1)a a a---= 2-ax c at aua a a-+-=2-ax c at a'll(1.2.2)andthereforethat (1.2.1)becomesa2,.,4-- = o.au a'llThewave equation inthis formhastheobvious solution,.,(u,v) = h(u) +b(v),(1.2.3)(1.2.4)where h(u) andb(v) are completelyarbitrary functions, unrelated to eachother andlimitedinformonlyby continuity requirements. Wethusarrive at6 TransverseWaves on a Strin,"Fig. 1.2.1 Arbitrary wave attwo successive instants oftime.d'Alembert's solution of thewave equation,7J(x,t) =ft(x- ct) + h(x + ct). (1.2.5)Whereasweexpectasecond-order linear ordinary differentialequationtohave two independent solutions of definite functional form, which may becombinedintoageneral solutioncontainingtwoarbitraryconstants, thewaveequation(1.1.3) hastwoarbitrary functions of x - ct andx +ct assolutions.Because the wave equationis linear, eachof thesefunctionscaninturnbeconsidered to be the sum of many other functions of x ct if this point of viewshould prove useful. For example, it is often convenient to subdivide acompli-cated wave into many partial, simpler waveswhose linear superposition consti-tutesthe actualwave.Let us nowexamine the properties of a solutionconsistingonlyof thefirstfunction7J(x,t) =ft(x- ct). (1.2.6)Figure1.2.1showsthewave at twosuccessiveinstantsintime, tlandt2. Thewave keeps its shape, and withthe passage of time it continually moves to theright. Aparticularpoint onthewaveat timeh, suchaspoint A 1 at thepo-sitionXl, has movedtopoint A 2 at theposition X2 at thelater timet2. Thetwopointshavethesamevalueof 7J; that is, ft(Xl - Ctl) = ft(X2- CI2). Thisfact implies that Xl - Ctl =X2- Ct2' HenceX2 - XlC = ,12- IIshowing that the wave (1.2.6) is moving in the positive direction with thevelocityc. Bya similar argument, h(x +ct) represents asecondwave pro-ceeding inthe negativedirection, independently ofthefirst wavebutwiththesame speed.Wehaveestablished, therefore, that thewaveequationpermitswavesofarbitrary but permanent shape to progress in both directions on the string withthewavevelocityc= (TO/>"O)1/2. Althoughthewaveequation(1.1.3) doesnot1.3 Harmonic or Sinusoidal Waves 'Jinitselfrestrict theamplitudeandformof thewavefunctions hand hthatsatisfyit, theconditionsunderwhichthewaveequationhasbeenderivedre-strict the wave functions applying to the string to a class of rather well-behavedfunctions. They must necessarily be continuous (!) and have rather gentlespatial slopes; that is, IA2, Ra is positive, which implies thatthe reflectedwave hasthe same phase as the incident wave, whereas ifAl O - ~ t h"'"1l(a)___2_m__f-\;-g2m-60 kglw"",IIlIOkg2mso kg(b)Im-(e)(d)l-._--2mProb.3.5.1- -/ayas before, the boundaryconditiononthe velocitypotential at the surface becomes~ = ~ a 2 t / >a2x ay TOat2whichreplaces(6.2.6)applyingwhengravity alone is acting.In the preceding section we foundthatthe boundary condition atthe sur-faceserves to establishthe formof the time functionT(t) occurring as a factorinthevelocitypotential andthat it doesnotinfluencethe formofthespatialfactors. Hencewe can adopt thevelocity potential (6.2.14)as suitable forpuresurface-tension waves and substitute it in (6.3.4)to obtain the equation for T(t).We findthat(6.3.5)d2T +(ToK3tanhKh)T= 0,dt2Poso that the time dependence again is simple harmonic, but nowwith thefrequencyw = (;: K3 tanhKh)1/2.The wavevelocity ofpure surfacetensionwaves is therefore(6.3.6)W (TO )1/2C =- = - K tanhKh .K Po(6.3.7)(6.3.8)The velocity is small compared with that of gravity waves unlessK is large, i.e.,waves of short wavelength. Inthis limit we can settanhKh= 1, sothatc = (T:Y/2.The speed of waves controlled by surface tension therefore varies asK1/2, whereasthe speedofwaves indeepwater controlledby gravity(6.2.19) varies asK-1/2Hence we expect that surface tension is the controlling factor for waves of shortwavelength(ripples), whereas gravity is the controlling factor for waves of longwavelength.Whenbothsurfacetensionandgravityareacting, we must restore thegravitational termtoBernoulli'sequation, givingfor theboundaryconditionaty= hPoPoTO a2T/ at/>- - + g(T/" + h) = - + constPo ax2at(y= h). (6.3.9)6.3 Effect 0/ Sur/au Tendon 193The boundary condition onq" inturn, takesthe form(6.3.10) (y= h).TO iJ3q, iJq, iJ2q,----g-=-Po iJx2iJy iJy iJt2The equation for T(t},foundby substituting(6.2.14)into(6.3.1O), is now(6.3.11)sothat the frequencyofthe wave is[ (ToK3) ]1/2W= gK + -;;; tanhKh .(6.3.12)For deep water, tanhKh =1, so that the expression for the wave velocity becomesc ==ToK)1/2.K K Po(6.3.13)Thisresult reduces, as it should, to(6.2.19) whenTO= 0andto(6.3.8) wheng =O. The velocityof waves on deepwater as a functionof wavelengthisshown inFig. 6.3.1. The dashedlines showhowthewavevelocitywouldvaryeither inthe absence of gravity or inthe absence of surface tension.WithbothIOr------c---------r-------.---------,_c_emin Fig. 6.3.1 Wave velocity of surface waves whenboth surface tension and gravity are acting(h X).194 Wall"" on aLiquid Sur/auacting, there is aminimum wave velocity

Cmin = -;;; = 23.5em/sec(l8e),occurring at thewavelength(6.3.14)('To )1/2Am== 211" -Pog1.73 em. (6.3.15)Theexistenceof aminimumwavevelocitymeansthatawindwill fail todis-turb the surface of water and will not set up a system of waves unless its velocityexceedstheminimum value just calculated. Additionalresults relatedto wavescontrolled by gravity and surface tensionmay be found among the problems.Problems6.3.1 Derive(6.3.14)and(6.3.15)andverifythenumerical results.6.3.2 Showthat (6.3.13), usedin conjunctionwith(6.314) and(6.3.15), gives the relation(6.3.16)This impliesthatforagivenC >Cmin. thetwocorrespondingwavelengthshavetheconstant>'m astheirgeometricmean.6.3.3 Use therelation(4.7.7), namely, (group = Cph - >'(dCph /d>.), in conjunctionwithFig. 6.3.1 todescribehowthegroupvelocityof water wavesdependsonwavelength. Showthat(6.3.17)bymakinguseof (6.3.16).6.3.4 Makean W-K diagramfor water waves when bothsurface tensionand gravityaretaken into account. Assume that the water is deep, so that tanhKh= 1. Discuss the phase andgroupvelocity ofthewaterwaves onthebasis ofthis diagram.6.3.5 Show how to modify the theory of pure surface tension waves when the plane interfaceis between two liquids of densitiesPo andand, in particular, show thatC =hK/(pO +6.4 Tidal Walle.. and theTide.. 1956.3.6 Extend the treatment of Prob. 63.5to include gravity, as well as surface tension, andshow that for "h;;: 1 (seeFig. 6.3.1),(PO - P ~ g "'0" )1/1C =, Po +P ~ ;+Po +P ~ 6.4 Tidal Waves and the Tides(63.18)In Sec. 6.2 we noted that so-called tidal waves, which are gravity waves havingawavelength muchgreater than the water depth, areespeciallysimple. Weshowedthat thehorizontal motionof thewaterisverynearlythesame at alldepths and that the wave velocityis independent of wavelength, though itdepends on water depth. It is possible togive an elementary treatment ofthesewaves,withoutuse ofthevelocity potential. Wethendiscussbrieflythecauseofnatural tidesinthe ocean, whicharebasicallytidal wavesgeneratedbythemoon and sun.(a) Tidal wavesWe supposethatwaterisconfinedtoalongcanal ofdepthhandunit width,as in our earlier treatment of gravity waves. Thexaxisis alongthecanal, andthe y axisvertical, withthe origin at thebottom ofthecanal. Sincethewatercanbeconsideredasincompressible, theequationof continuityrequires thatthe dilatationvanish,a ~ a."v . ,---+ - +- = 0,ax ay(6.4.1)where, asusual, ~ and."are thex andydisplacement components. Wenowsuppose thatthe waves inthe canal have awavelengththat is very great com-pared withthedepthofwater, sothat wecan assumethat thehorizontal dis-placement ~ is independent of yandmuchgreater thananyrelatedverticaldisplacement. Inthiscasewecanimmediatelyintegratetheequationofcon-tinuity (6.4.1) toobtain(6.4.2)Theconstant ofintegrationhasbeenset equal tozero, sincethevertical dis-placement." must vanish atthe bottom ofthe canal. The vertical displacementofthe water atthe surface isthereforeproportional tothe straina ~ / a x ,a ~.",,= -h-ax(6.4.3)196 Waves onaLiquid Surface----T------------I AI VIh B' C'_I B, C,:' I I I II I I II I I II I I I! ! IFig.6.4.1 Tidal wave in acanal.LetusnowapplyNewton'ssecondlawtoadisplacedelementofvolumeofthe water A'B'C'D', as shown in Fig. 6.4.1. The element has the equilibriumposition ABCD and hasthe dimensions.::lx and.::ly inthe xy plane and unity inthe z direction. As a result of the slope of the water surface, the pressure on thefaceC'D' oftheelement isgreaterthanthepressure onthefaceA'B' bytheamountaT/I>.::lP = Pog-.::lx.ax(6.4.4)We equatethenethorizontal forceontheelement -.::lP.::ly tothemass oftheelementPo .::lx .::ly times its horizontal accelerationto obtain the equationofmotion(6.4.5)Sincethemotionof thewaterisverysmall inthevertical directionincom-parisonwiththatinthehorizontaldirection, we canneglect entirely the verti-calcomponents of forceandacceleration, IfwenoweliminateT/I> using(6.4.3),we arrive atthe ordinarywave equation 1 -=--,ax2c2at2wherethewave velocity is given byc== (gh)1/2,(6.4.6)(6.4.7)inagreement withthevalue(6.2.20) foundearlier. Itisinterestingthat here,where we have made simplifying assumptions at the start, we are led to a differ-ential waveequation, whereasintheearlier moreexact treatment, nodiffer-6.4 Tidal Waves and theTides 197ential waveequationwasfoundtoexist. Bytakingapartial derivativeofthewave equation(6.4.6) withrespect tox andmaking useof(6.4.2), we see thatthe vertical displacement of waves in the canal satisfies an identical wave equa-tion. We have discussed the principal features of these tidal waves in Sec. 6.2.(b) Tide-generatingForcesBeforewecanrelatethewavemotionwehavebeendiscussingtooceantides,weneedtoseehowaheavenlybody, suchasthemoonor thesun, producestide-generatingforces ontheearth. For simplicity, let us consider theforcesproduced just by the moon, of mass M, located at adistance R fromthe centerof theearth. Weshall findit convenient totaketheearth'scenter tobetheorigin of a coordinate framewiththe z axis directed away fromthe moon,as inFig. 6.4.2. Aunit massat a point P, locatedontheearth, has its potentialenergy increased byGMrGM(6.4.8)becauseof thepresenceof the moon, where r is thedistancefromPto thecenter of the moon and G is the gravitational constant. Since the coordinates ofpoint Pare small comparedwithR, we can expandVMinpowers of x/R, y/R,and z/R,GM 1R [(1 + z/R)2 + (X/R)2 + (y/R)2]1/2GM GM GM- - + - z - - (2z2- x2_ y2) +R R2 2R3(6.4.9)M!IFig.6.4.2 The moon-earth system.198 Waves on aLiquid Surfaceo--t-----+o=-------+-- _zFig. 6.4.3 Tide-generating forces inthexzplane.Hence the moon's attraction foraunit massat pointPhasthecomponentsF,,=FII=F. =aVM---=axaVM---=ayaVM---=azGM--x+R3GM--y+R3GM 2GM--+--z+R2 R3(6.4.10)The first termin F. is considerably greater thanthe second, since the radius ofthe earthis aboutone-sixtieththedistancetothemoon. The first termis evi-dentlyequal totheaverageforceof themoononeachunit massoftheearthandis responsiblefor thecentripetal accelerationintheearth-moonsystem.An observer on earth cannot observe this term as a force, however, for the samereason that anastronaut cannot observe the gravitational attractionof theearth: it is precisely cancelled bythe free-fall accelerationthataccompanies it.Theremainingforcecomponents, whoseaveragesvanishfor theentireearth,havethe interesting distribution at the earth's surface as shown inthe xz planein Fig. 6.4.3a. Thecomponent of theseforces parallel totheearth'ssurface,shown inFig. 6.4.3b, is responsible forthe principalcomponents oftides.The sunproduces asimilar pattern oftide-generating forces,of abouthalfthe amplitude ofthose of the moon, symmetrical about the earth-sun axis. Themuchgreatermass ofthe sunismorethan offset by its greater distance away,which enters as aninverse cube intheexpressions for the tide-generating force.Theexistenceof two(not one) regionsofhightideontheearth atanytime,(6.4.11)(C==const), (6.4.12)6.4 Tidal Waves and theTides 199arises fromthemirror symmetrythat the forcesystem illustratedinFig. 6.4.3possesses withrespect tothexyplane.(c) Equilibrium Theory of TidesNewton proposed an explanation for the tides in the ocean known as theequilibriumtheory. As a model he imagined the earth to be covered byanoceanof constant depth, subject totheforce system(6.4.10), as well as thegravitational attraction ofthe earth. Equilibrium occurs when awater particleonthesurfaceof theoceanhasa constant potential energydue toboththemoon andtheearth, nomatter wheretheparticle is located.Ifwelet 1/ betheheight of thehypothetical oceanabovemeansealevel,thenVB =g1/ isthepotential energyperunit massofwater at thesurfaceofthe ocean due to the earth's gravitational field.The potential energy of the tide-generating force of the moon atthe earth's surface is given by the thirdterm in(6.4.9) with x2+ y2 +Z2= a2, where a is the radius of the earth. We can put thispotential term in better form by going to spherical coordinates with z =a cos8.Evidently for points on the earth's surface, 2z2- x2- y2= a2(3cos28 - 1), sothatthetide-generating potentialenergy becomes, GMV = - - a2(3cos28 - 1).M 2R3Hencetheequilibriumshapeof theoceancoveringtheearthisgivenbytherequirementthat, GMVM +VB =- - a2(3cos28 - 1) +g1/ = C2R3andthe height ofthe ocean abovemean sea level becomesGMa2C1/ = --(3cos28 - 1) + -.2gR3 g(6.4.13)(6.4.14)InProb. 6.4.2it isfoundthat C =O.Ifweintroducethemassof theearth,MB =a2g/G, we finallyhave that1M(a)31/ = - - - a(3cos28 - 1).2MBRInFig. 6.4.4 isshownaplotof (6.4.14) illustratingthetwotidal "moun-tains"ofthe equilibrium theory. Their height duetothemoonis about 0.6m,andthe height duetothe sun is about one-halfthis value(Prob. 6.4.3).The equilibrium theory ignoresthe dynamical aspects of thetides broughtabout bythe rotationofthe earthandthemotion ofthemooninits orbit. Itpredictshighwaterwhenthemoonpassestheobserver'smeridian, ononeor:MID Wa...... on aLiquid Sur/auxzFig. 6.4.4 The tidal mountains of theequilibrium theory.(6.4.15)theothersides oftheearth. Incontrast, highwater is foundto occur delayedmanyhoursafter the moonhaspassedthis meridian. Anexplanationof thedelayinvolvesthenotion oftidal waves. Nevertheless, the equilibrium theoryexplains the existence of two tides per day and howthe joint effect of the tide-generatingforces of thesunandmooncombine toproducespringandneaptides. Italsoexplains adiurnal inequalitythat arisesasaresult oftheincli-nation of the moon's orbit withthe celestial equator. When the moon passes theequator, the diurnal inequality due to themoon vanishes.(d) The Dynamical Theory of TidesAbout acenturyafter Newton, Laplaceproposedadynamical theoryof thetides. Asasimplifiedmodel toillustratehis theory, let usconsider that theearth has a canal of uniform depth encircling it at the equator and suppose thatwaterinthecanal isactedonbythetide-generating forces (6.4.10). Wecanfurther suppose that theearthdoes not rotate but that the moonrevolvesaroundit in24hr and50min, asit appears todotoanobserver onearth.The diagram inFig. 6.4.5 shows an arbitrary point P onthe equator, withthelongitude41, the moon with an angular displacement 8 from P traveling with anangular velocity w. From the diagram we see that 8= wt - 41. Let us substitutethis expression for8 inthetide-generating potential (6.4.11),whichthenreads, GMVM = - - a2[3cos2(wt - 41) - 1].2R8The tangential force on a unit mass of water in the canal at point P is thereforeo V ~ GMa .F. =- - = --3 cos(wt - 41) sm(wt - 41)a 041 R83GMa.= - - sm(2wt - 241). (6.4.16)2 R8Fig. 6.4.5 Coordinates for the dynamicaltheory oftides. I \I / "1/ 11wThis force contributes to the horizontal acceleration of water in the canal,iJ2UiJt2, andsomust beadded tothe term inthe waveequation(6.4.6)duetopressure. Withthis driving forcethewave equationbecomes(6.4.17)wherek==3GMa/2R3is theamplitudeof the tide-generatingforce per unitmass. In(6.4.17) wehavereplaceddxbya dq" which measures incrementaldistance along the equatorialcanal.We expect that the equation fortidal waves inthe canal withthe perturb-ing force(6.4.16)has aparticular solution ofthe form = sin(2wt - 2q,) (6.4.18)towhichmaybeaddedsolutionsof (6.4.6) representingfreeoscillationsandwaves of an arbitrary nature. On substituting(6.4.18) in(6.4.17), we findthatthe forcedtidal wavehasthe formka2 = (2 22) sin(2wt - 2q,)4c - waand, from(6.4.3), thatthetidal wave has avertical displacement kahTIl> = -h - = (2 22) cos(2wt - 2q,).a iJq, 2 c - wa(6.4.19)(6.4.20)Theequationsjust obtainedshowthat thedynamical theorypredicts, asexpected, two tides for each apparent revolution of the moon. If c2>w2a2,high202 WavE''' on aLiquid Surfauwater is inphasewiththe moon andthe tides are direct. However,if c2=::: w2a2,resonance occurs addthere should be tides of tremendous height. In Prob.6.4.4it isfoundthat thecritical depthforresonanceisabout 22km, whichiscon-siderably deeper thantheoceans. Hencethetidesshouldbeinvertedneartheequator. For acanal encircling the earthnear apole, the tides shouldbe direct,withresonanceoccurring foracanal at anintermediate latitude.Laplaceshowed theoreticallyfor aT. extendedocean coveringthe earth,that north-south flowcancels tidesat theintermediatelatitudebut that theequatorial tides should be inverted and those in polar regions, direct. In practicelandmasses considerably modifythetidal waves induced by themoon andthesun, asdoCoriolisforces arisingfromtherotationof theearth. Asa result,theory, coupled withthe knownmotion ofthe moon andsun, is ableto predictthe frequency but not the amplitude and phase of the important harmoniccomponents oftides at variouspoints ontheearth. Theactual phase andam-plitude of each component at any locationmust be foundby harmonic analysisoftheobservedlong-termbehaviorof thetide at that location. Oncethishasbeen done, it is then possible by harmonic synthesis to predictthe local tides atall futuretimes. Even so, variations inbarometricpressure over the ocean andtheactionof windcanalter theactual tidepatternina randomfashion.tProblems6.4.1 Verify the expansion(6.4.9)using(a) the binomial theorem and(b) Taylor's theoremforafunctionofthreevariables. Whichmethodis more efficient?6.4.2 Show that C= 0 in(6.4.13)by requiring that the average of 'lover the surface of theearthvanish.6.4.3 Findtheamplitudesofhighandlowtidesfrommeansealevel duetothemoonbytheequilibriumtheory. Assumethat MHI M=81andRia=60. Computetheratioofthesolartide tothe lunartide, given that M.unl MH=332,000 andR.unla=23,200.6.4.4 Findthedepthof anequatorial canal for whichthetidal-wavevelocityjust keepspace with the apparent motion of the moon around the earth. If the canal is only 2 miles deep,findthevertical andhorizontaldisplacements of the lunartide.6.4.5 Find at what latitudelunarresonance should occurin acanal 2 miles deep encirclingthe earth.tAninterestingdiscussionof tidal processesisgivenbyA. Defant, "Ebband Flow,"TheUniversity ofMichiganPress, AnnArbor, Mich., 1958.6.S Energy and PowerRelDtlons 2036.5 Energy and Power RelationsThepotential energyperunit areaassociatedwithasmall-amplitudegravitywave, neglecting surfacetension, isequal totheworkrequiredtodisplacethewater fromits level equilibriumstate. At apoint wherethewatersurfaceisdisplaced vertically upward an amountT/h,an infinitesimal element of horizontalarea dx dz andheight T/hhas amassPOT/h dx dz, andtheworkdoneinlifting itscenter of mass a distance hh abovethe equilibrium level is therefore(6.5.1)(6.5.2)The same amount of work is involved in displacing the surface whenT/h is nega-tive, instead of positive. Hence thepotentialenergy per unit area isd2WVI =dx dz =fPOgT/h2Forthegravitywavedescribedby(6.2.23), T/h = 71",sin(Kx - wI), sothat thepotential energy density becomesVI =fPogT/m2sin2(Kx - wI),whichhasthe average valueVI = !PogT/m2(6.5.3)(6.5.4)Thekineticenergydensityof thegravitywave (6.2.23) includescontri-butionsfromthevertical aswell asthehorizontal motionofthewater intheregionbetweenthe bottom and surface of thewater,ria .K 1 =fPO 10 (e +li2) dy. (6.5.5)If values of ~ and li are computed from(6.2.23)andw is replacedbythe valuegivenby(6.2.16), K1 takesthe somewhat complicated formKI= !PogT/m2{l +.2Kh h [sin2(Kx - wi) - COS2(KX - WI)]}, (6.5.6)smh2Kwhich, however, simplifies on averaging tobecomeK1 =!PogT/m2(6.5.7)Hence, though theaverage potential andkineticenergydensities are equal,their instantaneousvaluesarenot equal, a relationshipthat existsfor manytraveling waves. The total average energy density,E1= K1+VI = fPogT/m2, (6.5.8)isequal totheworkthat wouldberequiredtogivethesurfaceitsmaximumdisplacement against theactionof gravity. Sincetheequal averagepotential204 Waves on aLiquid Sur/auandkineticenergy densitiesjust computeddonotdepend onthe depthofthewater, they continuetoholdforwaves on extremely deep water, as well as forwaves on shallow water, i.e., fortidal waves.The power density in gravity waves, i.e., the energy passing in one secondthroughatransverseplaneof unit widthextending frombottomtosurface ofthewater,is evidently givenbythe integralJ" a ~P1= - p-dyo at '(6.5.9)where pis the time-dependent pressure in excess of the static-equilibrium value.Thetotal pressureat apoint (x,y), accordingtotheBernoulliequation(neg-lecting thej'll2 term),is givenbyaq,P= Po + Pog(h- y) +Po-'atwhereasthe static pressure atthis depthisP.tatic= Po +Pog(h- y).(6.5.10)(6.5.11)Hencetheexcess pressure atanypositionandtime, describedbythevelocitypotential q" is simplyaq,p =P- P.tatic =PO-'at(6.5.12)(6.5.13)(6.5.14)In terms of the velocity potential, the expression for the power density becomesP1= - Po (Oilaq, aq, dy.J( at axThe velocity potential (6.2.21) for the surface wave (6.2.23)takes the formcoshKyq, = TlmC -.--h COS(KX- wt)smhKwhenAis replaced by its equal, cl1m/sinhKh. Using this expression for q" we findthat(6.5.15)(6.5.16)The average power, therefore, is givenby151 =(jPogTlm2) [ jc ( 1+ S i ~ ~ : h ) JThe firstfactor is the averagetotal energy(6.5.8), andthe second factor is the6.5 Energy and PowerRelations 205group velocity ( 6 . 2 . 2 7 ~ ofthewavetrain. Wehavefoundthe samerelationtohold for other dispersive waves.Problems6.5.1 Make a reasonable estimate of the average amplitude and wavelength of typical waveson the ocean and compute the power they carry. Express the result inhorsepower per mile ofwavefront.6.5.2 Discuss the energy andpowerrelations for pure surface-tensionwaves.seven*Elastic Waves in SolidsInChap. 4wediscussedcertainspecial cases of waves insolids, corre-spondingtothesimpletypesofelasticdeformationconsideredinChap. 3. Inordertotreat more general cases, it is necessary toreformulate our descriptionofstressandstrain, expressingthesequantitiesas tensors ofthesecondrank.Themathematical techniquesrequiredhere arise inmany importantbranchesof physics, e.g., themotionof rigidbodies, theelectromagneticpropertiesofanisotropic materials, and relativity theory.7.1 Tensors and DyadicsTheintroductorydiscussionofelasticityandelasticwavesinChaps. 3 and4wasrestrictedtosimplestructures forwhichsymmetry of one sort oranother206generally allowedustouseonly asinglestraincomponentintheanalysis. Wenowdevelop aformal way of expressing all thecomponents of strain aswell asthose of stress as asingle entity, corresponding totheway avector expresses aphysical concept distinct fromits threecomponents. Wefindthat stressandstrain are in fact second-rank tensors, or dyadics, as suchtensors are called whenexpressedinthevector notationof Gibbs. Inthepresent sectionwestart byreviewing some basic notions underlying scalars and vectors and then show howa dyadic can be defined in a way that is consistent withthese notions.We com-pletethesectionwithabrief accountofsomeofthemathematical propertiesof dyadics.Many quantities of interest in aphysicaltheory can be expressed by singlenumbers that depend on position and time. We refer to descriptions of this sortas scalar fields; density and the velocity potential in a fluid are examples of suchfields. Suppose that we let q, =q,(x,y,z,t) standfor a particular scalar field,wherex, y, zare thecoordinatesofa point referredtoa reference (inertial)frame Sand t is time.With respectto a different frame S', the same scalar fieldis now expressed by a different functionq,' = q,'(x',y',z',t). For the two functionstorepresent the samephysical quantity, itisnecessarythat q, =q,' wheneverx, y, z and x', y', z' arethe coordinates ofthe same point in space. Wethensaythat q, isaninvariantfunctionofpositionandspeak of thefieldasaphysicalscalar field.Usuallythedefinitionof aparticular scalar fieldensuresthat ithavethisproperty of invariance. However, it is easyto exhibit ascalar fieldthat is not aphysical scalar field. For example, the xcomponent of thevelocity of aflowingliquidhas adefinitenumerical value ateachpoint in areference frameS, butits valueat anypoint dependsontheorientationof the x axis of S. Hencev",(x,y,z,t), whichis onecomponent of a vector velocityfield, does not itselfconstitutea physical scalar field, since it is not invariant withrespect toatransformationof coordinates. Thestatusofascalarfieldarrived at by somemeans that does not clearly establishwhether or not it is aphysical scalar fieldcan always be settled by testing its invariance with respect to a spatial rotationof the coordinate frame about acommon origin.A vector fieldis somewhat more complicatedthan ascalar field, since it in-volves both a magnitude and a direction that are assigned in some wayto eachpoint in space. If a mathematical representation of a vector field is to constitutea physical vector field, the vector at any point in space must be independent ofthe coordinate frame used in specifying the location of the point. Since the rec-tangular components of avectortake on values that dependonthe orientationof the axes of the reference frame, it is convenient to devise a test for invariancethat tellshowtherectangularcomponentstransformunderarotationof thecoordinate frame.We can establishthetestforinvarianceby examiningthetransformation398 Eltl.fticWalle.. in Solid.._.AJI""'"-------------------,....-Fig. 7.1.1 A vector rAreferredtotwo reference frameshaving acommon origin.ofadisplacement vectorOAbetweenanorigin0 commontothetwoframesand some fixed point A. Let us designate this vector by rA, with a magnitude rAandadirectionfixedinspace, say, towardthestar Arcturus(seeFig. 7.1.1).Withrespect toS, theterminus of rAhasthe coordinatesXA, YA, ZA,which arealsothecomponentsof thevectorrA. Withrespect toasecondframeS', theidentical vector=rA has the componentsIn terms of unit vectorsalongthetwo sets of axes,rA =iXA + jYA + kzA=++(7.1.1)(7.1.2)(7.1.3)If 1'11 is the cosine of the angle between the x' axis and the x axis, 1'21 the cosineof theanglebetweenthey' axisandthexaxis, etc., it isclear that theunitvector i has the components1'11, 1'21, 1'31 along the x', y', and z' axes. Extendingthis result tothe other two unit vectors in S,we can writethati = 'Y11i' + 'Y2J' + 'Yuk'j = 'Y12i' + +k='Y13i' + 'Y23j' + 'Y33k'.If we now substitute (7.1.3) in (7.1.1) and compare the resulting expression with(7.1.2),we findthatthe set of equations= 'Y11XA +'Y12YA +'Y13ZA='Y21XA +'Y22YA +'Y23ZA='Y31XA +'Y32YA +'Y33ZA(7.1.4)expresses the transformation of the components of a displacement vector under asimple rotation of axes. We now assert that any set of three functionsV,,(x,y,z,t),(7.1.5)'J.l Tensors andDyadics Z09Vu(x,y,z,t) , andVz(x,y,z,t) thattransforms under a rotation of axes according tothe scheme (7.1.4)constitutes the components of a possible physical vector field.Insteadof definingaphysical vectorfieldasonethatassociatesamagni-tude and direction with each point in space, we see thatwe can define it equallywell as afieldspecifiedbythreefunctionsofposition andtime,oneassociatedwitheachcoordinate axis. Thethreefunctions, however, cannot bearbitrarilyspecified butmust be suchthattheytransform upon rotation of thecoordinateaxesas dothe componentsof adisplacement vector. This wayof definingaphysical vector field by its transformation properties is the one that can be mosteasily generalizedtodefinetensor fieldsand, incidentally, istheapproachusedinthe formal mathematical discussion of n-dimensionalvector spaces.We definea second-ranktensor fieldas one that associates a functionofpositionwitheachoftheninepairsofcoordinateaxes, xx, xy, ... ,zz. Thenine functionsTu(x,y,z,t), TXl/(x,y,z,t) , etc., cannot be arbitrarily chosen if theyaretorepresent a physical tensor field, but theymust transforminthesamewayasthenine coordinatepairs, xx, xy, etc. If welet thenumber 1 standforthesubscript x, 2 forthe subscript y, and 3 forthe subscript z, thetransforma-tion for atensor field, corresponding to(7.1.4)for avector field, takesthe form3 3T:m = L L 'Y1&'YmIT.I..-11-1Wecanwrite the tensor having thecomponents T rr, Try,.. , as thedyadicT = iiTrr + ijTxu + ikTxz + jiTux + ... , (7.1.6)where the nine unit dyads, ii, ij, etc., enablethe tensor to bewritten as a singleentity, just asthethreeunitvectors, i, j, k, enable avectortobe written asasingle entity V= iVx+ jVu+ kVz. We shalloften finditconvenienttowritethe dyadicinthe formof a3X3 matrixt(7.1. 7)without the unitdyads appearing.The most important mathematical operation with a dyadic consists in dot-ting avector intoit, either fromthe left side or fromtherightside. Wedefinethe operations on the unit dyads,i (ii) = (i . i)i = i, i . (ji) = (i . j)i = 0,t Following P. M. Morse andH. Feshbach, "Methods of Theoretical Physics,"McGraw-HillBook Company, NewYork, 1953, we usethe term dyadic as anunequivocal designation forasecond-ranktensor, just asthetermvector designates a first-ranktensor. Manyauthors, how-ever, usetheunmodifiedtermtensor as implying secondrank andreserve theterm dyadic forthe notation(7.1.6) that uses unit dyads explicitly, as opposedtothematrix notation(7.1. 7).:lIO ElosticWaves in Solidsetc., as obvious extensions of ordinary vector analysis. Hence we findthatV T = V"(iT",,,, +jT"'1I +kT",,,)+VII(iTII", +jTl1li +kTII,,)+ V,(iTu +jT'1i +kTzz)= i(V",T",,,, + VIITII", +V,T,,,,) +Similarly(7.1.8)(7.1.9)Accordingtotheabovedefinitions, dottingagivenvectorintoadyadicfromone orthe other of its sides gives risetoone orthe other oftwonewvec-tors, (7.1.8) and(7.1.9), generally differing in magnitude and direction fromthegiven vector but withmagnitudes linearly related to the magnitude of the givenvector. The two new vectors are equal if Tis a symmetric dyadic, i.e., if T"'II = Til""etc. Only inthis event do V and Tobey the commutative law of multiplication.One can showthat a symmetric dyadic Scan always be reduced to diagonalor normal formbyproperlychoosingtheorientationof thecoordinateaxes,whicharethencalledtheprincipal axes ofthe dyadic. Innormal form(S"''''00)S= S"""ii +S1IIIjj +Szzkk= 0 SIIII 0 .o 0 Szz(7.1.10)The off-diagonalcomponents, suchas S"II' all vanish.A dyadicAisanantisymmetricdyadicwhenA"'II = -A llx,All' = -A'II'Au= -A"", and Au= AIIII= Azz =O. For suchadyadicA V= -V A= i(A"'IIVII - AuV,) += -AXV= VXA,where(7.1.11)(7.1.12)isthe vector representation of the antisymmetric dyadic A. It is easy to verifythatthe components of A, as definedby (7.1.12),transform as components of avector (Prob. 7.1.2). Thusthecross-productoperationof ordinary vector alge-braisequivalent toadot-product operationinwhichoneof thevectorsisre-placedby anantisymmetric dyadic.We definethe transpose of adyadictobe adyadic satisfyingTI. V= V T,or VTI = T V, (7.1.13)whereVisanarbitraryvector. EvidentlyT,,/ = Tx;', but Tx/ = Til"" etc. Anarbitrarydyadic such as (7.1.6) canbe separated intosymmetric andanti-symmetricparts by usingtheidentityT== T+ Tt +T- Tt.2 2'J.l Tensors andDyadics 211(7.1.14)Ifwewrite S for thesymmetricpart andAfor theantisymmetricpart, it isevident that( TQ j(TzlI + Tllz)HT+TQ)S = j(TlIZ + TzlI) TIIIIj(Tliz + TzlI)j(Tzz +Tzz) j(TzlI +Tllz) Tzzand j(TzlI- TlIZ)HTQ- Tu)0j(Tliz- TzlI) j(Tzz- Tzz)j(TzlI- Tllz) 0(7.1.15)(7.1.16)(7.1.17)The vector representing Acan evidently be foundby placing avector-productsymbol (X) betweenthe elements of the dyads intheoriginal tensorT andmakinguseof thefact that i X i =0, i X j = -jX i = k, etc., and multi-plying the resulting vector by j.The dyadic1 ! Dis calledthe idemfactor, andplays thepart of aunitdyadic, thatis,V1 = 1 V= V. (7.1.18)Itispossibletodefine tensorsof higher rankbyassociatingfunctions ofx, y, Z,t withthe 27 (=33) coordinate triples xxx, xxy, ... ,ZZZ, the 81 (=34)coordinatequadruples, etc. A tensorissaidtobeofrankr whenit has3rele-ments. A scalar can bethought of asatensor of zero rank; an ordinary vector,atensor of first rank;adyadic, atensor of second rank;etc.Veryoftenavectorequationcanberecast intoaforminvolvingdyadicswithusefulresults. We can illustratethe procedure by an example drawnfrommechanics. Theangular momentumofarigidbodyrotatingwiththeangularvelocity wis expressedbytheintegralJ = f r X(wX r)dm, (7.1.19)where r istheposition ofthemasselementdm. If weexpandthetriplevectorproduct inthe usual manner andthen factor outthe angular velocity w,J=f[wr2- (w' r)r] dm= w f (r21 - rr)dm= w' I. (7.1.20)212 ElasticWalles in SolidsThe symmetric dyadic1==f(r21- rr)dm (7.1.21)is the moment-oj-inertia dyadic, having the nine components In= f(y2 + Z2) dm,I : I : ~ = I ~ " , =- f xy dm, etc. The diagonal components, I",,,,, I ~ ~ , Iu, are recognizedas the moments oj inertia about the three coordinate axes. The off-diagonalcom-ponents, suchas I " , ~ , areknownastheproductsoj inertia. Theaboveexampleillustrates how a dyadic can arise naturally in an analysis that initially involvesonly scalar andvector quantities.Inendingthis brief introductiontodyadics, let us recall againthat thedefinitionofatensorfieldstateshowthefieldtransformsunderarotationofcoordinate axes. Suchadefinition assuresus thatwe are dealingwith amathe-matical descriptionwhichis independent of theorientationof thecoordinateframe. Sincethebasiclawsofphysicspresumablypossess this independence,it follows that any valid quantitative physical law can be expressed by an equa-tionthatissomesortof atensor equation-of zerorankif itis ascalar equa-tion, of first rankif it is a vector equation, of secondrankif it isa dyadicequation, etc. Furthermoretheequationexpressing thephysical lawmust notcontainanyindicationof theorientationof theaxes of the referenceframe.Hence when the equationis transformed bya rotationof axes, sayfromaframe Sto a frame 5', itmust keep the same functional formin 5' as in S.Problems7.1.1 (a) Express the unit vectors I', J', k' in terms of I, i,k, and show thatthe transforma-tioninverseto(7.1.4) isJ'j =2:'YijX;, wherex, =x, x. =y, andJa=z (b) Makeuse oftheproperties oftheunit vectors(suchas 1'1= I, I' i = 0, etc.) toestablish that there aresix3relationsof thetype L 'Yij'Yik = Ojkconnectingtheninedirectional cosines involvedin the;=1transformation bet\\een Sand S' (Ojk is theKronecker delta). Interpret geometrically the factthat thereforeonlythreeof the 'Yi/Scanbechosenindependently. (c) Showthat thedeter-minant of 'Yij equals 1. Interpret the sign.7.1.2 Prove thatthethree components Az== Au., Au ==A.., A. ==Azu of an antisymmetricdyadictransform asthe components of a vector andhence enahle A to he treated as a physicalvectorfieldwhenit is afunctionofx, y, z, andt.7.1.3 Let 0, beaunit vectorofvariabledirection\\ ithanorigin at thecoordinateorigin.Let thepositionvectorr fromthe originhedeterminedbythe equationr = S' 0"7.Z Strainas aDyadic Z13whereSis a symmetricdyadic. When0, variesindirection, its terminusdescribesa unitsphere. Show thattheterminus of r thendescribes anellipsoid.Hilll: Assumethecoordinatesystemis oriented suchthat S isdiagonal.7.1.4 Investigatethe geometrical si,:(nificance ofthe equationrSr= 1,where r = Ix + jy + kz andS is a symmetricdyadic.7.1.5 A vector has one scalar invariant-itsmagnitude or length-whenits components aretransformedby arotationof axes. Showthatasymmetricdyadic SischaracterizedbyIhreescalar invariants' (a) its trace, I, = S II + S22 + Saa; (b) the sumof its diagonalminors,ISll S'21 IS22 S231 + ISaa S31 I;12= S2' S22 +S32 Saa SI3 Sll(c) its determinant, I a = ISiil.7.1.6 Let S be any symmetricdyadicwiththethreeinvariantsI" 12, andI a, asdefinedinProb.7.1.5. When referredto principalaxes, S may be writteninthe form S= USI + jjS2 +kkSa. Showthat thethreeprincipal components S" S2, andSasatisfythecubicequationsa - IIS2+ I 2S - I a = O.7.1.7 Showthat theninedirectionalcosines Yii cannot beconsideredthecomponentsofa(physical) dyadicbyexaminingtheir transformationproperties. Showthat twosuccessiverotationsYibY;'j are in factequivalent to the single rotationY;i, whose components are foundby thematrixmultiplicationof"Iii andY;j'7.1.8 IfVisaphysical vectorfield, showthat thedivergenceofV, V V, isaninvariantscalar field. Hinl: By performing anaxis rotation show thatavz avo av. av: a v ~ av;-+-+-=-+-+-OX oy oz ox' oy' oz'when each side is calculated atthe same point in space. It is also instructiveto apply the testfor invariance to the two products V . Wand V X W, where Vand Ware both physical vectors.7.1.9 If AI, A2, andAa arethreearbitraryvectorsthat arenot coplanar, showthatanyarbitrary dyadic T canalwaysbe expressedasthe sumofthree dyadsT= A,B, + A2B2 +AaBaby suitably choosingthethreevectorsBI, B2, andBa.7.2 Strain as a DyadicWehavealreadydefinedandmadeconsiderableuseofthecomponentsof thestraindyadicinChap. 3. Wenowwish toseehowthedyadic notationdis-cussedintheprecedingsectionenables us totreat strainas a singlemathe-214 Ela..ticWa ..e.. in Solid..matical entity. Let usstart byassumingthat wehaveanundistortedelasticmediumat rest inan inertial frame. The position vector r = ix + jy + kzservestolocate apoint intheunstrainedmedium. When a(small)straintakesplace, themediuminitially at r suffers adisplacementp= iHx,y,z) +h(x,y,z) + kr(x,y,z), (7.2.1)(7.2.2)andthemedium at aneighboring point r + dr suffersthe displacementp + dp =i ( ~ + o ~ dx +o ~ dy +o ~ dZ) + j(- ..) + k(...).ox oy ozBysubtracting(7.2.1) from(7.2.2) wefind that therelativedisplacement ofthemediumisgivenbythe equationdp = drVp, (7.2.3)where dr hasbeen factoredoutonthe left side,leavingthe dyadicVpexpress-ingthestrain. Theninecomponentsof thestraindyadicVparemost simplyexhibitedbywritingthe dyadicinmatrix form,o ~ 07/ orox ox oxVp=o ~ 07/ or(7.2.4)oy oy oyo ~ 07/ oroz oz ozWerecognizethediagonal componentsasthethreetensionstrains E,n:, E1I1I, En,definedinSec. 3.1, withasumthatisthedilatation(3.2.4)(J = E;n: +EIIII +En,(7.2.5)whichwe knowtobe aninvariant ofthe dyadic(Prob. 7.1.5).The strain dyadic(7.2.4)in generalis not symmetric, but it can always bedecomposedintothesumof asymmetric and anantisymmetric partusingtheidentity (7.1.14). Let us denote the symmetric part by Eand the antisymmetricpart byep. Inmatrix formthetwodyadicsmay be writteno ~!e7/ +o ~ )!er+o ~ )ox 2 ox oy 2 ox oZE=! e ~ + 07/)07/1 er07/)(7.2.6)2 oy ox oy "2 oy + oz)! e ~ + or) ! e7/ + or)or2 oz ox 2 OZ oy ozo!(01; _ 01/)2 oy ox! ( o ~_or)2 oz ox! (01/ _o ~ )2 ox oyo!(or _o ~ )2 ox oz! (or _01/)2 oy OZo7.2 Strain as aDyadic 215(7.2.7)Let usnowregarddrasasmall but finitevectorofconstant magnitudebutof variabledirection. Itsoriginisat apoint Pinthemediumlocatedbythe vector r. Its terminustherefore describes a small sphere surrounding P. Asa result of a small elastic strain, the point P suffers a displacement p, carrying itto some newpositionQ, as illustratedinFig. 7.2.1. Points onthe small spheresurrounding Pnowbecome anonsphericalsurface surroundingQ, as describedby the vectordr +dp = dr(1 + E+ cj.(7.2.8)We can interpret(7.2.8) by first noting that 1 + E is a symmetric dyadic,sothat theterminusofthevector dr +dp, withanoriginat Q, describesanellipsoid(seeProb. 7.1.3). Accordingto(7.1.11) and(7.1.12) relatingtoanti-symmetric dyadics, theremainingterm dr cj> can be writtendr cj> = epX dr,Fig. 7.2.1 A small strain.dr+dp(7.2.9)J16 EllIsticWaves in SolidsQFig. 7.2.2 The small bodily rotation represented bycI- XdrwhencI- is smaIl.wherethe vectorep == i! (or _(71)+j! ( o ~ _or) +k!(071_ o ~ )2 oy oz 2 OZ ox 2 ox oy= j-VXp, (7.2.10)crossed into dr fromthe left, is equivalent tothedyadic cI> dotted into dr fromtheright. Evidently(7.2.9) expressesasmall (bodily) rotationofthemediumabout thepoint Q(seeFig. 7.2.2andProb. 3.3.7). No elastic distortion of themediumisconnectedwiththerotationdyadic cI>. ThedyadicEis termedthepurestraindyadic, todistinguish itfromthestraindyadic vp, whichincludesrotationaswell asdistortion.Wecansummarizeourpresent findingsbystatingthat themost generalsmall strain is one that causes:(1) A translationp of eachpoint in amedium.(2) A small rotationep ofthemedium about eachpoint.(3) A smallelastic distortionthat changes a sphere surrounding each point intheunstrainedmedium into anellipsoid.In the case of a homogeneous strain, items 2 and 3 are independent of position inthemedium. Insuchanevent theprincipal axesofthepurestraindyadicdonotchange withposition, and it is generallymost convenientto choose coordi-nateaxes x, y, z inthedirectionoftheprincipal axes. Thepure straindyadicthentakesthe simple formE= UEl +jjE2 +kkE3, (7.2.11)whereEl, E2, andE3 aretheprincipal extensionsofthemedium. Inthecaseofsmall strains we can usuallyneglect any small rotation of the medium, as givenby(7.2.9), indesignatingthedirection ofthe principalaxes.Thethree independent off-diagonalcomponents ofE, such asE =! (or +(71),1/< 2 oy oz(7.2.12)'1.3 Stress as aDyadic 21'1areone-halftheshearingstrainsinthethreecoordinateplanes, asdesignatedbythe subscripts ofthecomponents. InSec. 3.3wediscussedsheardistortionina planeand defined theshearingstrainin the yz plane by (3.3.3) to be'YII' = 'Y'II = or/oy + oT/loz. Hencewe cannowwrite thatEZII=Ellz =hZl/ = hllz =~ G: + :DEll'=E'II = hll' = h'lI =~ ( : ~ + ::)E.. =Ex.= h .. = h .. =~ G: +: ~ }Anyset of sixfunctions of positionE..(X,y,z) , EZII(X,y,Z) , ... , E(X,y,z)thattransform properly, andhence are components of a possiblephysical sym-metric dyadic,do not necessarily describe a possible state of strain of an elasticbody. The strain components must also be consistent with their definition (7.2.6)in terms of space derivatives of the displacement components ~ , T/, r. It is foundthat there exist six equations, called equations of compatibility, that the six straincomponents must satisfyidentically.t For simplicityit is oftenpreferabletoformulatethebasic differential equations forelastic wavesintermsofthedis-placementp ratherthanintermsofthe straindyadicE. We shallsee howthiscan be done in Sec. 7.5.Problems7.2.1 Show that an arbitrary strain deformation Ecanbe written as the sum of a pure shearand a hydrostatic compression,E= (E- j81) + j81,where 8 isthe dilatation.7.2.2 Express the strain existing in a stretched rod as a dyadic. Do the same for a dilatationand for a shear distortionintheyz plane(refertoSees. 3.1to 3.3).7.3 Stress as a DyadicInSees. 3.1and3.3we definedthevarious components of stress. For example,f.. is atension stress-aforceperunit area-actingacrossaplanenormal tothe x axis and directed along the x axis; simiiarly,jzlI is a shearing stress-againa force per unit area-again acting across a plane normal tothe x axis butnowdirected along the y axis, etc. For the rotational equilibrium of the medium, wet See, forexample, A. E. H. Love, "ATreatise ontheMathematicalTheoryof Elasticity,"4th ed.,sec. 17, Dover Publications, Inc., NewYork, 1944.21B EliuticWOIIU In SoUdsyB~ - - - - - - - - - - 9 I > A - - ~ : rztig. 7.3.1 Tetrahedral element in astressed elasticmedium.Only the stress components onthe face OBC are shown.showedthat j%ll=jflZ. Hencewestronglysuspect thattheninestress compo-nents, ofwhichsix areindependent, constitute asymmetric dyadic. Toshowthatthisisindeedtrue, letus calculatethe stress f nonaplane surfacewhosenormal, specified by the unit vectorD1= I cosa + j cosfj + kcos'= 11 + jm + kn, (7.3.1)is inclinedtothe coordinate axes.Figure7.3.1shows suchaplane interceptingthecoordinate axes at A, B,C, thus forming a small tetrahedron in whichthe stress components do not varyappreciably. (Thetetrahedroncanbemade small bychoosingtheposition ofthe origin 0suitably.)As usual for a closed surface, the outward-drawn normalsaretaken aspositive. If S isthe area of the triangle ABC, then lS, mS, and nSare theareas of the trianglesOBC, OAC, andOAB, respectively. Sincethemediumwithinthetetrahedronmustbe in staticequilibrium, theforceactingon the tetrahedron across the face ABC must equal the vector sum of the forcesacting onthe tetrahedron across the otherthree triangular faces. We thus havethatf n = I(lj= + mjll' + nju) + j(lj%ll + mjll1l + nj'lI)+k(lj,. + mjll' + nj), (7.3.2)wherethe area Shas been canceled fromthe equation.Theresult just foundcanbeput ina moreuseful formbymakingthesubstitutions 1= D1 0I, m= D1 j, n= Dl k.We then may write (7.3.2)f n = Dl 0iiJ= + 010ii/ZII + D10ikjD+ Dl 0 jiJfIZ + Dl 0 jjjll1l + Dl 0 jkjllz+ Dl kiju + 010kif'lI + 010kkj.., (7.3.3)(7.3.5)'1.3 Stress as aDyadic 219where 01 occurs dottedfromthe left intoeach oftheninedyads. Onfactoringout01, the stress acting acrossthe plane ABC becomesfn= 01 F, (7.3.4)where(fzz fZIl fZ.)F== fllz filII fll'f.z f'lI f isthestress dyadicexpressedasamatrix. Wehavethussucceededinshowingthat the nine stress components constitute a symmetric dyadic and incidentallyhave developed a useful formula, (7.3.4), for finding the stress on any surface.As a simple example of a stress dyadic,let us express the hydrostatic pres-sure Pin anonviscous fluidasadyadic. It is evidentthatP= - fzz = - fll/l = -frelates pressure to the three tension stresses. Hence thestress dyadic for ahydrostaticpressure isP = -P1, (7.3.6)where 1istheidemfactor (7.1.17). The forceperunit areaonasurfacewhoseorientation is specified by01 istherefore(7.3.7)Thisequationshowsthat the surfacealwaysexperiencesaforceperunitareaonit directedopposite to the (outward) normal of thesurface. Hydrostaticpressure isthusseentoberepresentedfundamentallybyadyadic;it isnot asimple scalar, asis often implied inelementary discussions.Since stressis asymmetricdyadic, itcanbereferredtoprincipalaxesforwhichthethree shearing stresses vanish. In such aneventF =iih + jj/2 + kkfa,(7.3.8)where h, f2, and faare theprincipal tensions alongthethree principal axes. Forexample, the elongation of arod,asdiscussedinSec.3.1, hasthe stress dyadic(7.3.9)where h= fl isthe appliedtension(3.1.1) and /2=fa= O.Consider next anelasticmediuminwhichthestressdyadicvaries frompoint to point for some reason. In such a situation there is generally a net forceacting on elements of the medium, since the forces on opposite faces of a volumeelement nolonger balanceeachother. Let usendeavor tocomputethebodyforce per unit volume thatarises inthisway.220 ElasticWa...... in SolidsHBDF1i/XI Ax/ Gx TIfIA,p}!-----I------fzEI Ix / / ITI II:y0)-----------1-1-.,..---------_x1 (I /zI (-j/-- ------- xzFig. 7.3.2 Element of astressedelasticmediumwhenabody force is present.First consider the net force in the x direction on the volume element& centered atthe point P having the coordinates x, y, z, using Fig. 7.3.2toguideour thinking. At thefaceEFGHtheforceduetothetensioninthexdirection iswhereas atthe faceABCD the forceduetothe same tension is(alu "'''' ax 2 .(7.3.10)Hence the net forceinthe xdirectionduetothe variation in I",,,,isal",,,,axNext consider theshearingstress11/"" OnthefaceBCGFtheforceinthex directionis(r +all/'"&JI/'" ay 2 'whereasthe force onthe opposite face ADHE is_(I - all/'"&1/'" ay 27.3 Stress a$ a Dyadic 221Hence thenet force inthex direction duetothe variation in fllxisafllz ax ay &.ay(7.3.11)(7.3.12)(7.3.13)Similarly, that duetothe variation in fzzisaf.z ax ay &.azOn adding thethree contributions(7.3.10) to(7.3.12), we findfor thetotal netforce inthe xdirection!:.Fz =(afzz+ afllz + afz) ax ay &,ax ay azwith similar expressions forthe components inthe yandz directions. Wethusfindthat thereisanet elasticbodyforceperunit volumeinthemedium, asgiven byF1 = i(afzz+ afllz + afz) + j (afzll+aflill + aflI)~ ~ ~ ~ ~ ~+ k(afz + afllz + af,,)ax ay az=vF, (7.3.14)where F isthe stress dyadic(7.3.5). The quantity V F isthedivergence of thedyadic F; it is evidently a vector.Ifanelasticmember isinequilibrium, ordinarilytheonlybodyforceisthat due to the gravitational attraction of the earth. We then have thatVF + Pog =0, wheregis the (vector) acceleration of gravity. Thetotalbody forceonthe elastic member inequilibrium isbalancedby externalforcesappliedtothe surface ofthemember. Inthe case of anonviscous fluidhavingthe stress dyadic(7.3.6), the forceper unitvolume becomesF1= V P = -V, (P1) = -VP,which agrees withthe calculation(5.1.4).Problems(7.3.15)7.3.1 Writethepressure at anydepthbelowthesurfaceofaliquidasadyadic. FindtheforcedFon an element of area dS of a submergedbody. Integrate this force overthe surfaceof thebodyto establish Archimedes'principle. Hint:UseGauss' theorem foradyadic.f dS' T = fv .Tdvas an aidinevaluating the integrals.(7.3.16)zn ElasticWaves inSolids7.3.2 Solve Probs.3.3.1and3.3.2by using(71.5)to rotatethey and z coordinate axes 45aboutthex axis. Hint:InitiallyF=I.,(jk + kj)andE=h.,(jk + kj)*7.3.3 Use(7.3.4)andGauss' theoremfor adyadic(7.3.16) to establish(7.3.14).7.3.4 Define the mean presSlIre in a stressed elasticmediumto beP= -j(fzo +I +Itt),an invariant of the stress dyadic.Show that any stress dyadic Fcan then be written as the sumof apure shear dyadicandadyadicrepresentingmeanpressure.7.4 Hooke's LawInChap. 3 we appliedHooke's lawtoindividualcomponents ofthe stress andstraindyadicsanddefinedanumberofdifferent but relatedelasticconstantsfor an isotropic medium.We now wishto exhibit Hooke's law as a dyadic equa-tion inwhich eachstress component is a linear functionofthe nine strain com-ponents. Suchanequationmaybewritten in component form3 3lij = L LCijklEkl, (7.4.1)k-ll-lwhereCijklis one ofthe 81components of a fourth-ranktensor(or tetradic). Be-cause of spatialsymmetries of one sort or another, thenumber of independentelastic moduli is not 81 but reduces to21 for the most anisotropic crystal and toonly 2 for an isotropic medium (see Probs. 7.4.1 and7.4.2). Let us now see how(7.4.1) canbewrittenfor anisotropicmediumfor whichmanyof thec'sareeither zero or equal toone ortheother oftwoindependentmoduli.Insteadof applyingtherestrictionsofsymmetryto(7.4.1) directly, it iseasier to start afresh, making use of the fact that the principal axes for the stressand strain dyadicsnecessarily coincide in an isotropic medium. With respecttoprincipalaxes, thetwodyadics have only diagonalcomponents, so that we canwriteHooke's law inthe simple formIzz = aEzz +bEw + bEn11/11 = bEzz + aE + bEnIn= bEzz +bE + aEn .(7.4.2)whereaandb areelasticmoduli. Tomakethetwoelasticmoduli agreewiththose generally adopted, known as Lame coefficients, (7.4.2) can be altered to readIzz =>'(Ezz +E1/1I + En) +2P.Ezz11/11 = >'(Ezz + EI1II + En) +2P.E1/1IIn =>'(Ezz +EI1II + En) +2P.En .(7.4.3)'1.4 Hooke'$Law UJWe shall presently showthatthe Lame coefficient IL is identicalwiththe shearmodulusdefinedby(3.3.5)andthatthe other Lame coefficient is givenbyA =B- ilL,(7.4.4)whereBis thebulkmodulusdefinedby(3.2.6). Asasingleequationamongdyadics, the component equations(7.4.3) takethe form(7.4.5)whichreplaces (7.4.1) for isotropicmediabut doesnot reveal that theelasticmoduli arebasicallycomponentsof a tetradic. Although weobtained(7.4.5)with both Fand Ereferredto principal axes, the equation continuestohold forarbitrary axes, since it expresses a relationship among symmetric dyadics.Let us now establish that IL is indeedthe shear modulus previously definedand that Ais related toBand IL by (7.4.4). Ifweexamineanyof theoff-diagonal terms of (7.4.5) when principal axes are not used, we find, forexample, that(7.4.6)whichagrees with the earlier definition of theshear modulus (3.3.5). Ifwethenset Fequal tothestressdyadicfor hydrostaticpressure(7.3.6), wefindthateach ofthe diagonalcomponents becomes(7.4.7)In view of the definition of the bulk modulus Bby(3.2.6), we arrive at (7.4.4).Hooke's lawforanisotropicmedium canbeeasilywritteninaninvertedformexpressingstrainas a functionof stress. All we need todois towrite(3.1.13), namely,1 0" 0"Ezz = + Yf u- Yf1l1l - y fn0" 1 0"E1I1I = - Y fzz +Yf1l1l - Yfn0" 0" 1En = - y fn- yf"" +yf(7.4.8)in dyadic form. We recall that Y is Young's modulus (for a rod) and 0" is Poisson'sratio. The absence of off-diagonalcomponentsin(7.4.8)showsthat the stress-strain system isreferredtoprincipal axes. Asasingle dyadicequation, (7.4.8)becomes(7.4.9)whereF== -!U",,,, +filII + f ..) (7.4.10)(7.4.11)is the mean pressure in the medium. When the stress-strain system is not referredto principal axes, (7.4.9) continues to hold, with off-diagonal components such as1+0'Ella = -y-flla.Comparing with(7.4.6), we findthe relation amongthe elastic constantsy = 2}.l(1 + 0'), (7.4.12)(7.4.13)whichwas earlier established as(3.3.6).If wenowformthesumofthediagonal elementsof (7.4.9), Le., wetakethe invarianttrace of the equation(Prob. 7.1.5), we findthat3(1 + 0') 90'0=E",,,, +EIIII +E.. =- y F + yF.On rearrangement the equationbecomesyF=- 0 =-BO.3(1- 20')(7.4.14)Hence we have establishedanother relationamongthe elastic constants,y=3B(1- 20'), (7.4.15)whichappearedas (3.2.8). The pressureinvolvedhereis the meanpressure(7.4.10),which differs fromthe hydrostaticpressure in a fluid, forwhichf",,,, = filII = f ..= - p.Problems7.4.1 Show(a) that therearea maximumof 21independent elasticconstants, usingthefact that bothstressandstrainare symmetricdyadics, and(b) thatthere existsa symmetryamong the constants arising fromthereciprocity relationsa/ij a//d-=-,aEll aEijwhere ij andkl standfor pairsofvaluesof x, y, z.7.4.2 Giveargumentsestablishingthat thenumberofelasticconstantsreducestotwo foranisotropic mediumandthereforethat Hooke'slawcanbewrittenintheformof (7.4.2).(7.5.1)'1.5 Waves in anIsotropicMedium 2257.4.3 Express each side of Hooke's law(7.4.9) as the sum of a pure shear term and a hydro-staticterm, usingthe resultsofProbs. 7.2.1and7.3.4. Hooke'slawcannowbe separatedinan invariant manner into two equations,one involving shear, the other hydrostatic compres-sion. Carryout this separation andshowthat it leadstothe two relations among the elasticconstants(7.4.12)and(7.4.15).7.5 Waves in an Isotropic MediumWearenowpreparedtoderivea waveequationfor elasticwaves inanex-tended homogeneous isotropic mediumin which the perturbinginfluence ofsurfacescanbeignored. Seismicwaves in theearthconstituteanimportantexample of such waves. In our idealized analysis we ignore the static body forceof gravity and suppose thatthemediumhasa uniformdensityPo.Anelasticwaveinthemedium causesaspatialandtemporal variationinthestressdyadicF, whichinturnproducesatime-varyingbodyforceF1 perunit volume, as given by(7.3.14). The elastic body force isthen responsible fortheacceleration02p/ ot2ofthemass Poinaunit volume. Newton'ssecondlawthereforetakesthe form02pF1= V F=Po-'ot2To obtain awave equation for the displacementp, weneedto express F interms of the spatialderivatives of p, usingHooke's law(7.4.5),F = >'(V p)1 + 21lE.The divergence ofthefirst term in(7.5.2)isV [(V p)1] = vv p,andthe divergence of the secondis(Prob. 7.5.1)V E= V Mvp + (Vp)l] = j(V Vp + VV p).Hence(7.5.1)becomes02p(>' +Il)VV, ~ + IlV, Vp= Po -,ot2(7.5.2)(7.5.3)(7.5.4)(7.5.5)which is oftenput intothe alternative form(7.5.18)ofProb. 7.5.2.Werecognize (7.5.5) assomeformof avector waveequation. Tomakefurther progresslet usmakeuseofHelmholtz' theoremthat anyvector field-subject tocertainmathematical restrictions-canbeexpressedas thesumofanirrotational fieldhaving a vanishing curlandasolenoidal fieldhaving avan-ishing divergence (see Appendix A). This theorem suggests that we separate the226 ElasticWaves in Solidsdiscussionof (7.5.5) intotwocases: irrotational waves, for whichVX e =0,and solenoid waves, forwhichV e =O.(a) [rrotational WavesWhenVX evanishes, VX (VX e) =vv . e- V. Ve= 0, so thatVV e = VVeThe waveequation(7.5.5) then becomes102eVVe=--'CI2ot2where2 A +2p.CI == ---PoY(l - 00) B + tp.(l +00)(1- 2oo)po Po(7.5.6)(7.5.7)(7.5.8)(7.5.10)We have expressed the elastic constant A+2p.involved in the wave velocityCIintermsof morefamiliar constants (seeProb. 7.5.3). Theelasticconstant isthat for pure linearstrain(seeProb. 3.1.3).We recognize(7.5.7)as an ordinary wave equation in three dimensions forthevector displacement e. Whenit applies toaplanewave travelinginthexdirection, it is equivalent tothethree scalar waveequations0 2 ~ 1 0 2 ~-=--ox2CI20t2iJ21/1021/(7.5.9) -=--ox2CI2ot202t 102t-=-_.ox2CI2ot2Wehavediscussedthesolutionofwaveequationssuchas (7.5.9) inconsider-abledetail inChap. 1 andhaveappliedthisknowledge tothree-dimensionalacousticwavesinfluids inChap. 5. Here, foraplane wave ofarbitrary shapetraveling in thepositive x direction, the solutions havethe forml;(x,t) = /I(x- Cit)1/(x,t) = h(x - Cit)t(x,t) =fa(x- Cit),wherethethreewave functionsmustsatisfy thevanishing-curlcondition( ot 01/) ( o ~ ot) (01/ o ~ )VX e =i - - - +J - - - + k - - - =O.oy OZ OZ ox ox oy(7.5.11)1.5 Waves in an IsotropicMedium 221Since ~ isnotafunctionof y or z, itisnecessary thatOTJ-=0oxot-=0ox '(7.5.12)showingthat theyandz componentsof thewave (7.5.10), that is, faand fa,vanish(orare atmost constantdisplacementsthatdonot concernus). Hencethe only solution of (7.5.7) that representsaplane wave traveling in the x direc-tionisoneinwhichthewavedisplacement isinthedirectionof wavetravel.Therecanbenotransversemotionof themediumthat travelswiththewavevelocityCl. Suchawaveisusuallycalledalongitudinal wave, hencethesub-script I onthe wave velocity. Thename compressional waveis also used. WhenPoisson's ratio has the typical value of 0.3,the longitudinal wave velocity in anextendedmediumis about 16percent greaterthanthatoflongitudinal waveson a slender rod of the same material. For a fluid, for which the rigidity modulusp. isnecessarilyzero, thewavevelocity(7.5.8) isidentical with (5.1.9) foundfromthe scalar waveequation forpressure.(b) Solenoidal WavesWhenV. evanishes, thewaveequation(7.5.5) becomes immediately102eVVe=--'clot2where(7.5.13)(7.5.14)p. yC,2 == - = -,---,.---Po 2(1 +u)PoThewavevelocityCI isthatfoundinSec. 4.5for torsional (shear) wavesonarod or tube of circular section. Except fora different wave velocity, the presentwaveequation isthesame asthat for longitudinal waves.Let us again look at the equations for a plane wave traveling in the x direc-tion.Thewaveequation(7.5.13)can be written asthreescalar wave equationssimilar to(7.5.9), with solutions similar to(7.5.10). The only change consists inreplacingCl byc,. Now, however, insteadof thevanishing-curl condition, thedivergenceV . emustvanish, thatis,o ~ OTJ otV e = - + - + - =O.ox oy OZSinceTJand tare functionsof xand t, not of yandz, wefindthatO ~ = O.ox(7.5.15)(7.5.16)Hencetherecanbenowavedisplacementinthedirectionofwavetravel: the(7.5.17)228 Elastic Waves in Solithwaveis entirelytransverse, withonlyyandzcomponents. Sucha waveingeneral iscalledatransversewave, andit mayalsobecalledashearor dila-tationless waveinthe present instance.Irrotational, orlongitudinal, wavesinvolveavibrationin only one direc-tion, andaresaidtopossessonedegreeof freedom. Solenoidal, or transverse,waves can vibrate independently intwo directions(orpolarizations)andthere-forepossess twodegreesoffreedom. Longitudinal andtransversewaveshavedifferent velocities in anelasticmedium, asgiven by(7.5.8) and(7.5.14), withtheratio~ = [2(1 - u)J l/2.CI 1 - 2uThus,whenu.., 0.3, cilc, .., 1.7. Althoughthetwokindsof wavespropagateindependentlyinahomogeneousmedium, at aninterfacebetweentwomediathereisusuallyapartialconversionof onekindofwaveintotheother. Whathappensinaparticular caseisdictatedbytheboundary conditionsthat mustbe satisfied attheinterface: continuity ofthe displacementvector 9 and of thestress dyadicF.An extended !>olidmediumboundedby asurface, e.g., theearth, can sup-port elasticsurfacewaves somewhat similar tothesurfacewaves ona deepbodyof water. Thewavedisplacement is restrictedtothematerial near thesurface, fallingoff exponentiallywithdepth. It isfoundthat asurfacewaveconsists of amixture ofthesolenoidal andirrotational wavesthatcanexistintheinterior ofthesolid. Thewavevelocity of asurfacewaveis somewhat lessthanthetransversewave velocityCI,the amount depending on Poisson's ratio.tSeismicwaves arisingfromlocal movements intheearth's crust involvesurface waves, aswellas waves passing throughthe interior ofthe earth. Sincethe energy involved in sUlface waves spreads outintwo dimensions, instead ofthree, surfacewavescontributegreatlytothedestructiveness ofearthquakes.Bystudyingthetimeofarrival ofvariouscomponentsofseismicwavesfromdistant earthquakes, it is possible togaininformationregardingthephysicalpropertiesof thematerial makinguptheinteriorof theearth. For example,thereappears tobe a sharpdiscontinuityinpropertiesat a depthof about2,900 km. Belowthis depththere is no evidence forthe propagation of atrans-versewave,whichis consistent withthehypothesisthattheearthhas a liquidcore,probably of compressed metallic iron. tt For a discussionof the theory of surface waves, seeL. D. LandauandE. M. Lifshitz,"Theoryof Elasticity,"sec. 24, Addison-Wesley Publishing Company,Inc., Reading, Mass.,1959tFor further informationregardingthisapplicationof wavetheory, seeK. E. Bullen, "AnIntroductiontotheTheoryof Seismology," CambridgeUniversityPress, NewYork, 1963.7.6 Ener,y Relations ZZ9Problems7.5.1 Verifythe vectoroperations usedinestablishing(7.5.3)and(7.5.4).7.5.2 Show thatthe wave equation(7.5.5)can beput inthe alternativeformate(x +21)VV e - IVX(VXe) = po-'att(7.5.18)whichexhibitsdirectlythe partsof the wave equationthat vanishfor solenoidal andforirrotational waves.7.5.3 Showthat(7.5.19)Showthat whenanexternal bodyforceperunit volumeFl.ispresent, theequationfor theelasticdisplacement (7.5.5) becomes(7.5.20)(Thisequationbecomesanequation for elaslicequilibriumwhen e isnot afunction of time.Itssolutione(x,J,z) must thensatisfyspecifiedconditions at theboundaryofa body. Thestress and strain at each point of the body can be calculated knowing e. We thus have formu-latedthe problem of elastic equilibriumin afundamentalway.)7.6 Energy RelationsExternal work must be expended to increase the elastic distortion of a medium.Whenthe stress-strain relation issingle-valued, the work done in producingthedistortioncanberecovered, atleast inprinciple, byreversingtheloadingpro-cedure and allowing the medium to do work on its surroundings as it returns toitsundistortedstate. Wecanthendefineanelasticpotential energyequal tothe work done in (reversibly)distortingthe medium. This potentialenergy canbethought of aslocalized inthemedium andis described by ascalar functionof positionVI, thepotential energy per unitvolume.Let usseewhat formVI takesfor ahomogeneousisotropicmediumthatobeys Hooke's law. The stress at any point is given by the stress dyadicF, andthe strain bythe strain dyadicE, thetwo beingrelated by Hooke's law(7.4.5)or (7.4.9). For convenience we canassumethat the mediumunder consider-ation is in a state of uniform stress.Perhaps the simplest way of finding the expression for the potential energydensityVIconsists in imagining thatarectangular block of material of dimen-:IJO ElasticWaves in Solids J:z:Fig. 7.6.1 A rectangular block stressed bytensions appliedto its faces.sionsa, b, e isstressedbyuniformtensionsappliedtoitsthreepairsof faces.The edges of the block are then parallel to principal axes, with respect to whichstressandstrainshear componentsvanish. ThesituationissketchedinFig.7.6.1.Suppose that at some stage in the process of applying tensions to the facesof theblock, thethreeprincipal tensionshavethevalues h,Thetotalforces, exerted oneach of the three pairs of faces of the block, are then= be!: = =(7.6.1)If E;,anddenote the corresponding principal extensions, the block has beenelongated inthe three coordinate directions by the amounts, = ax a =aE1,(aTJ)' ,TJ = ay b =bE2,(at)' ,t = az e =eEa(7.6.2)(7.6.3)Theexternal workrequiredtoincreasetheseelongationsbyadditional infini-tesimal amounts drl', dt' is clearlydW= F;+F;dTJ' +dt'= abeU:+dE; +(7.6.4)(7.6.5)(7.6.6)7.6 Ener,y Relations 231wheresubstitutionshavebeenmadefrom(7.6.1) and(7.6.2). Thetotal workdone in straining theblock canbe calculated fromthe integral3W= abc JLf:dE:.i-lTheintegral (7.6.4) isalineintegral, whosevaluedependsontheinitialand finalstress-strain state, and not on intermediate states, provided the stress-strainrelationis single-valued, which is certainlytrue if the mediumobeysHooke's law. The integration is easily carried out by a well-known artifice whenthe stress-strainrelationis linear. Let h, h, fabethefinal values oftheprinci-paltensions, corresponding to the principal extensions El, E2, E3. We now imaginea process in which each stress component, and therefore each strain component,increases fromzero to itsfinal valuein proportionto someparametera, whichincreases fromzeroto unity. Accordingly, wecanwritethatf ~ =ah d E ~ =El daf ~ = ah d E ~ =E2 daf ~ =afa d E ~ =E3 da.The integral (7.6.4) then becomesW = abc(jlEl +hE2 +faE3) hi ada.The potential energy density, which equals to the work done per unit volume indistortingthemedium,istherefore givenbyW 1 3Vi = - = - LfiEi (7.6.7)abc 2 i-lsince hi ada =j. Thisequation, andthevariousother forms intowhichitcan be put, constitutes an equation of state foran elastic solid. Although(7.6.7)contains noexplicit reference toHooke's law, it is avalidequationonlyformedia having alinear stress-strain relation.Wecanwrite(7.6.7) in aparticularly elegant formifwedefinethedoubledot (or scalar) product oftwo dyadics Aand Bbytheequation3 3A :B== r r aijbij = a:z;:z:b:z;:z: + az1lbz1l+ . . . + anb...i-lj-l(7.6.8)Each component of the dyadic Ais multiplied by the corresponding componentofthe dyadicB, and the sum oftheninetermstaken.The definition is directlyanalogoustothedefinition ofthe dotproduct oftwovectors,232 ElasticWavesinSolidsThe doubledot product (7.6.8) is easily showntobe a scalar invariant; i.e., itsvalue does not depend on the orientation of the coordinate axes (see Prob. 7.6.1).Wethereforecanwrite(7.6.7) intheform(7.6.9)Since the double dot product is a scalar invariant, (7.6.9) continues toholdwhennonprincipal axesareused, whereupon Fand Ehaveoff-diagonal compo-nents. Inthiscase, thepotential energy density inexpanded formreadsVi = Mf.r.rEzz +f1/uE1/1/ +fzzEzz + 2U1/zE1/z +fzzEzz +fzuEz1/)]'If nowwereplace Fby itsvalue(7.4.5),F = >..(V!l)1 + 2J.lE,wefindthat(7.6.10)(7.6.11)(7.6.13)Vi = ix'(v, !l)E:1 + J.lE:E= ix'(Ezz + E1/1/ + Ezz)2+ J.l[Ezz2+E1// + Ezz2+ 2(E1/z2+Ezz2+ EZ1/2)], (7.6.12)wherewehavemadeuseofthe fact thatE: 1= Ezz + E1/1/ + Ezz = V !l.Evidentlythe double dot product of the idemfactor anda dyadic gives thetraceof the dyadic,andincidentally confirmsthe factthatthetrace is a scalarinvariant (seeProb. 7.1.5).Alternatively, wecan replace Eby itsvalue(7.4.9),E= l+uF+3u p1y y'where P isgiven by(7.4.10). Inthiscasewefindthat1 +u [2 2 2 (2 2 2)]Vi = ~ fzz +f1/1/ +fzz + 2f1/z +fzz +fZ1/U- 2Y Un +f1l1/ +fzz)2, (7.6.14)sinceF:1=fzz +f1/1/ +fzz=-3P.It is left as anexercise (Prob. 7.6.3) toshowthat the expressions foundinChap. 3for potential energydensities associatedwithvarious simplestress-strain systemsarecontainedinthesegeneral expressions(7.6.9), (7.6.12), and(7.6.14).7.6 Ene"yRelations 2JJTokeepthediscussionofelasticityassimpleaspossible, wehavesofarmadenodistinctionbetweenthe almostequal adiabatic andisothermal elasticconstants of solids. A quantitative discussion ofthis distinctionrequiresacon-siderable excursion intothefield ofthermodynamics andcannotbeundertakenhere.The elastic potentialenergy of a solidjust computed isknowninthermo-dynamics as a(Helmholtz) freeenergywhenit appliesto isothermal conditions.withisothermal elasticconstants, andasaninternalenergywhenit appliestoadiabatic conditions, withadiabaticelasticconstants.Iftherateof distortionof an elastic body is too fast for isothermal conditions but too slow for adiabaticconditionstoprevail, noelasticequationofstateexists. Thevalueofthelineintegral in (7.6.4)then depends on the path followed as the distortion progresses.Forproblemsinvolving elasticwaveswecannormally assumethat conditionsareadiabatic, whereas for problemsinstaticelasticity, theconditionsareiso-thermal. Since accurate values oftheelastic constants arenearly alwaysmeas-uredusing standing- ortraveling-wavetechniques, theelasticconstants soob-tained are adiabatic values.Isothermal values can then be calculated, if needed,using certain formulasfromthermodynamics. tProblems7.6.1 Show that the double dotproduct of two dyadics, as definedby(7.6.8), gives a scalarinvariant of thetwodyadics. Ilin': Rotatethecoordinateframeand useresultsquotedinProb. 7 1.1.7.6.2 Showthat when VI canbeexpressedasafunctionofthestresscomponents !i;, thecorresponding strain components are given by fij = aV./a!ij.Similarly,ifV,is expressed as afunction of the strain components fi;, showthat!ij = aV.!afij.Establish the reciprocity rela-tions a!ij/aftl =a/kl!afij usedin Prob. 7.4.1.7.6.3 Show that the potential energy densities (3.1.16), (3.2.10), and(3.3.8) obtained earlierare specialcases ofthemore generalequations ofthis section.7.6.4 Discuss the energy content of sinusoidal plane waves traveling in the positive xdirec-tioninasolidisotropicelastic medium. Treat separatelyirrotational wavesandsolenoidalwaves.7.6.5 Show thatPI = F.aeattSee, for example, Landau and Lifshitz, op. cit., sec. 6.(7.6.15)(7.7.2)(7.7.3)(7.7.4)ZJ4 ElasticWavesinSolidsgivestheinstantaneouspowerflowin anelastic\\aveinasolid, whereF isthe stress dyadicandae!a/ isthedisplacement velocity. Applythisresult tothewavesdiscussedinthepre-vious prohlem andshow for each sort of wave thatthe average power flowequals the averagetotalenergydensitytimesthewavevelocity.*7.7 MomentumTransport by a Shear WaveInSec. 1.11, andagaininSees. 4.1d and5.3a, weexaminedthemechanism bywhichparticularelasticwavestransportlinearmomentum. A closeconnectionwas foundbetweenthe transport of momentumandthe transport of energyby atraveling wave. It would appear that in a dispersionless medium these twoaspects ofwavemotion arealways related by anequation ofthe formP= cE= c2g, (7.7.1)where P= averagerateof energy flowE= average energy densityg = averagelinearmomentum densityc= wave(phase)velocity.Herewewishtoexaminehowashear(solenoidal) wavein anextended elasticmedium, described forinstancebythesolutionl1(X- ct) ofthewave equation(7.5.13), transports momentumas impliedby(7.7.1). Themechanismis notimmediatelyobvious, sincethemotionof themediumisbasicallytransverse.apparentlywithno forcecomponentinthedirection of wavetravel.Let us start our analysis with a simplifiedderivationof the transversewave equation02110211P.Ox2= po ot2 'whichistheycomponentof(7.5.13). Whentheonly displacementcomponentis 11, whichisafunctionof onlyx(and t), thepurestrain dyadic(7.2.6) takesthe formE=! 011(ij + ji) = j-y(ij + ji),2 oxwhere'Y== 011/0X isthe shearing strain, discussed in Sees. 3.3 and7.2. Since thedilatation(7.5.15) vanishes, Hooke's law(7.4.5)givesthe forcedyadicFOl'] (" +00)= p. ox IJ Jl.The forceper unit volume acting onthemedium, according to(7.3.14), isthen, 02l'] 02(}F1 = v' F =Jp. ox2 =Po ot2 ' (7.7.5)(7.7.6)7.7 MomentumTransport by aShearWave 2JSwherewe have equated the forcedensity tothemass of a unit volumetimes itsacceleration.