iNonlinearacousticsthroughproblemsandexamplesOleg V. RudenkoSergey N. GurbatovClaes M. Hedberg2009Order this book online at www.traord.comor email orders@traord.comMost Traord titles are also available at major online book retailers. Copyright 2010 O. V. Rudenko, S. N. Gurbatov, C. M. Hedberg.All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written prior permission of the author.Printed in Victoria, BC, Canada.: 978-1-4269-0544-5 (sc): 978-1-4269-0545-2 (dj): 978-1-4269-0546-9 (e-book)Our mission is to e ciently provide the worlds nest, most comprehensive book publishing service, enabling every author to experience success. To nd out how to publish your book, your way, and have it available worldwide, visit us online at www.traord.comTraord rev. 2/9/2010 www.traord.comNorth America & internationaltoll-free: 1 888 232 4444 (USA & Canada)phone: 250 383 6864fax: 812 355 4082ContentsIntroduction 31 Simplewaves 102 Planenonlinearwaveswithdiscontinuities 323 NonlinearwavesindissipativemediaandBurgersequation 594 Sphericalandcylindricalwavesandnonlinearbeams 745 Highintensityacousticnoise 886 Nonlinearnondestructivetesting 1087 Focusednonlinearbeamsandnonlineargeometricalacoustics 1278 Varioustypesofnonlinearproblems 145References 168Subjectindex 171iiiivForewordNonlinear acoustics is based mainly on results obtained more than 30yearsago. Theseresultshaveappearedinmonographs[1-5]andwerediscussedinspecial problem-orientedprograms andlectures onthetheoryofwaves[6]. Thenextstageofthenonlinearacousticsdevel-opmentisassociatedwiththeonsetof thewideuseof itsconceptsandmethodsinappliedelds[7-9]. Thelatterrequirednumericalso-lutionsofnonlinearequationsdescribingone-dimensional waves[13],andbeams[14]. Wavesininhomogeneousmedia[15] andrandomlymodulated perturbations [13-15] were studied. Modern studies in fun-damentalandappliednonlinearacousticsaredescribedinnewbooksandreviews(see,forexample[16-21]).Nowadays, nonlinear acoustics is considered to be a well-developedbranchofscienceandtechnologyprovidingresultswhichcanbead-vantageouslyemployedbyspecialists workingindierent elds. Itmaybeuseful forundergraduateandpostgraduatestudents, aswellasthespecialistsinneighbouringscienticdirections,tohaveatext-book which would allow them, within a comparatively short period oftime, to grasp the fundamentals of nonlinear acoustics and to becomeactive users of the methods developed here concerning simplications,calculationsandnumericalestimates.Webelievethatasequenceof problemsproperlyorderedfromalogical viewpointcanberegardedamongthemostecientwaysofgraspingnewideas. Theproblemsdealtwithinthisbookcomeingroups. Asarule, therstprobleminagroupisaimedatcarefulexaminationof animportanttheoretical aspectandisfollowedbyacomprehensivesolution. Thesubsequentproblemsservethepurposeofmasteringthemethodsofcalculationsandmakingestimates. The12simplest ones have nothing but an answer,whereas more complicatedproblems have both an answer and an explanation. The most diculttasks are being privileged by having a solution. When a group consistsofproblemsofthesametypeasolutionisonlyprovidedfortherstone, whiletheothersinthegrouparemeanttobeconsideredinasimilarmanner. Wehavetriedtoadheretothislayoutwhereitispossible.This bookof problems stems fromthe programs oeredtothestudentsof theMoscowandGorky1UniversityAcousticsChairsaswell as to the students of the radiophysics Department of the MoscowState University Physics Faculty. Many new problems were suggestedbytheauthorsinthecourseof preparingthemanuscript. Sections1,2,3arejoint eorts of theauthors. Problems 1.4, 1.7, 1.9, 1.19-1.24, 2.2, 2.6, 2.10, 2.15, 2.17, 3.2, 3.4, 3.6-3.13weresuggestedbyS.N. Gurbatov in their initial version while the others are due to O.V.Rudenko;section4isbasedonthesuggestionsofO.V.Rudenkoandsection5isacontributionfromS.N. Gurbatov. Thewholebodyoftheproblemshasbeenperfectedforalongtimebymeansof cross-checkingandeditorialactivity. TheauthorsareverygreatfultoV.A.KhokhlovaandV.A.Gusevwhotooktheburdenofscienticeditingofthemanuscript. Itwasaccordingtotheirsuggestionsthatcertainsolutions were modied. Since courses of nonlinear acoustics are beingdelivered in various universities and technical colleges in Russia, Swe-den, USA,Germany, France, GreatBritain, China, Japanandmanyothercountries, wedarehopethatthisbookwill bereader-friendlyand add to the preparation of specialists in the eld. We are ready todiscuss every remark or suggestion forwarded to us by those interestedandaimedatimprovingandsupplementingourfutureeditions.O.V.Rudenko,S.N.Gurbatov,andC.M.HedbergTheforewordandchapters1-5weretranslatedfromRussianbyD.G.Sorokin.1Since October 11, 1990 again Nizhny Novgorod3INTRODUCTIONForthesubsequentsolvingofnonlinearproblems,itisrstneces-sarytogivesomeinformationaboutlinearacousticwaves.ThesystemofequationsdescribingtheuidorgaswithaccountforshearandbulkviscositiesconsistsoftheNavier-Stokesequationofmotion_ut+ (u) u_ = p + u +_ +3_grad div u, (I.1)thecontinuityequationt+ div (u) = 0 , (I.2)andtheequationofstatewhichforacousticwaveshastheformofaPoissonadiabatp = p () = p0_ 0_. (I.3)TheEulerianapproachtocontinuousmediaisusedintheformu-lationof thesystem(I.1)-(I.3). Inthecontext of this approachallvariables the pressure p, the density and the velocity u are func-tionsof coordinatesof animmovablereferenceframe, andof timet.Often,another approach is used,namely the Lagrangian one which ishelpful forsolvingone-dimensional problems(seeProblems1.1-1.5).TheLagrangiandescriptionof thecontinuous mediumuses coordi-nates of liquid particles measured at a denite initial moment of time,insteadofthecoordinatesofanimmovablereferenceframe. TheLa-grangianapproachisthebasiconeinthetheoryofelasticityofsolidsandismainlyconcernedwiththepositional relationshipof dierentinternal sectionsof solidswhichisresponsiblefortheappearanceofstressandstrainelds. Atthesametime,therigiddisplacementsofsolidbodiesarelessinteresting.Lettheundisturbedstateofauidbe = 0,p = p0,andu = 0.Disturbancescausedbythewavearedenotedbytheprimedlettersandareputintothesystem(I.1)-(I.3)p = p0 + p

, = 0 +

. (I.4)4Thedisturbancesareconsideredtobesmall:p

p0

0 [u[c0 1 . (I.5)Hereisasmall parameter. Apowerseriesexpansioninwillbeusedtosimplifythedierentialequations. Theequilibriumsoundvelocityisc0=_p_=0=_p00. (I.6)The ratio of the particle velocity to the sound velocity is known asthe acoustic Mach number M. One can see from equation (I.5) that isofthesameorderastheacousticMachnumber. Substituting(I.4)intothesystem(I.1)-(I.3),onecanreduceittotheform0ut+p

u _ +/3_grad div u == ut (0 +

) (u) u (I.7)

t+ 0 div u= div(

u) , (I.8)p

= c20

+12_2p2_2+ ... A

0+12B_

0_2+ ... . (I.9)The linear terms are collected in left-hand-sides of equations (I.7) and(I.8),andthenonlineartermscontainingpowersandproductsofthevariables describing disturbances are collected in the right-hand-sides.Inkeepingonlythelinearterms,thesystem(I.7)-(I.9)isreducedto0ut+p

u _ +3_grad div u = 0, (I.10)

t+ 0 div u= 0 , (I.11)5p

= c20

. (I.12)Thedeviationofdensityinequation(I.11)canbeexcludedbyusingequation(I.12). As aresult, apair of equations is derivedwhichconsistsofthefollowingone:p

t+ c200 div u= 0 (I.13)andequation(I.10),whichconnectsthetwovariablesp

and u.Inaccordance withthe Helmholtz theorem, anarbitraryvectoreld(inourcasethevelocityeld)canberepresentedasthesumofpotentialandvortexcomponents:u = ul +ut, ul= , ut= rot

A. (I.14)Thefunctionisaso-calledscalaroracousticpotential,andthefunction

Aisthevectorpotential. Substituting(I.14)intoequation(I.13),andtakingintoaccounttheidentitydiv rot

A = 0,wederivep

t+ c200 div ul= 0 . (I.15)Onecanseethatthepressurevariation(aswell asthevariationofdensity(I.12))iscausedonlybythepotential (theacoustic)compo-nentofthevelocityeld. Thevortexcomponentdescribestheshearmotion where the medium behaves as an incompressible liquid. It fol-lowsfromequation(I.12)thattheshearmotionisdescribedbythediusionequation:utt=0ut. (I.16)Todescribethepotentialnaturalmodeofthemedium,thefollowingequationcanbederivedfromequation(I.10):0ult+p

= bul, b _ +43_. (I.17)Apply the gradient operator to equation (I.15), then dierentiate equa-tion(I.17) withrespect totimeandsubtract this fromtheformer.6Thereafteronecaneliminatethepressurevariationandndasingleequationforthepotentialcomponent ul:2ult2c20ul=b0tul. (I.18)Let us now consider a plane wave where all parameters depend onone of the coordinates (for example, x) and time (t). From the poten-tiality condition it follows that the velocity vector has only one nonzerocomponent, which is its projection on the x-axis. This component willbe denoted hereafter as u. Consequently, the acoustic wave is a longi-tudinalwaveforwhichtheparticlesofthemediumvibratealongthedirectionof itspropagation. Foraplanewaveequation(I.18)takesform(seeProblem3.1)2ut2 c202ux2=b03ut x2. (I.19)Forexample,letamonochromaticwavepropagatealong-axis:u(x,t) = Aexp (i t + ikx) , (I.20)wherekandarethewavenumberandthefrequency. Aftersubsti-tutionof(I.20)in(I.19),thedispersionlawisobtained:k2=2c20_1 ibc200_1, k = c0_1 ibc200_1/2. (I.21)Theminussigninthesecondformulain(I.21)correspondstoawavetraveling in the positive direction of the x-axis, and the minus sign toawavetravelinginthenegativedirection. Thedimensionlesscombi-nationofparametersinsidethebracketsin(I.21),whichcontainstheeectiveviscosityb,isassumedtobesmall:bc2001 . (I.22)Thenthedispersionrelation(I.21)takesformk =c0+ ib22c300. (I.23)7Substituting(I.23)in(I.20),wederiveu(x,t) = Aexp_b22c300x_exp_i t + ic0x_. (I.24)Onecansee,thatthewaveamplitudedecreasesaccordingtothelawA(x) = Aexp (x), b22c300. (I.25)The combination of parameters in (I.25) is the absorption coecientof thewave. If absorptionoverthedistanceof onewavelengthissmall,orinotherwords,=b22c3002 c0= bc2001 , (I.26)then the assumption (I.22) is true. Really, a wave can be denedas aprocesswhichcantransportenergyoverdistancesmuchgreaterthanthewavelength. Themosttypicalandimportantcaseispreciselytheonecorrespondingtoinequality(I.26).If the viscosity (wave absorption) is negligible, one can derive fromequations (I.12),(I.15),(I.17) the following linear relations for acousticdisturbancestravelinginpositivedirectionalongx:p

c200=

0=uc0. (I.27)For the wave propagating in negative direction the particle velocity informula(I.27)containsaminussign. Consequently, anydisturbance(acousticpressure, acousticdensityorparticlevelocity)isdescribedbythesamewaveequation. Asimplerequationcanbederivedifwerestricttheproblemandconsideronlythewavepropagatinginonedirection, say, inpositivedirectionofthex-axis. Replaceinthedis-persionrelation(I.23)thewavenumberandfrequencybydierentialoperatorsaccordingto(I.20):k ix, i t. (I.28)8Asecondorderequationisobtained(insteadofthethird-orderequa-tion(I.19))ux+1c0ut= 2ut2, b2c300. (I.29)Itismoreconvenienttoreplace(x,t)withthenewsetofvariablesx, = t x/c0. (I.30)Bytheintroductionofatimewhichmovestogetherwiththewaveatthesoundvelocity(I.30),weremovethefastprocess(wavepropaga-tion) and watch the slow distortion (evolution) of the wave caused byweak viscosity. In the accompanying coordinate system (I.30) equation(I.29)takestheformofthecommondiusionequation(see(3.2)):ux= 2u2. (I.31)Ageneralizedformof thisequationwithaccountfornonlinearityistheBurgersequation(3.8).Usingtheapproachdescribedabovewecanget another impor-tant linear equation which governs the evolution of spatially boundedwavebeams. Thedispersionrelationforthethree-dimensional waveequationformediawithoutabsorptionis2c20= k2 k2x + k2y + k2z. (I.32)Herekx,ky,andkzareprojectionsofthewavevectorontheaxes.Let thewavebeampropagatealongthex-axis whileweaklydi-vergingorconverging. Itmeansthattheprojectionskyandkzaresmall in comparison with kxand the following approximate relation isvalidc0= kx1 +k2y + k2zk2x kx +k2y + k2z2kx,kx ( c0kx) =c02_k2y +k2z_. (I.33)9Byreplacingthewavevectorandfrequencycomponentswithoper-ators inthesecondformulain(I.33) (byanalogywiththeformula(I.28))kx ix , ky iy, kz iz, i t, (I.34)wederivex_ut+ c0ux_ =c02_2uy2+2uz2_. (I.35)Passing on to the accompanying coordinate system (I.30), one obtainstheevolutionequation(see(4.20)):2ux =c02_2uy2+2uz2_. (I.36)Ageneralizationof thisequationforthenonlinearcaseleadstotheKhokhlov-Zabolotskayaequation(4.23).10 Chapter 1Chapter 1SIMPLEWAVESProblem1.1Show that a set of equations of hydrodynamics in Lagrangian vari-ablesforaone-dimensional planemotionhasasolutionintheformofsimpleRiemannwaves. Reducethissettoasinglenonlinearequa-tion for the variable (x, t) - which is the displacement of the mediumparticlesfromtheirinitialpositionx.SolutionThe reference equations of hydrodynamics in Lagrangianrepresentationhavetheform[3],[4]02t2= px, 0= (1 +x), p = p() = p0( 0). (1.1)The rst equation is a generalization of the second Newtons law withrespect to a continuous medium. The second one (the continuity equa-tion)isthelawof conservationof masswrittendownindierentialform. The thirdone is the equationof state giveninthe formofaPoissonadiabat for fast processes of compressionandrarefaction(ascomparedwiththermodiusion)accompanyingthesoundpropa-gation.With a simple wave is understood a wave formulation (a nonlinearone, speakinggenerally)inwhichall thevariablescanbeexpressedthroughoneanother withthehelpof somealgebraicrelations. If,however, the relations of variables contain integrals or derivatives, thewavewillnotbeconsideredassimple;fromaphysicalviewpointthismeans the appearance of dispersion, i.e. the dependence of even a smallperturbation behaviour on its spectral composition. As the latter twoequationsof(1.1)areformulatedas = 0/(1 +x), p = p() = p(0/1 +x)), (1.2)Simple waves 11the density and pressure are expressed as functions of only one variable/x. Thisimpliesthattheset(1.1)hasasolutionintheformofsimplewaves.Consider the equationof state (1.1) inthe formof anadiabat.Then(1.2)yieldsp = p0(1 +/z). Substitutingthisrelationshipintotheright-handsideof therst equationin(1.1) results inthenonlinearEarnshawequation2t2= c202/x2(1 +/x)+1, (1.3)wherec0=(p0/0)1/2istheequilibriumsoundvelocity. Theequa-tion(1.3) has anonlinearityof ageneral typeandcanformallybeused to describe strong disturbances. However, it is required, that thedenominatorin(1.3)will notvanish, i.e. (/x ,= 1). Innonlin-earacousticswearedealingwithweaklynonlinearwavesforwhich[/x[ 1.Problem1.2Considering weak nonlinearity, simplify the Earnshawequation(1.3)andretainonlythetwoprincipalnonlinearterms.SolutionMakeuseoftheapproximaterelationship(1 +x)(+1) 1 ( + 1)x+12( + 1)( + 2)(x)2. (1.4)Substitutetheexpansion(1.4)intotheright-handsideof Earnshawequation(1.3). Rewriteitintheform:2x2 1c202t2= ( + 1)x2x2 12( + 1)( + 2)(x)22x2. (1.5)Theleft-handsideof(1.5)correspondstoatypicallinearwaveequa-tion. Theright-handsideisresultingfromthegeneral typeof non-linearityexpansioninterms of power nonlinearities whichcontainsquadraticandcubicnonlinearterms.12 Chapter 1Problem1.3A nonlinear medium occupies a half-space x > 0 and at its bound-aryx=0aharmonic signal =Asin t withangular frequencyisprescribed. Findoutwhichfrequenciesthatcanappearduringwavepropagationinamediumduetoquadraticandcubicnonlinear-ities, byanalysingtheequation(1.5)usingthemethodofsuccessiveapproximations.SolutionBy considering nonlinear eects to be weak, the right-hand side in equation (1.5) can be neglected in the rst approximation.A solution to the linear wave equation in the form of a wave travellinginthepositivedirectionoftheaxiswillbeexpressedas(1)(x, t) = Asin (t x/c0). (1.6)In order to nd a solution to the second approximation one has to sub-stitute(1.6)intotheright-handsideofthenonlinearequationwhichwillconsequentlytaketheformF=12( + 1)A2(c0)3sin 2+18( + 1)( + 2)A3(c0)4(sin 3+ sin ), (1.7)where =t x/c0is thetimeintheaccompanyingcoordinatesystem which travels with the wave at sound velocity c0. The equationof the secondapproximationwiththe right-handside (1.7) canbewrittenas2(2)x21c202(2)t2= F(t x/c0). (1.8)It is evident that F has the meaningof anexcitingforceinaninhomogeneouswaveequation(1.8). Itgeneratesnewwavesatthefrequencies of the secondharmonic 2(quadratic nonlinear eect)andthethirdharmonic3(cubicnonlineareect). Apartfromthis,the cubic nonlinearity gives additional contribution to the wave at thefundamentalfrequency(theself-actioneect).Simple waves 13Problem1.4Indicatewhichfrequencies that emergeinaquadraticnonlinearmedium (in the rst approximation) provided a biharmonic signal =A1 sin 1t + A2 sin 2t is givenas input. Consider inparticular thelimitingcase1 2.SolutionInanalogywithproblem1.3itcanbereadilyshownthatthesecondharmonics21, 22ofthereferencefrequencywavesare generated in the medium, as well as perturbations at sum 1 +2and dierence 12frequencies. For 1 2, the second harmonicalone will be generated inasmuch as the dierence frequency excitationeciencytendstozero(seealsoProblem1.7).Problem1.5Simplify equation (1.5) by using the method of slowly varying pro-leretainingonlyaquadraticnonlinearterm.SolutionThemethodof slowlychangingproleallowsasig-nicant simplicationof thenonlinear partial dierential equationsdescribingthe process of propagationof intense waves. Simpliedequationswillnaturallylendthemselvestoeasiersolutions. Thecon-cept of the method is as follows. Given the absence of nonlinear terms,a solution to equation (1.5) will look like a sum of two travelling wavesof arbitrary form: = (t x/c0) +(t +x/c0). The wave with pro-le () propagates in the positive direction of the x-axis whereas thewavetravelsinthenegativedirection. Wetakeaninterestinthepositive one. When a weak nonlinearity is observed and the right-handsideoftheequationdiersfromzero, thewaveformwillceasetobeconstant-itwill deformasitpropagates. Providedthenonlinearityis weak,the wave prole is changing slowly. i.e. along with the fastdependenceofthefunctionon=t x/c0, aslowdependenceofonxmustdevelop= (= t x/c0, x1= x). (1.9)14 Chapter 1Here 1isthesmallparameteroftheproblemconformingtothesmallness of nonlinear terms in equation (1.5) as compared with linearones ( + 1)[x 2x2[/[2x2[ ( + 1)[x[ 1. (1.10)Thefactthat [x[ 1hasbeenusedinpassingfromtheEarnshawequation(1.3)tothesimpliedequation(1.5). Ifweassume,inpar-ticular, that the displacement variation is in accordance with the har-moniclaw=Asin (t x/c0)thesmallnessconditionwill acquiretheform ( + 1)A/c0= ( + 1)2A/ 1. (1.11)This means that the particle displacement amplitude A must be smallcompared to the wave length . In other words, the ratio of the oscil-lation speed amplitude u0 to the sound speed c0, u0/c0 (acoustic Machnumber),hastobeasmallquantity. Therefore,thesmallparameteroftheproblemwillbetheacousticMachnumberM= u0/c0.Let us in equation (1.5) pass from x and t to new variables x1andinlinewiththeassumption(1.9). Calculatethederivatives2t2=22,x= 1c0+ x1,2x2=1c2022 2c0

2x1+ 22x21. (1.12)Substituting (1.12) into equation (1.5) and neglecting all the terms oftheorderof 2, 3andof thehigherinnitesimal orders(oneneedstotakeintoconsiderationthattheright-handsideoftheequationissmallcomparedwiththeleft-handone)yieldsux=

c20uu, (1.13)where u=/ =/t is the oscillationspeedof the mediumparticles,and = ( + 1)/2isaparameterofacousticnonlinearity.Simple waves 15The equation (1.13) in nonlinear acoustics is called an equation ofsimplewaves. Itisnoteworthythatthisisanequationoftherstor-derratherthanofthesecondorderastheoriginalone;therefore,theproblemhasbeensimpliedtoalargeextent. Derivationofequation(1.13)fromthehydrodynamicsequationsintheEulerianrepresenta-tioncanbefoundelsewhere,forexamplein[4],[6].The value of the nonlinear parameter canbe expressedas=( +1)/2 = 1 +B/2A,where = cp/cvis for gases and A , Bare theseriesexpansionfactorswhenexpandingthepressureuctuationsinterms of the density increment p

= A (

/0) +(B/2)(

/0)2+. . .:C 0 20 40 60 80 100 3.1 3.5 3.7 3.8 4.0 4.1Table1.1: NonlinearityparameterfordistilledwaterForexample, seawateratS=3.5%andt=20Chas=3.65.For water with steam- and gas-bubbles it depends on the bubble size,thebubbleconcentrationandthefrequency, andcanbeashighas5103. Afewothertypesof liquidshavethevaluesshowninTable2.1.Medium Valueoffor20 CMethanol 5.8Ethanol 6.3Acetone 5.6Glycerol 5.4Transformeroil 4.2Petrol 6.6Table1.2: Nonlinearityparameterfordierentliquids16 Chapter 1Problem1.6At the boundary of a nonlinear medium, x = 0, the particle veloc-ityfollowsthelawu(x=0, =t)=u0 sin(t). Solvingthesimplewave equation (1.13) by the method of successive approximations, de-ne the law of the second harmonic amplitude variation with increaseindistancex.SolutionFromtheequationforsimplewavesweobtainequa-tionsintherstandsecondapproximationsu(1)x= 0,u(2)x=

2c20

(u(1))2. (1.14)Thesolutionoftherstapproximationu(1)=u0 sin issubsti-tutedintothesecondequation(1.14). Integratingit, providedthatu(2)(0, ) = 0 (no second harmonic at the medium boundary), we ndthatu(2)= (/2c20) u20xsin 2. (1.15)Itis clearthatthe secondharmonic amplitudeincreaseslinearly withthex-coordinate. Thedistancex = xS= c20/(u0) = /(2M), (1.16)atwhichthesecondharmonicamplitudeformallyattains1/2oftherst harmonic amplitude is dened as a characteristic nonlinear length,orthediscontinuityformationdistance. Inreality, however, solution(1.15) holds true for distances xxSsince, under conditions ofconsiderable energy transfer from the rst harmonic to the second one,the solutions obtained by the method of successive approximations areinexact. Formula (1.15) implies that for acoustic signals, which alwayshaveasmall Machnumber(M 1), thenonlinearlengthxS .Inotherwords, tohaveitsproleandspectrummarkedlydistorted,awavehastocoveradistanceequaltomanywavelengths . Thisisexactly what we call the slowness of prole variation over the scalesontheorderof(seeProblem1.5).Simple waves 17Problem1.7At a boundary x = 0 the perturbation is a sum of harmonic signalsu(0, t) = u1 sin 1t+u2 sin 2t. Solving the simple wave equation (1.13)with the aid of successive approximations, nd the amplitudes u+anduof combinationharmonics 1+2and1 2. Compare thegenerationeciencyforthesumanddierencefrequencies.AnswerInanalogywithProblem1.6wendthatu=

2c20u1u2(12)x,uu+= [12[1 + 2< 1. (1.17)Problem1.8Show that an exact solution to the simple wave equation (1.13), forthearbitraryshapeperturbationu(x=0, t)=(t)atthenonlinearmediumboundary,isgivenbytheimplicitfunctionu(x, t) = (+ ux/c20). (1.18)Obtainformula(1.18)bythemethodof characteristicsknownfromthe theory of quasilinear partial dierential equations of the rst order.SolutionDierentiating(1.18)onendsux=(/c20)u

1 (/c20)x

,u=

1 (/c20)x

, (1.19)where the prime denotes a derivative with respect to a complete argu-ment of function . Substituting (1.19) into the simple wave equation(1.13) yields identity. Solution by the method of characteristics is de-scribedinProblem2.2. Itisinterestingtondoutthewaynonlineareectsarehiddenintheimplicitrelation(1.18). Expressing(1.18)inaseriesforsmallxweobtainu () +

() c20ux + . . . () +

c20x

()(). (1.20)It is seen that the second term is quadratic in the function , i.e. it de-scribesquadraticnonlineareects. Thetermsthatfollowcorrespondtononlinearitiesofhigherdegrees.18 Chapter 1Problem1.9Byusingtheimplicitsolution(1.18)tothesimplewaveequation(1.13),considertheevolutionofthelinearprolereferenceperturba-tionu(x = 0, t) = (t) = (t t). (1.21)Discussthecases> 0and< 0.Solution Substituting(1.21) intothe general formula(1.18)yieldsu(x, ) = (+ u(x, )x/c20t), (1.22)and,therefore,u =( t)1 x/c20. (1.23)Thus, for anydistance xthe prole remains linear in, the onlychangebeingits slopeanglewithrespect tothe-axis. Whentheslopeispositive(>0)thesolutionisvalidovertheniteintervalx xS.Problem1.11Usingasolutionofthejoiningtype(1.23),considertheevolutionofasingletriangularpulsewithduration2T. Forx = 0theproleisapproximatedbythepiecewiselinearfunctionuu0= 0 (< 0, > 2T),uu0=T(0 < < T),20 Chapter 1Figure 1.1: Problem1.10 Graphical analysis of the process of distor-tionofoneperiodofaRiemann(simple)waveduringitspropagationina nonlinearmedium. Theshape ofthe initialsignalis sinusoidalintimeanditsevolutionisdescribedbythesolution(1.28).uu0= 2 T(T< < 2T). (1.29)Considerthecasesu0>0andu0 2T)uu0=T (1 u0c20T x)1, (0 < < T u0c20x)uu0=2T T(1 +u0c20T x)1, (T u0c20x < < 2T) (1.30)Thesolutionintheformof asimplewaveholdsuptothedistanceSimple waves 21x=xS=c20T/(u0)wheretheleadingedgebecomesvertical. Theproledistortionprocessforu0> 0isdepictedinFigure1.2.Figure 1.2: Evolution of the prole of an initial single triangular pulseatitspropagationinanonlinearmedium.Problem1.12Findthespectrumofasimplewaveinanonlinearmediumifthewave at the input is given as u0(0, ) = u0() where is a functionwhichisperiodicinitsargumentwithperiodT= 2.SolutionOne needs to calculate the coecients Cnused in theFourierseriesexpansionof theimplicitsolution(1.18)tothesimplewaveequation(1.13)uu0= (+ zuu0) =+

n=Cn(z) exp(in). (1.31)Here z= (/c20)u0x is a dimensionless distance. The expansion coef-cientsareequaltoCn(z) =12_T(+ zuu0) exp(in) d() . (1.32)22 Chapter 1TherstintegrationbypartsprovidesCn(z) =12in_Texp(in)d =12in_Texp(in[z()])d() .(1.33)Informula(1.33)wepassedtothevariable =+ zu/u0, whichleadsto = z(), andnowtheintegral containsanexplicitfunctionof. PerformingthesecondintegrationbypartswecometoCn(z) = i2nz_[exp(inz()) 1] exp(in) d. (1.34)Asz 0wecanexpandtheexponentundertheintegral (1.34)intoaseriesandobtainanevidentresultofthelinearapproximationCn(z) =12_() exp(in) d= Cn(z= 0) = constant , (1.35)i.e. nointeractionof harmonics occurs andtheparameters inthemediumareequaltotheirreferencevalues.Problem1.13Making use of the answer in the previous problem (formula (1.34))ndthe harmonics amplitude dependencies ondistance z =x/xSprovided a harmonic signal u(0, ) = u0 sin is prescribed. Find thepowerlawsoftheamplitudegrowthforz 1.SolutionLet us exploit a mathematical identity from the theoryofBesselfunctions[22],exp(iz cos ) =+

k=ikJk(z) exp(ik). (1.36)Withthehelpof identity(1.36), theexponentunderintegral (1.34)canbeexpressedasexp(inz sin ) =+

k=Jk(nz) exp(ik). (1.37)Simple waves 23Afterthis,theintegralcanbereadilycalculatedasCn(z) = inz+

k=Jk(nz)nk= inzJn(nz). (1.38)Deningthereal FourierseriesexpansioncoecientsAnforcos nandBnforsin n:An(z) = Cn + Cn= 0,Bn(z) = i(CnCn) = 2Jn(nz)/(nz) , (1.39)weobtainthewell-knownBessel-Fubinisolutionuu0= sin(+ zuu0) =

n=12Jn(nz)nzsin(n). (1.40)DependenciesoftheharmonicamplitudesBnonthedistancearevi-sualizedinFigure1.3. Usingtherst termsof theBessel functionexpansionintotheseriesJn(x) (x/2)n/n!, (1.41)wecanwriteBn (nz/2)n1/n!. (1.42)Moreexactpowerapproximationsandtabulatednumericalvaluesofharmonicamplitudesaregivenin[10].Problem1.14Calculatethevariationwithdistanceof thedierencefrequencywaveamplitudeprovidedabiharmonicsignal u/u0= sin 1t +sin 2tis specied at the input x = 0. It is assumed that 1= (N +1) and2= N,whereN> 1isaninteger.24 Chapter 1Figure 1.3: Problem 1.13 Distance-dependent amplitudes of the rst,secondandthirdharmonicsofaninitiallysinusoidalwave. Thespec-tral contentisgovernedbytheBessel-Fubiniformula(1.40).SolutionInasmuch as the dierence frequency isequalto 12= , we are only interested in the coecient C1(z) in (1.34). Usingtherelationship(1.37)forabiharmonicsignalweobtaineiz sin(N+1)+iz sin N=

k=Jk(z)eik(N+1)

m=Jm(z)eimN. (1.43)Substituting this expression into (1.34) shows that the integral diersfromzerofork(N+ 1) +mN= 1alone. Thisispossibleonlyforthevaluesk = 1,m = 1and,therefore,wendthatC1= iJ21(z)/z, A1= 0, B1= 2J21(z)/z. (1.44)Simple waves 25Problem1.15ForN 1,withintheconditionsoutlinedforthepreviousprob-lem, dene the dierence frequency wave amplitude at a distance equalto the discontinuity formation length z= zS. Make a comparison withtheresultduetothemethodofsuccessiveapproximations(seeProb-lem 1.6) and nd out the way this amplitude depends on the frequencyratiobetweenand1.AnswerzS 1/(2N),B1 zS/2 /(41)Problem1.16Calculatethebehaviour of theamplitudes of thelowfrequencyharmonics whichareintroducedinanonlinear mediumas aresultoftheself-demodulationofthereferenceamplitude-modulatedsignalu/u0=(1 mcos t)sin(Nt), whereN 1isaninteger, andmisamodulationstrength.Answer InanalogywithProblem1.14aresultisachievedinthe form of a series containing the Bessel function products. The maintermsoftheseserieshavetheformC1= 2iz J1(z)J0(m2 z)J1(m2 z), C2=i2zJ0(2z)J21(mz). (1.45)For small z we obtain expressions corresponding to the solution by themethodofsuccessiveapproximationsB1 mz/2, B2 m2z/4.Problem1.17Consider interaction of a powerful low-frequency wave with a weakhigh-frequency signal u(0, t)/u0= sin t +msin Nt, (m 1, integerN1). Inwhat manner does theweaksignal amplitudevaryinspace?26 Chapter 1SolutionTaking the smallness of m into consideration, formula(1.34)yieldsCn(z) m2_sin(N) eiNz sin eiNd im2 J0(Nz). (1.46)HenceAN= 0andBN= mJ0(Nz). Thesolution(1.46)holdswithintherangebeforethediscontinuityformation(forz< 1). AsN 1theBessel functionargument in(1.46) canbealargequantity; inthis casetheweaksignal amplitudewill oscillateinspacetoeven-tuallydiedowngradually. Thiseectof nonlinearsuppressionof ahigh-frequency signal due to an intense low-frequency disturbance (e.g.noise)isofinterestinsomepracticalsituations.Problem1.18Usingthemethodof successiveapproximations, analyzethede-generateparametricinteractioninsimplewaves. Forareferenceper-turbationu/u0=sin 2t + msin(t + )withm 1, deneunderwhich phase shift the weak signalis amplied and under which itissuppressed.SolutionParametric amplication of weak signals in the eld ofan intense pump wave is considered to be an important nonlinear eectfrom a practical viewpoint. If the pump frequency is 2while that ofa signal is , the process is dened as degenerate; it is sensitive to thephaseshiftbetweenthesetwowaves. Inproblem1.6theequations(1.14) of the rst and the second approximation are formulated. Recallthatu(1)isareferenceperturbationinwhich =t x/c0isplacedinsteadof t; u(2)is thesolutionof thesecondapproximationtobefound. HoldingtheFourier-componentontheright-handsideof anequationforu(2)atthesignalfrequencyyieldsu(2)x=

2c20mu20 sin( ). (1.47)Thesolutionatthesignalfrequencyhastheform1u0(u(1)+ u(2)) = msin(+ ) m2 z sin( ). (1.48)Simple waves 27Hence it can be seen that the signal amplitude for z 1 behaves like2[C1(z)[ = m[cos2(1 z/2)2+ sin2(1 +z/2)2]1/2m(1 z cos(2))1/2. (1.49)Providedthephaseshift varies from0to, amplicationis ob-servedwithintherange/4 3/4suchthatthemostecientamplicationof thesignal occurs for =/2. Withintheranges0 /4and3/4 thesignal issuppressed, mostlyat=0andat=. Itseemsuseful tosolvethisproblemwiththespectral representation (1.34) of the simple wave equation solution (inanalogywithProblems 1.14and1.17) rather thanresortingtothemethodofsuccessiveapproximations.Problem1.19FindaFourier-transformofasimplewaveu(x, )C(x, ) =12_u(x, )eid (1.50)assumingthattheperturbationvanishesas .SolutionEmployingthegeneral solution(1.18)foraFourier-transformofasimplewave,oneobtainsC(x, ) =12_(+

c20ux) eid . (1.51)Likeinthesimilarproblem1.12,inwhichaperiodicsignalwasdealtwith, one needs to pass over to a new variable = +(/c20)x u. Then= (/c20)x(),andfor(1.51)wehavetheexplicitexpressionC=12_()(1

c20xd()d) ei((/c20)x())d . (1.52)Aevenmoreconvenientformcanbeachievedbyintegrating(1.52)twicebypartsandtakingintoaccountthat() = 0:C(x, ) =12i(/c20)x_[ei(/c20)x()1] eid . (1.53)28 Chapter 1Asx 0formula(1.53)yieldsC(x, ) =12_() eid= C0() , (1.54)beingtheFourier-transformofthereferenceperturbation.Problem1.20Proceedingfromsolution(1.53) totheprevious problem, ndauniversal behaviour of the Fourier-transform within the low-frequencyrange. Demonstrate that if n>1andC0() nas 0, auniversal asymptoticspectrumbehaviourisformedduetononlinearinteractions betweenthe spectral components inthe low-frequencyrange( 0).SolutionWithinthelow-frequencyrangetheexponentinthesolutioncanbeexpandedintoaseries. RestrictingourselvestothetermswhicharequadraticinwecometotheexpressionC 12_() eid +i4x c20_2() eid . (1.55)TakingtheFourier-transformpropertyintoaccount12_2() eid=_C0()C0( ) d, (1.56)wereduce(1.55)totheformC(x, ) = C0() +i2

c20x_C0()C0( ) d . (1.57)It follows that for the reference spectra of the type C0() n, n > 1thelow-frequencywavespectruminanonlinearmediumisdescribedbytheuniversalexpression[C(x, )[

2c20x, =12_2() d=_[C0()[2d.(1.58)Simple waves 29Problem1.21Proceedingfromexpression(1.53)forthesimplewavespectrum,ndthe Fourier-transformof asignal conformingtothe sinusoidalperturbationattheinput = u0 sin 0t.AnswerUsing the Bessel functions relation (1.36) and the prop-ertyofthe-function() =12_+eitdt , (1.59)onecanwriteC(x, ) = iu0

k=Jk(kz)kz( k0), (1.60)wherez=(/c20)0u0x=x/xS. AftertheinverseFourier-transformofformula(1.60)wearriveattheBessel-Fubinisolution(1.40).Problem1.22Find the spectrum components Cint(x, ) resulting from the inter-actionofanintensepumpwaveu1(t)withaweaksignalu2(t).u(x = 0, t) = (t) = u1(t) + u2(t)u1(t) = u0 sin 0t, u2(t) = b sin t. (1.61)SolutionNeglectingtheweaksignal self-action, theexponentin(1.53) canbeexpandedintoaseriesintermsof u2. Restrictingoneselftoalinearterm:C(x, ) = c20/(2ix)_[ei(/c20)xu1()1] eid +12_u2() ei(/c20)xu1()eid . (1.62)30 Chapter 1HerethersttermdescribestheFourier-transformofthepumpwavewhilethesecondoneisforthespectrumCint(x, )resultingfromthenonlinear interaction between the signal and the pump. Using relation(1.36) for Bessel functions and the ltering properties of the -functiononeobtainsfrom(1.62)Cint(x, ) = i b/2

k=Jk[

c20(k0)u0x]( + k0)Jk[

c20(k0 + )u0x]( + +k0) . (1.63)Problem1.23Employingtheresultof thepreviousproblem, considerthecaseofalow-frequencypump0 (thisproblemisageneralizationofProblem1.17). Describethesignal spectrumforvariousinteractionstagesandevaluatethesignalspectrumwidth.Solution For 0the nonlinear interaction leads to ahigh-frequencysignal modulationandappearanceof componentsatfrequencies = + k0(k=0, 1, 2, . . .) near thesignal fre-quency. FortheFourier-transform(1.63)oneobtains,respectingthatk0 L,Cint(x, ) = ib2

k=Jk( 0z)( + k0)Jk( 0z)( + +k0),where z=

c200u0x = x/xS. (1.64)It can be readily seen that (1.64) describes the spectrum of a signalwithharmonicphasemodulationu2(x, ) = b sin[+ (/c20)xu0 sin 0]. (1.65)Therefore, for 0theinteractioncanbeinterpretedasalow-frequencyphasemodulationof asignal createdbyapowerful pumpSimple waves 31wave. Asthewavespropagate, themodulationdepthincreases. For(/0)z 1 two harmonics = 0 dominate in the spectrum. For(/0)z 1, however, thespectrumbroadenssubstantially. Usingthe Bessel function asymptotic behaviour for large arguments [22] onecanestimatetheeectiveharmonicnumberinthespectrum(1.64):k z(/0). Thecorrespondingspectrumwidthis z 0.Problem1.24Usingtheresultof (1.63), considerthecaseof ahigh-frequencypump( 0). Interpretthenonlinearinteractionprocessfromaphysicalviewpoint.AnswerNonlinearinteractionleadsinthiscasetoappearanceoftwospectralcomponents=k0 neareachoneofthepumpwaveharmonics= k0;Cint(x, ) = ib2

k=Jk(kz)[(+k0) (++k0)]. (1.66)32 Chapter 2Chapter 2PLANENONLINEARWAVESWITHDISCONTINUITIESProblem2.1Determine the maximal distance, i.e. the range limits within whichthesolution(1.18)u=(+ ux/c20) of thesimplewaveequation(1.13)holds.SolutionInProblem1.8thefollowingderivativehasbeencal-culatedu=(/c20)u

1 (/c20)x

. (2.1)The maximal distance xSis when the derivative (2.1) is innite, whichhappenswhen1

c20x

(+ ux/c20) = 0. (2.2)Vanishingof denominator(2.2)in(2.1)correspondstothefactthatin a certain prole point at a distance xSthe derivative (2.1) turns toinnity, i.e. thetangentlinetothispointbecomesvertical; inotherwords,thisistheformationofadiscontinuity(ashock). Thesoughtprolepointcorrespondstothemaximumvalueoffunction

,i.e. isobtainedfromthecondition

=0. Therefore, thetwoconditions:

=0and(2.2)will allowsolvingtheposedproblem. Inpracticalsituations it seems convenient to make use of the fact that a solution ofthesimplewaveequation(1.13)canbewrittenexplicitly(1.25)withrespectto(u, x)= 1(u)

c20ux. (2.3)Ithasthefollowingproperties: a)atthedistancex=xSaverticaltangent line to the curve u(xS, ) appears; b) a discontinuity is formedintheinexionpointleadingtothefollowingcoupleofequationsu= 0,2u2= 0. (2.4)Plane nonlinear waves with discontinuities 33Problem2.2Solvingthesimplewaveequation(1.13)bythemethodofcharac-teristicspresentsanevidentillustrationofthesingle-valuednesscon-ditionformulatedinthepreviousproblem. Denewhichportionofthe reference perturbation prole u(x = 0, ) = () which will toppleoverrst,andatwhatdistance.SolutionA set of characteristic equations for the partial dier-entialequation(1.13)hastheformx= (/c20)u,ux= 0, (2.5)with(x = 0) = 0andu(x = 0, 0) = (0). Here0(u)isapointintheaccompanyingcoordinatesystemfromwhichacharacteristicfortheperturbationucomesout(Figure2.1).Thesolutionofsystem(2.5)= 0(/c20) (0)x (2.6)describes a family of straight lines in a plane (, x) with dierent slopesdependingonu=(0). It is worthwhilenotingthat (2.6) is theexpression (2.3) written in other notations. The time interval betweentheneighbouringcharacteristicsinaccordancewith(2.6)variesasd= d0[1 (/c20)

(0)x]. (2.7)Therefore, thewavewilltoppleoverontherstintersectionofchar-acteristics(seeFigure2.1)andwhendvanishes. ThistakesplaceatthedistancexS=c20max

(0). (2.8)The neighbourhood of the prole point in which a maximum of deriva-tive

isachievedwillbethersttotoppleover.34 Chapter 2dt0Figure2.1: Explanationof theturnoverprocessof aRiemannwave.The straight lines are the characteristics (2.6) andthe intersectionof lines is evidenceof theappearanceof multi-valueddependenceofparticlevelocityontime.Problem2.3Find the distance at which a discontinuity is formed for a nonlinearsimplewavespeciedattheinputintheformof theunipolarpulseu(x = 0, t) = u0et2/t20.SolutionTherstwayWritethesolutionofthesimplewaveequationforthegivenunipolarpulseastheexplicitfunction= (x, u):= t0_ln(u0/u) ux/c20. (2.9)Plane nonlinear waves with discontinuities 35Here we use the minus before the root since adiscontinuityisalwaysformedintheleadingfront(inourcasefor< 0).Itshouldsatisfytheconditions(2.4):u=t02u[ln u0u ]1/2

c20x = 0, (2.10)2u2= t02u2[ln u0u ]1/2+t04u2[ln u0u ]3/2= 0. (2.11)From (2.11) we nd that a discontinuity is formed for u = u0/e. Sub-stituting this value into (2.10) yields the distance xS=_e/2(c20t0/u0).ThesecondwayFollowing the pattern described in Problem 2.2, calculate a derivativeofthereferenceperturbationform

(0) = 20t20u0e20/t20. (2.12)Amaximumof thefunction(2.12) is attainedfor 0=t0/2andequals2u0/e t0. Formula(2.8)forthisvalueimmediatelysuppliestheresultxS, whichcoincideswiththeexpressionobtainedthroughtherstway.Problem2.4FindthecoordinatexSof thediscontinuityformationinahar-monicreferencewaveu(x=0, t)=u0 sin t. Determinetheprolepoints()wherediscontinuitiesappear.AnswerA discontinuity is formed for = 2n at the distancexS= c20/u0, (n = 0, 1, 2, . . .).Problem2.5Findthecoordinateof shockformationxSinthestep-wisedis-turbanceu(x=0, t)=u0 tanh(t/t0). Determinetheprolepoint()wherethediscontinuityappears.36 Chapter 2Answer Adiscontinuityformsfor =0atthedistancexS=c20t0/u0.Problem2.6Aperturbationat theinput is consideredtobeasuperpositionof harmonic oscillations with two incommensurable frequencies u(x =0, t) = u1 sin 1t + u2 sin 2t. DeterminethedistancexSatwhichtherstdiscontinuityappears.AnswerTherstdiscontinuityappearsatthedistancexS=c20(1u1 + 2u2). (2.13)Problem2.7Atwhichdistancefromahigh-powerultrasoundsourcedoestheshockforminwater? TheintensityoftheultrasonicwaveisI=10W/cm2, the frequency is f= 1 MHz, the density is 0= 1 g/cm3, thesoundvelocityisc0= 1.5105cm/s,andthenonlinearityis = 4.AnswerUsing the formula given in the answer to Problem 2.4,thesolutioncanbeevaluatedtobexS=c202f_c002I 25cm. (2.14)Problem2.8Determinetheintensityofawavepropagatinginwaterforwhichadiscontinuityformsatthedistance10m. Thefrequencyisf= 200kHz.AnswerI= 0.5c500(2fxS)2 0.15W/cm2.Plane nonlinear waves with discontinuities 37Problem2.9Evaluatethevibrationvelocityamplitude, theparticledisplace-ment, the particle acceleration, andthe magnitude of the acousticMachnumberforProblems2.7and2.8.AnswerFor Problem 2.7: velocity u0= (2I/c00)1/2 36 cm/s,displacement 0= u0/ 6106cm, acceleration a0= u0 2108cm/s2,andMachnumberM= u0/c0 2.4104.ForProblem2.8: u0 4.5cm/s, 0 4106cm, accelerationa0 5106cm/s2,andMachnumberM 3105.One cansee, that the displacements of particles are verysmalleven in high-power ultrasonic elds; they are on the order of molecularscales. On the other hand, huge accelerations exist, up to 106g, wheregisaccelerationof gravityonearth(g9.8m/s2). Machnumbersaresmall, andthisfactwasalreadyusedatthesimplicationofthenonlinearequationsinProblems1.2and1.5.Problem2.10Aplanemonochromaticwavepropagates inair. Writethefor-mulafor the shockformationdistance throughthe soundpressurelevelN(dB)andfrequencyf. EvaluatetheMachnumberandshockformation distance for N= 140 dB (this level corresponds to the noiseofaheavyjetaircraft)andf= 3300Hz. Theadiabaticpowerindexforairis= 1.4.SolutionIt is common for atmospheric acoustics to characterizethesoundintensitybythelevelofroot-mean-squarepressureN(dB)relative to the reference pressure p= 2105Pa. The peak pressure isp

=2p10N/20. The shock formation distance for a plane monochro-maticwaveisdeterminedbytheformula(1.16), whereM=u0/c0,andu0isthepeakmagnitudeof vibrational velocity. Withaccountfor thedenitionc20=p0/0, where0is densityof air andp0isatmospheric pressure (p0 105Pa), we write the Mach number in the38 Chapter 2formM= p

/(c200) = p

/(p0). Consequently,theMachnumberandtheshockformationdistanceareevaluatedasM=2 pp010N/20 2103,xS=c0f

+ 1 12

p0p10N/20 6m. (2.15)Problem2.11Make a simple geometrical construction based on the conservationof momentumtransferredbyasimplewavewhichremovesthenon-single-valuednessoftheoverhangproleatdistancesx>xS(Figure2.2).uSt12SFigure2.2: Constructionof ashockfromamulti-valuedproleof aRiemannwaveinaccordancewiththeruleofequal areas.SolutionMakesurethat themomentuminalimitedsimplewave(u 0for )isindependentofxforx < xS_+0ud= 0_+(+

c20ux) d=0_+() d(

c20x) = 0_+() d. (2.16)Plane nonlinear waves with discontinuities 39Thegeometrical meaningoftheconservationlawisthekeepingofaconstantareabetweenthewaveprole(x, )andthe axis. Aftertheoverhangformation(x>xS), thisareahastoberetainedinas-muchas themediumregiontakenupbythewavemotionremainsclosed (not aected by external forces). Consequently, a discontinuityin a non-single-valued wave prole has to be introduced such that thecut-o areas S1and S2(see Figure 2.2) are equal. In fact, the area S1is added to the prole whereas the area S2is subtracted from it. Pro-videdthatS1= S2,theareaunderthecurveisequaltothereferencevalue_+() d.Problem2.12Demonstratethatacompressiveshockwave,ajumpbetweenthetwoconstant values u1andu2withu2>u1, is stable, i.e. is notsupposedtochangeitsformduringpropagation.SolutionLet for simplicity u1= 0, u2> 0. At a reference pointx = 0 a wave has a discontinuity located at = 0 within the accompa-nyingcoordinatesystem. Fordistancesx>0thedistortedproleisconstructed graphically by the method outlined in Problem 1.10. It isevidentthattheprolebecomesnon-single-valuedforarbitrarysmallx(dashedlineinFigure2.3a). Thisnon-single-valuednessisremovedinaccordancewiththeruleofequalityofareas(seeProblem2.11).Asaresultoneobtainsajumpofthesameshapeandmagnitudebutwith a front slightly displaced forward. This means that the compres-sionwaveisstable. Thefrontdisplacementwithintheaccompanyingcoordinate system = t x/c0testies that the positive (with respectto the unperturbed level u1= 0) jump u2 travels with supersonic speedc = c0+u2/2 - the faster the higher the shock jump. It is worthwhilenotingthattherarefactionshockwave(u22. Itisof interestthatforz 1, uS c20/x,whichdoesnotdependontheinputsignal amplitude. Thesolutiontotheproblemissetforthin[4]and[6].Plane nonlinear waves with discontinuities 41Problem2.14Using the results of the previous problem, ndthe shape of asinusoidal input at the distances z= x/xS> 2. Calculate the spectralcompositionandthe energydensityaverage over the period: E=0u2= 0T1_T0u2(x, ) d .AnswerThewaveacquiresthesawtoothproleuu0=11 + z(+ sign()), . (2.17)Itsspectrumisuu0=

n=12n(1 + z) sin n. (2.18)Duetodiscontinuityformationandtheirnonlineardamping(whichgrows higher for larger u0), the harmonic amplitudes decrease fol-lowingthepower lawAn n1. Theenergydensitydecreases asE= 20u20/3(1 + z)2andforz 1itdoesnotdependontherefer-enceperturbationamplitudeu0.Problem2.15MakinguseoftheplottingsdescribedinProblems1.10and2.11,followtheevolutionof therectangular-at-inputpulse() =AforT< < 0,and() = 0outsidethisinterval. Findanasymptoticformofthepulseasx .Answer Theinitial shapeofthepulseanditsshapeforthreecharacteristicdistancesaredepictedinFigure2.4.For x(/c20)A/T 1 the pulse acquires a universal triangular shapewithaslopewhichisindependentofAandT:u = c20x, (S< < 0) u = 0 , (< S, > 0). (2.19)HereS(x) = [2AT(/c20)x]1/2isthecurrentpulseduration. Itcanbeproventhatforanyxthepulseareaisequal toAT, whichisinlinewiththeconservationofmomentum.42 Chapter 2 ttSTAu0Figure 2.4: Problem2.15 Evolutionof the shape of aninitiallyrectangular pulse. Because of the motion of the leading shock front thepulsebecomestriangular. Thereafter,thepeakvaluedecreasesandanincreaseinitsdurationtakesplace.Problem2.16Perform a graphical analysis of the nonlinear evolution of a bipolarsoundpulseconsistingoftwosymmetrictriangularpulses(seeProb-lem1.11)withduration2T0andareaSforthefollowingcases: a)ararefactionphasefollowedbyacompressionphase; b)acompressionphasefollowedbyararefactionphase.Answer It is evident fromFigure 2.5, that, for case a) thepulseistransformed intoaso-calledS-waveofconstantduration2T0;and, forcaseb)thepulseistransformedintoanN-wavedurationofwhich 2T(x) grows with x. Increase of the curve number in Figure 2.5correspondstoincreaseofdistance.Plane nonlinear waves with discontinuities 43t ta bl 2 34l 23uuFigure2.5: Problem2.16Theevolutionof theshapeof abipolarpulseisdierent inthecompressionandrarefactionareas. ApulsecantransformintoanS-wave(a)ortoanN-wave. Highernumberofcurvecorrespondstoincreaseinthedistancetraveledbythewave.Problem2.17FindtheasymptoticbehaviouroftheFourier-transformsfor(/c20)Sx/T20 1inthepreviousproblemsconditionsbyemployingthelinearproleevolutionresultsinProblem1.9. Discusscharacter-istic properties of the spectra structure in the high and low frequencyranges.Answera)ThespectrumoftheS-wave:[C(x, )[ =2T20T0(1 sin T0T0)c20x. (2.20)b)TheN-wavespectrumisself-similar:[C(x, )[ =2T2T [ cos T sin TT[c20x, T= [2S

c20x]1/2. (2.21)Spectra(2.20)and(2.21)aredisplayedinFigure2.6.Within the high-frequency range the spectra drop according to thepower lawwhichis associatedwiththe presence of discontinuities.44 Chapter 2uT0uT02 l|C(uT ,x)j0|C(uT ,x)j0x>xl 2x< x3 636abFigure 2.6: Asymptotic behavior of the spectra of bipolar pulses at largedistances in a nonlinear medium of initial: S-wave (a) and N-wave (b).FortheS-pulseallthespectrumcomponentsdiminishas 1/x. FortheN-pulsethespectrummaximumis observedtograduallymoveinthedirectionof lowfrequencies; at highfrequencies thespectralcomponents decrease as 1/x, whereas for low frequencies it looks like[C(x, )[ x. Thespectral densitygrowthatlowfrequenciesisrelated to the parametric energy pumping under nonlinear interactionofhighfrequencyharmonics.Plane nonlinear waves with discontinuities 45Problem2.18Formulate a set of equations describing the evolution of the proleofasimplewavecontainingadiscontinuity.Solution Obtainadierential equationdescribingthe shockfront motion within the accompanying coordinate system. Consider adiscontinuityatadistancexwhichhasthecoordinateS(x)(Figure2.7).t(x)ttt(x+Ax)t1 2A'ABB'SSuuul2Figure 2.7: Inthederivationofequation(2.23)theruleofequalareasdescribingtheshockmotionwithintheRiemannwaveproleisused.Thevibrationvelocityimmediatelybeforethefront(pointA)isu1, andimmediatelyafter the front (point B) it is u2. Whenthedistance increases byx, the point Amoves toA

whichhas thecoordinate 1= S(x) (/c20)u1x and the point Bmoves to B

withthecoordinate2=S(x) (/c20)u2x. Theruleofequal area(seeProblem2.11)impliesthatthenewcoordinateofthediscontinuityS(x + x) =12(1 + 2) = S(x)

2c20(u1 + u2)x. (2.22)46 Chapter 2Passingtothelimitin(2.22)asx 0,wecometotheequationdSdx=

2c20(u1 + u2). (2.23)Therefore, the front travel speed in the accompanying coordinate sys-tem depends only on the perturbation values u1 and u2 at the disconti-nuitywhich,generallyspeaking,dependonthedistancex. Inasmuchasu1andu2notonlybelongtothediscontinuitybuttothesimplewaveproleaswell,thesolutionin(2.3)willbevalidforthem,i.e.S(x) = 11(u1)

c20u1x, (2.24)S(x) = 12(u2)

c20u2x. (2.25)Herefunction1describesthesimplewaveprolebeforethedis-continuity, and2for after the discontinuity. 11,2are the inversefunctionsof 1,2. Thethreeequations(2.23)through(2.25)forthethreeunknownsS(x),u1(x),u2(x)furnishacompletesetforsolvingtheproblem.Problem2.19Makinguseoftheequations(2.23)through(2.25)inthepreviousproblem, nd the change with distance of the jump magnitude and thedurationofatriangularpulsewithashockwaveattheleadingedge.Forx=0thepulseisprescribedasfollows: u/u0=1 /T0(0zS=1(orx>xS). Thediscontinuityamplitudeisobservedtoincreasefor1 1), (2.36)is constant beforetheshockformationandincreases monotonicallyaftershockformsbecauseitmoveswithvariablesupersonicspeed.Problem2.24This problem, and those which follow (2.24-2.28), are based on theresultsobtainedbyD.G.Crighton2.Theyare concernedwithsimple waves inacubicallynonlinearmediumwhosepropagationisgovernedbytheequation(1.13)ux= u2u. (2.37)Findasolutiontothisequationandperformagraphicanalysiswithrespecttoasinusoidalsignal.SolutionEquation (2.37) can be solved by the method of char-acteristics. Or,bymeansofadirectsubstitutiononecanverifythatitissatisedbytheimplicitexpressionu(x, ) = (+ u2x). (2.38)Inthecaseofasinusoidalsignalwewrite= arcsin(u/u0) u20x(u/u0)2. (2.39)Inordertographicallyanalyzethenonlineardistortionof thewaveprole, make use of the expression (2.39) and proceed in analogy with2See original paper: I.P. Lee-Bapty, and D.G. Crighton.Phil.Trans.Roy.Soc.London, A 323, 173-209, 1987; and the general-ization for diracting beams: O.V. Rudenko, and O.A. Sapozhnikov,J.Exp.Theor.Phys.(JETP) 79(2), 220-228, 1994.Plane nonlinear waves with discontinuities 51Problem1.10. Asdistinctfromthequadraticallynonlinearmediumfor whichthedistortedproleis achievedthroughgraphicsumma-tion(seeFigure1.1) of thereferenceproleandthestraight line,here-inlinewith(2.39)-thestraightlinehastobereplacedwithaparabola. TheappropriateconstructioniscarriedoutinFigure2.9.Abreakinanon-single-valuedprolewaseectedfollowingtheruleof theareaequality; thecut-oportionsarecross-hatched. Itisev-ident that thewaveproleacquires asaw-toothshape. Unlikeforaquadratically-nonlinearmedium,however,wherethesawiscom-posedoftriangularteethinthecubicmediumtheteethofthesawremindsofatrapezoid.Problem2.25Findthecoordinateofthediscontinuityformationinasinusoidalreference wave propagating in a cubically-nonlinear medium. Identifytheprolepointswherethediscontinuitiesappear.SolutionLet us employ formula (2.39) of the previous problem.DenoteforconvenienceV= u/u0, = ,andz= u20x: = arcsin V zV2. (2.40)ProceedingasinProblem2.3(therstmethod)andconsideringforsimplicitythersthalf-period(0 < < )wearriveatV= (1 V2)1/22zV= 0,2V2= V (1 V2)3/22z= 0.(2.41)ThesetwoequationsimplythatadiscontinuitystartstoformintheprolepointwhereV= 1/2atadistanceofz= 1.Problem2.26Formulateadierentialequationofthetype(2.23)describingthediscontinuitypropagationinacubicallynonlinearmedium.52 Chapter 2utu/u0Figure 2.9: Graphical analysis of the distortion of a Riemann (simple)waveinacubicallynonlinearmedium.Solution Consider anon-single-valuedportionof the proledepictedinFigure2.10.Accordingtothe lawof conservationof momentumthe shadedareasS1andS2havetobeequal,thatisddx_u2u1[(u) S(x)] du = 0 , (2.42)whereu1isthevalueoftheparticlevelocitybeforethefront,andu2isthevalueimmediatelyafterthefront. SubstitutingthesolutionofPlane nonlinear waves with discontinuities 531uuut(u)SS212ttS(x)Figure2.10: Thepositionof shockinacubicallynonlinearmediumusedinthederivationofequation(2.45).thecubicsimplewaveequation(2.37)intheform= 1(u) u2x (2.43)into(2.42)andcalculatingtheintegraloneobtains(3u3ddxu)[u2u1= 0. (2.44)Hencethedesiredresultfollows:dSdx= 3(u21 + u22 + u1u2). (2.45)Thefrontpropagationvelocitywithintheaccompanyingcoordinatesystemgovernedbyequation(2.45)dependsonlyonthevaluesof u1,u2of thediscontinuity(which, generally, candependonx). How-ever, in contrast to the equation for a quadratically nonlinear medium(2.23), equation(2.45)describesamorecomplexmotion. Therefore,54 Chapter 2inasmuchasu21 + u22 + u1u2> 0foranyvalueofu1,u2themotionofanydiscontinuityacrossacubicallynonlinearmediumisalwayswithsupersonicspeed(naturally,asbecomingmorerigidinnonlinearme-diaforwhich>0). Amongall discontinuitiesunderxedu2theslowestisthatforwhichu1= u2/2; itsmotionisdescribedbytheequationdS/dx = (u2/2)2.Problem2.27Usinggraphicanalysisoftheprole(seeProblem2.24)andequa-tion (2.45), demonstrate that in a cubically nonlinear medium a slowjumpforwhichu1= u2/2andu2> 0(seethepreviousproblem)isstable. Showalsothatajumpwithaperturbationvaluebeforetheshockwavefrontequaltou1< u2/2isunstable,i.e. itsshapegetsdistortedonpropagation.SolutionSubstitutingthevaluesof u1= u2/2andu2>0intoequation(2.45)yields dS/dx = (u2/2)2. ThismeansthatthefrontcoordinateS(x)isS(x) = (u/2)2x (2.46)at a distance x (provided that S(0) = 0), coinciding with the positionof apoint Aonthe simple wave prole whichcorresponds totheperturbationu1= u2/2before the front. As is showninFigure2.11theadvancedfrontpassesexactlyoverthepointAbelongingtoparabola= u2x.Because of this the wave at a distance still has the form of a jumpbetweenthevaluesofu1= u2/2andu2. Thus,thiswaveisstation-ary,i.e. itsformdoesnotchangeonpropagation.As is evident fromthe similar constructioninFigure 2.12, thereference jumpbetweenthe values of u10isunstable. AnABportionofthesimplewaveproleemergesbeforethefrontandtheformofthewholewaveatadistancexdiersfromthereferencestep.Ithastobenotedthatararefactionjumpwiththefollowingpa-rameters at discontinuity: u1> 0, u2= u1/2 turns out to be a stablePlane nonlinear waves with discontinuities 55uu = -u 2/21u2AtFigure2.11: Astableslowcompressionshockinacubicallynonlin-earmedium.slow wave. This fact results from the equation (2.45) invariance withrespect to the transformation u1= u2, u2= u1or, more generally,from the invariance of (2.37) with respect to the substitution u u.Problem2.28Demonstratethatequation(2.37)forthesimplewavesinacubi-callynonlinearmediumhasasolutionofthetypeu(x, ) = f(x) ( S(x)) . (2.47)Deneunknownfunctionswhichdescribe: f(x)-thevariationofthecharacteristic wave amplitude; and, S(x) - the wave motion as a wholein the accompanying coordinate system. Find also the stationary pro-leshape,i.e. thefunction().56 Chapter 2tABuuu12Figure2.12: AnunstablecompressionshockinacubicallynonlinearmediumSolution Substituting (2.47) into the simple wave equationgivestherelationshipf

f

Sdd= f32dd. (2.48)In(2.48)thevariableshavetobeseparated. Thisispossiblepro-videdthex-dependentfunctionsareequaltof= u0(1 +x/x0)1/2, S= 0 ln(1 + x/x0) . (2.49)Here x0and 0are arbitrary positive constants. In deducing the func-tions f(x) and S(x) the following boundary conditions were employedfordeniteness: f(0)=u0- thereferenceamplitudeisequal tou0;and, S(0)=0- thereferencefrontpositionistheoriginof theac-companyingcoordinatesystem. Substitutingtheobtainedsolutionsof (2.49) into(2.48) yields anordinarydierential equationfor thefunction(202u20x0)dd= 1 . (2.50)Plane nonlinear waves with discontinuities 57Thesolutionof(2.50)hastheform= C + 20 ln [[ u20x02. (2.51)Astheformula(2.51)denesanon-single-valuedandunboundedfunction (), solution (2.51) can not correspond to a wave possessingphysical meaning. However, an arbitrary integration constant Cin thesolution(2.51)permitstodisplacethefunction( C)alongthe axisandobtainaperiodicsequenceofbranches. Connectingthesebranches bydiscontinuities, D.G. Crightonmanagedtoconstruct aperiodictrapezoidal saw(seeFigure2.13)representinganasymp-toticform(underz=u20x 1)of thereferencesinusoidal signal(seeFigure2.9).-AA2A-2A-2A-AA2A-rdu(tt )s0Figure2.13: Theasymptotic(atlargedistances)proleof aperiodicsignal havingsinusoidal shapeat theinput of acubicallynonlinearmedium. Eachsaw-toothissimilartoatrapezium.ItisworthwhilenotingthatthefrontsdepictedinFigure2.13ap-pear as an alternating sequence of two discontinuities with parametersu1= A,u2= 2Aandu1= 2A,u2= A. Boththediscontinuities,58 Chapter 2asisshowninProblem2.27,areslowstableshockwavespropagat-ingwiththesamevelocity.Nonlinear waves in dissipative media and Burgers equation 59Chapter 3NONLINEARWAVESINDISSIPATIVEMEDIAANDBURGERSEQUATIONProblem3.1Makinguseof themethodof slowlyvaryingprole(seeProblem1.5),simplifythelinearequation2ut2 c202ux2=b0

3ut x2, (3.1)describing propagation of sound in a viscous heat-conducting medium(see[4],[6]). Here=4/3 + + (c1v c1p)isadissipationfactorwhere ,are bulk and shear viscosity,respectively,and is thermalconductivity. Obtain a solution to the resulting equation for sinusoidalandunipolarpulsedinputsignals.SolutionItisassumedthatdissipativeeectsleadtoslowdis-tortionof theprolewhichallowsonetopassovertotheaccompa-nyingcoordinates =t x/c0, x1=x. Theterms2, 3, . . . aredisregardedwhilethetermsoftheorderof0cancel eachother. Asaresult, thetermsof thesameinnitesimal order1remain, whichmakeuptheparabolictypeequationux= 2u2, =b2c300. (3.2)The general solution of (3.2) conforming to the arbitrary shape initialperturbation u(x = 0, t) = u0(t) is expressed with the aid of the Greenfunctionu(x, ) =_+u0(

)G(x,

) d

, G =exp(2/4x)4x. (3.3)60 Chapter 3Fortheharmonicperturbationu0= a sin(t)oneobtainsu(x, ) = a exp(2x) sin(t) (3.4)whichisanexponentiallydampedwave. Thequantityxdisthein-verse of the damping factor (xd 1/(2)) and is called characteristicdampinglength. Theconditionxd impliesthattheamplitudeofthewave(3.4)dropsinsignicantlyoverdistancesontheorderofthewavelength. Therelation/xd=bc200 1 , (3.5)isasmallparameteroftheproblem; itisoftheorderoftheratiooftheright-handsidein(3.1)toanyoneofthetermsontheleft-handside. Thepresenceof thesmall parameterjustiesthetransitionfrom(3.1)to(3.2).For a unipolar pulse having characteristic duration t0at the input,thewidthofthefunctionG(x, )isobservedtogreatlyexceedt0atthedistances4x/t20 1andformula(3.3)issimpliedtou(x t20/4, ) = G(x, )_u0(

) d

. (3.6)Therefore,forlargedistancesthepulsetakesontheasymptoticformoftheGaussiancurve.Problem3.2Deduce anevolutionBurgers equationdescribingthe slowpro-cessesofthewaveproledistortionduetononlinearanddissipativepropertiesofthemedium.SolutionUsing the method of slowly varying prole, the simplewaveequations(1.13)andtheparabolicequation(3.2)werederivedearlierux= uu,ux= 2u2, (= /c20, = b/(2c300)) , (3.7)Nonlinear waves in dissipative media and Burgers equation 61describing the prole evolution due to nonlinear and dissipative eectsseparately. Inasmuchastheseeectsareweaktheyaredescribedbyindependent terms in the reference equations. As a consequence, non-linear anddissipativetermswill enter thesimpliedequationaddi-tivelyintheformofindividualsummands. Thus,ageneralizationofequations(3.7)resultsinux= uu+ 2u2, (3.8)which is called Burgers equation. If we pass in (3.8) to the dimension-lessvariablesV= u/u0, = , z= u0x = x/xS(3.9)-whereu0isthecharacteristicperturbationvalue(forexample, theharmonicwaveamplitudeorpeakperturbationinthepulse)andisacharacteristicfrequencyofaperiodicsignal(ortheinversepulseduration)-theequationacquirestheformVz= VU+ 2V2. (3.10)Herethenumber =b2c00u0=12Re=u0

, (3.11)is asingle dimensionless set of parameters enteringequation(3.8),therebycompletelydeningtheevolutionprocess.Sometimes, theacousticReynoldsnumberRe=(2)1isusedinsteadof . Onecanwriteas arelationbetweencharacteristicnonlinearanddissipativelengths; =xSxd=c20u0/2c300b2=1u0/12. (3.12)Itisevidentthatthequantityestimatestherelativecontributionof nonlinear anddissipativeeects intothewaveproledistortion.For 1nonlinearitydominates, whereas for 1dissipationprevails. Aconsistentderivationof theBurgersequation(3.8)fromtheequationsofhydrodynamicsisoeredin[4].62 Chapter 3Problem3.3Assuming that the sound absorption coecient in water is denedbythevalue=0.61017s2/cmandthatitinairamountsto=0.51014s2/cm, estimate the acoustic Reynolds number in problems2.7,2.8,2.10.Answera)Re 22,b)Re 13,c)Re 300,Problem3.4Let(x, )beacertainknownsolutiontotheBurgersequation(3.8) that corresponds to the boundary condition (x = 0, ) = 0().FindasolutionconformingtothesuperpositionoftheconstantowwithvelocityV0= constantonthereferenceperturbation0,i. e.u(x = 0, ) = V0 + 0() . (3.13)AnswerThissolutionisgivenbyformulau(x, ) = V0 + 0(x, + V0x) . (3.14)The propagation speed of the wave downstream is c c20V0= V0higherthanthatinanunperturbedmedium.Problem3.5Findastationarysolutionof theBurgersequationsatisfyingtheconditions of a symmetric jump u( ) = u0and u( ) =u0. Employthe transformation(3.14) inthe previous problemtoconstructastationarysolutionwhichsatisestheconditionsu( ) = u1andu( ) = u2> u1.Answer Thestationarywaveissoughtintheformu(x, )=u(+ Cx) wheretheconstant Cis denedfromtheconditions as . Intherstcasethestationarysolutionhastheformu = u() = u0 tanh(u0/2) = u0 tanh(/2) , (3.15)Nonlinear waves in dissipative media and Burgers equation 63whichdescribes asymmetric shockwave travellingwiththe soundvelocity. Thefront thickness is inverselyproportional tothejumpvalueu0. AssumingV0=(u1 + u2)/2, u0=(u2 u1)/2from(3.13)and(3.14)weobtainforthemovingshockfrontu =u1 + u22+u2u12tanh[u02(+ u1 + u22x)] . (3.16)It seems useful to make sure that the travel velocity of the weak shockwave front (3.16) does not depend on its thickness and coincides withthevelocity(2.23)ofthediscontinuitypropagation.Problem3.6Demonstrate that the Burgers equation, through the change of thevariablesu =S, S=2ln U (3.17)oru =2 ln U, (3.18)(Hopf-Colesubstitution)isreducedtothelineardiusionequation.FindthegeneralsolutionoftheBurgersequation.SolutionFromequation(3.8),obtaintheequationforSSx 2(S )2= 2S2, (3.19)whichafter the transitionof (3.17) to U is reducedto the linearparabolicequationUx= 2U2, (3.20)coinciding in form with (3.2). The solution of this equation subject tothe boundary condition U(x = 0, t) = U0(t) will be formulated similarto(3.3)U(x, ) =14x_U0(t) exp[(t )24x] dt . (3.21)64 Chapter 3Takingthechange(3.17)intoaccountU0(t) = exp[ 2S0(t)] , S0(t) =_tu0(t

) dt

, (3.22)thechainof transformations(3.22) (3.21) (3.18)providesthegeneralsolutiontotheBurgersequation,i.e. furnishestheexpressionof the eld u(x, ) in an arbitrary section x through the reference eldu0(). Letusshowonemoreforminwhichthegeneral solutioncanbewritten. Using(3.18)weobtainfrom(3.21),(3.22)u(x, ) =_txexp[12F(, t, x)]dt_ exp[12F(, t, x)]dt, (3.23)whereF= S0(t) (t )22x, S0(t) =_tu0(t

) dt

. (3.24)Problem3.7On the basis of the general solution of the Burgers equation arrivedatinthepreviousproblem,considertheevolutionofaharmonicref-erencesignal u0(t)=a sin(t). Investigateitsasymptoticbehaviourasx .AnswerMakinguseoftheexpansion[22)exp(z cos ) = I0(z) + 2

n=1In(z) cos(n) , (3.25)whereInaremodiedBessel functions, onecanobtainfrom(3.18),(3.21)and(3.22)u(x, ) =2aRe

n=1n(1)n+1In(Re) exp(n22x) sin(n)I0(Re) + 2

n=1(1)nIn(Re) exp(n22x) cos(n).(3.26)Nonlinear waves in dissipative media and Burgers equation 65Here the parameter combination a/2 has the meaning of the acous-ticReynoldsnumber(3.11). For2x 1theexponentsin(3.26)dropmarkedlywithnresultinginthattherstharmonicistheonlyoneremainingu(x, ) 2aRe I1(Re)I0(Re) exp(2x) sin() . (3.27)Forsmall andlargeReynoldsnumbers, usingtheasymptoticBesselfunctions, a harmonic damping as per the linear acoustics laws can beformulatedu(x, ) exp(2x) sin()_a Re 14/ Re 1. (3.28)Inthe latter case the harmonic amplitude does not dependonitsreferencevalue.Problem3.8Employingthegeneral solutionoftheBurgersequation, considerthe unipolar pulse evolution by approximating it as a -function at theinput: u0=A(t). IntroducetheReynoldsnumberforthisproblemanddiscussthelimitingcasesRe 1andRe 1.AnswerThesolutionhastheformu(x, ) =214x(eRe1)e2/(4x)1 +12(eRe1)(1 + (/4x)), (3.29)where(z)=(2/)_z0exp(t2) dtistheerrorintegral andRe=A/(2). ForRe 1theresultin(3.29)coincideswiththelinearsolution(3.6). ForRe 1equation(3.29)impliesthatthepulsehasauniversaltriangularformu(x, ) _ /(x) , T< < 00 , < T , > 0(3.30)where T=2Ax is the pulse duration. To derive formula (3.30) onehastousetheasymptoticsofthefunction(z)asz .66 Chapter 3Problem3.9Let(x, )beacertainknownsolutionof theBurgersequationsatisfyingtheboundarycondition(x=0, t)=0(t). Analyzetheinteraction of this wave with the linear prole of the ow (see Problem1.9)onthebasisofthegeneralformoftheBurgersequationsolution(seeProblem3.6)fortheboundaryconditionu(x = 0, t) = t + 0(t) . (3.31)Examinethecasesof> 0and< 0.Answer Nonlinear interaction with the linear prole bringsabout the change of the characteristic amplitude and frequency as wellasofthewave(x, )evolutionrate. Thesolutioncanbewrittenasu(x, ) =1 x+11 x(x1 x,1 x) . (3.32)For > 0 and x 1 the characteristic amplitude and frequency ofthewaveincreasewithoutbound.Problem3.10Using the saddle-point method, nd an asymptotic solution to theBurgers equation (3.8) for large Reynolds numbers ( 0). Interpretthissolutiongraphically.Solution The formula for the general solution(3.23) of theBurgersequationincorporatesintegralsofthetypeI=_+f(t) exp(12F(, t, x)) dt . (3.33)As 0 the main contribution to the integral will be from the neigh-bourhoodsof thepointsinwhichthefunctionFhasitsmaximum.LettKbeoneofthesepoints,whichcanbefoundfromtheequationFt= 0 , 0 =tKxu0(tK) . (3.34)Nonlinear waves in dissipative media and Burgers equation 67Intheneighbourhoodof thispointthefunctionFcanbeexpandedintoaseriesconnedtoquadratictermsF(, t, x) FK + F

K(t tK)2/2 , (3.35)where FK= F(, tK, x) , F

= x1u

0(tK) < 0. Then the integral(3.33)canberenderedasasumofcontributionsinthesaddle-pointsI=

IK, IK= f(tK)4[F

K[exp(FK2) . (3.36)As 0 one term in this sum will be prevailing, which corresponds tothe absolute maximum of function F. Therewith, the general solution(3.23)yieldstheasymptoticresultu(x, ) =t(x, ) x, (3.37)where t(x, ) is the coordinate of the absolute maximum of the func-tionF(, t, x) = S0(t) (t )22x, S0(t) =_tu0(t

) dt

. (3.38)Theabsolutemaximumseekingprocedurelends itself well tocleargraphic representation. It is evident that the coordinate t(x, ) is therst point in which a mobile straight line h descending from innity inparallel with the abscissa axis t touches the function F. It seems moreconvenient,however,toconsiderthersttouchingpointbetweenthefunctionS0(t)andtheparabola(, t, x) = h +(t )22x, (3.39)descending(ashdecreases)onthefunctionS0(t)(seeFigure3.1).68 Chapter 3o(t,t,x)t (t,x) tS (t)0`tFigure3.1: Graphical interpretationof theloweringof parabolaasamethodforndingtheabsolutemaximumofthefunction(3.38).Problem3.11Usingtheasymptoticsolutionof theBurgers equationobtainedinthepreviousproblem,analyzetheevolutionofaunipolarpulsebyapproximatingitwitha-functionattheinput: u0(t) = A(t).SolutionForthefunctionS0(t)denedbyformula(3.38)wehaveS0=A(t)where(t)istheHeavisidefunction. Agraphicprocedure for nding the absolute maximum coordinate is for this casedepictedinFigure3.2.Fixthedistancex, i.e. theparabolawidth(3.39). If>0, thentheparabolawill evidentlytouchthestepbyitscentret= , i. e.t(x, ) = ;inthiscase,accordingto(3.37)theeldu(x, ) 0forall >0. For 2or2x > 2themain term in the Fay solution becomes the rst one in the series (3.46)andthewavetakesontheformu =4e2xsin() = umax(x) sin(). (3.47)Formula(3.47)coincideswith(3.28)anddescribesthenonlinearsat-urationeect: nomatterhowlargethewaveamplitudeu0iswhenentering the nonlinear medium, at distances more than two lengths oflineardampingx>2/(2)=2xdthewaveamplitudecanneverexceeedumax=4e2x=2bc00exp(b22c300x). (3.48)Nonlinear waves in dissipative media and Burgers equation 73Problem3.16UndertheconditionsoftheProblemsa)2.7,andb)2.8;estimatethedissipativelengthxd=1/(2)=2c300/(b2)andndthemax-imal intensityof awave whichis transmittedover adistance 2xd.Assumeforwaterthat= 0.61017s2/cm.Answer a)xd 42m, Imax 104W/cm2, b)xd 1km,Imax 4106W/cm2,74 Chapter 4Chapter 4SPHERICALANDCYLINDRICALWAVESANDNONLINEARBEAMSProblem4.1Consider converging, spherically-symmetric waves in the linear ap-proximation. The reference shape of perturbation u0(t) is specied onthespherical surfaceof radius r0 (whereis acharacteristicwavelength). Usingthemethodof slowlyvaryingprole(Problem1.5),simplifythelinearwaveequationu 1c20

2ut2= 0, , u =2ur2+2r ur. (4.1)SolutionPassingtotheaccompanyingcoordinatesystem=t+(rr0)/c0, r1= r and neglecting the small term 2one obtains2ur+1ru+c0rur= 0 . (4.2)The ratioof the thirdtermtothe rst one inequation(4.2) is aquantity on the order of c0/(r0) /r. Consequently, the third termis small everywhereexcept intheimmediateneighbourhoodof thefocusr=0. Neglectingthethirdtermin(4.2)asimpliedequationisobtained:ur+ur= 0 . (4.3)Itsconvergingwavesolutionforrdecreasingfromr0to0isu(r, t) =1ru0(= t +r r0c0) . (4.4)Itincreasesinnitelyasthewaveconvergestothefocalpointr = 0.Spherical and cylindrical waves and nonlinear beams 75Problem4.2DerivethemodiedBurgersequation(see(3.8))usingageneral-isationof thesimpliedequation(4.3)byanalogywiththemethodusedinProblem3.2. Assumebothnonlinearanddissipativeeectsto be weak. As a result the wave prole is slowly distorted during thewavepropagation.Answerur+ur uu+ 2u2= 0 . (4.5)Here= /c20,= b/(2c300),asinProblem3.2.Problem4.3Transform the modied Burgers equation (4.5) for spherical wavesusingdimensionlessvariablesU= uu0rr0, = , = u0r0 ln(r0/r) . (4.6)Comparing the obtained result with (3.10), determine the formal anal-ogybetweenconvergenceandcoordinate-dependentdissipation.AnswerU= UU+ exp(/z0)2U2(4.7)Here=/(u0)istheinverseReynoldsnumber(see(3.11), z0=u0r0isthedimensionlessinitialradiusofthewavefront. Onecansee, that Burgers equationinthe form(4.7) reduces the problemfor the spherical wave inahomogeneous mediumtoaplane wavepropagating in a medium with dissipation decaying exponentially withdistance(increasingfrom0to ).76 Chapter 4Problem4.4DerivethemodiedBurgersequationforacylindricallyconverg-ingwave,byanalogywithProblems4.1and4.2.Answerur+u2r uu+ 2u2= 0 . (4.8)Hereweusethesamenotationasin(4.5).Problem4.5Transformequation(4.8)usingthechangeofvariablesU= uu0_ rr0, = , = 2u0r0(1 _r/r0) . (4.9)Showthemeaningoftheobtainedequation(likeinProblem4.3).AnswerU= UU+ (1 2z0)2U2. (4.10)Itisevidentthatequation(4.10)isequivalenttotheBurgersequa-tion for the plane waves in a medium where dissipative characteristicsdiminishaccordingtoalinearlawasrchangesfromr0to0(inthiscase increases from zero to 2z0where z0= u0r0is a dimensionlessreferenceradiusofthefront).Problem4.6Find the distance that must be covered by the reference harmonicsphericallysymmetricwaveinanondissipativemediumtoinduceadiscontinuityformationinthewaveprole. Considera)convergingandb)divergingwaves.SolutionInasmuch as for = 0 equation (4.7) coincides with anordinaryequationofsimplewavesthecoordinateofthediscontinuitySpherical and cylindrical waves and nonlinear beams 77formationrSintheinitialharmonicwavehastobederivedfromthecondition = u0r0[ ln(r0/rS)[ = 1 (see Problem 2.4). Here the caser0< rS< correspondstothedivergingwave,whereas0 < rS< r0isforthewaveconvergingtofocusr = 0. Thedistance [rSr0[thatistobecoveredbythewavetobecomediscontinuousamountstoa) [rSr0[ = r0[1 exp(1u0r0)] , (4.11)b) [rSr0[ = r0[exp(1u0r0) 1] , (4.12)Itisclearthatfortheconvergingwaves(4.11)theinequality [rS r0[ (u0)1, i.e. adivergingwavehastocoveralargerdistanceinordertoreachadiscontinuousstate. Thechangeofthenonlineardistortionbuild-uprateisrelatedtothefactthatinconverging spherical waves the amplitude increases as rfalls o (fromr0to0)whereasindivergingwavesitdecreasesas rgrowsfromr0to.Problem4.7Determine if a discontinuity can always be formed in a converginginitiallyharmonicwavepropagatinginanondissipativemedium.SolutionInacylindricallyconvergingwavetheconditionfordiscontinuityformation,accordingto(4.9)hastheform= 2u0r0(1 _r/r0) = 1 . (4.13)Since 0 < r < r0, the maximal value is attained for r = 0 amountingto 2u0r0. Providedthe parameters onthe radiatingcylindricalsurfacearechosensuchthat u0r0 1 is valid. This meansthatforz0>1doubleshockformationsareobserved. Aninitiallynarrowfront broadens duetodissipation. Its thickness attains themaximum value (2/)z0 exp(1/z01) at the point r = r0 exp(1/z01). Thenthenonlinearactionbecomesmorepronouncedagain, andthefrontthicknesstendstozeroasthewaveconvergesatitsfocus[10].Spherical and cylindrical waves and nonlinear beams 79Problem4.9Usingthequasioptical approximationof thetheoryof diractionandthe methodof slowlyvaryingprole (Problem1.5), deduce asimpliedequationforthebeaminthelinearapproximation.SolutionLet us proceed from the linear wave equation writteninCartesiancoordinates2ux2+2uy2+2uz2 1c202ut2= 0 . (4.17)Letthewavepropagatealongthebeamaxisx. Inthequasiopticalapproximation a harmonic signal is usually considered. In this case thewaveamplitudeisassumedtochangeslowlybothalongthex( x)axisandacrossthebeam(y, z)u = exp(it + ix/c0)A(x1= x, y1=y, z1=z) . (4.18)If broadband signals or nonlinear propagation - where the signal spec-trumisenrichedwithharmonics- areconsidered, thewavecannotbe treated as harmonic. Both its prole and spectrum are assumed tobeslowlychangingduringpropagationandformula(4.18)shouldbegeneralizedu = u(= t x/c0, x1= x, y1=y, z1=z) . (4.19)Substitute (4.19) into (4.17). The terms of the order of 0cancel eachother andthe2terms aredisregarded. As aresult all remainingtermsareof thesameinnitesimal order1. Thesetermsformthesimpliedequation2ux=c02 u, =2y2+2z2. (4.20)For harmonic signals u =Aexp(i), (4.20) yields the familiarparabolicequationdiractiontheory2iAx= A, = /c0. (4.21)80 Chapter 4Problem4.10Usingthemethodofthepreviousproblem, deducethesimpliedKhokhlov-Zabolotskayaequationfromthenonlinearwaveequationu 1c20

2ut2=

c30

2u2t2. (4.22)SolutionOntheassumptionof theslowchangeof thewaveproleandthebeamshape(4.19)oneobtains(ux

c20uu ) =c02 u. (4.23)ThisistheKhokhlov-Zabolotskayaequation. If thetransversecoor-dinatedependenceisneglected(u = 0)(4.23)willbetransformedintothesimplewaveequation(1.13). If nonlinearityisdisregarded( =0) (4.23) will be transformedintothe equationof the lineartheoryof diraction(4.20). Equation(4.23)describesthewaveac-countingsimultaneouslyfornonlinearanddiractioneects. Amorerigorousderivationof(4.23)fromtheequationsofhydrodynamicsisdiscussedin[4,7].Problem4.11Acting as in Problem 3.2, obtain the expression for the dimension-less complex of parameters, N, making it possible to assess the relativecontributionofnonlinearanddiractioneectstothedistortionofawave.SolutionLetasignalatinputx = 0bedescribedbythefunc-tionu(x = 0, t) = u0f(r/a)(t) . (4.24)Herer= y, zarethecoordinateswithrespecttothebeamcross-section,aischaracteristicbeamwidth;andu0andarethecharac-teristic amplitude and frequency. Taking (4.24) into account we switchSpherical and cylindrical waves and nonlinear beams 81tothedimensionlessvariablesofthetype(3.9)V= u/u0, = , z= u0x = x/xS,

R = r/a.(4.25)Equation(4.23)willbereducedtotheform(Vz VV ) =N4 V . (4.26)Here is the Laplace operator with respect to the normalized coor-dinates

R. The only parameter entering equation (4.26) is the numberN=2c302a2u0=122M(a)2. (4.27)OnecanwriteNasaratioofnonlinearanddiractionlengthsN= xS/xD=c30u0/a22c0. (4.28)Hence it follows that for N 1 nonlinearity prevails, whereas dirac-tiondominatesforN 1.Problem4.12Calculateinthelinearapproximationthevariationofcharacteris-ticsofthecircularharmonicGaussianbeamu(x = 0, r, t) = u0exp(r2/a2) sin(t) (4.29)duetodiraction.SolutionFor the beams with round cross-section, equation (4.20)willacquiretheform2ux =c02 (2ur2+1rur) . (4.30)Asolutionto(4.30)subjecttotheboundarycondition(4.29)canbeachieved via the method of separation of variables or by using integral82 Chapter 4transformations. Directsubstitutioncanbeusedaswell tocheckifthesolution(4.30)hastheform[6]u =u0_1 + x2/x2Dexp(r2a211 + x2/x2D) sin(+ arctan(x/xD)r2a2x/xD1 + x2/x2D, (4.31)wherexD=a2/(2c0)isacharacteristicdiractionlength. Theso-lution (4.31) describes the transformation of the reference plane waveintoasphericallydivergingone. Thebeamaxisamplitudefallsoasumax= u0 (1 +x2/x2D)1/2. (4.32)Forx xDtheamplitudedecreasesasumax u0xD/xfollowingthe x1lawofasphericallydivergingwave. Thebeamwidthgrowsasa(x) = a(1 + x2/x2D)1/2. (4.33)Forx xDanda(x) ax/xD,i.e. thewidthincreaseslinearlywithx,andthewholesoundeldislocalizedwithinaconewiththeapexangle 2a(x)/x 4c0/(a). Inadditionithastobenotedthatthewavephaseat thebeamaxis acquires ashift of arctan(x/xD).Thissuggeststhatthewavepropagationspeedatthebeamaxisissomewhathigherthanthatoftheplanewaveatthesamefrequency.Astheangularfrequency(4.29)grows, thediractionprocessattenuatesandallthementionedeventsshowupatlargerdistances.Problem4.13Making use of the solution (4.31) of the previous problem, demon-strate that a broadband signal (pulse) changes its shape in the far eldregion(x xD). Thediractionleadstoaproleshapedierentia-tiononthebeamaxis.SolutionAll of the reference signal harmonics are described byexpression(4.31),whichforx xD,r = 0takestheformu u0()xDxsin(+ /2) =a22c0xu0() cos() . (4.34)Spherical and cylindrical waves and nonlinear beams 83Thesignalformisdenedbyasumofallharmonics(4.34)u =a22c0x_u0() cos() d=a22c0x_u0() sin() d.(4.35)Thelastintegralisthereferenceformofthepulse_u0() sin() d= u(x = 0, ) . (4.36)Comparing(4.36)and(4.35)wendu(x xD, ) =a22c0x u(x = 0, ) , (4.37)i.e. thefarregionsignallendsitselftodierentiation.Problem4.14Show that the compression and rarefaction regions of the nonlineardiractingwavearedistortedtodierentextentsothatthereferenceharmonicsignal proleturnsnon-symmetricasitpropagates. Makeuseofthefactthatdierentharmonicsappearoutofphasewithre-specttoeachother.SolutionToobtainacomprehensivesolution,letusformulatethewaveproleapproximatelyasasumof therstandthesecondharmonicsonlyu = A1(x) sin[+ 1(x)] + A2(x) sin[2+ 2(x)] . (4.38)It is evident that thesecondharmonichas anamplitudeA2whichissmall comparedtoA1. Astheharmonicfrequencyishigher, thediraction phase shift 2 for this harmonic is less than 1 (see Problem4.12). Taking these factors into account the graphic summation of thetwosinusoids(4.38)willindeedyieldanonsymmetricprole(Figure4.1).Thecompressionregionisshortenedandmoresharp,whereastherarefactionregionis extendedandmoresmooth. Harmonics inter-feresuchthatthepositivepeakvalueoftheperturbationexceedsitsreference(x = 0)value.84 Chapter 412uutFigure4.1: Explanationof theasymmetryinthedistortionof half-periods of compression and rarefaction of an initially harmonic signal.Theasymmetryappearsbecauseofthedierentratesofdiractionofdierent harmonics. The resulting prole u is constructed as a sum oftwothincurvesfortherst(1)andsecond(2)harmonics.Problem4.15Employing the model equation (4.22) and the method of successiveapproximations, calculatetheamplitudeof thedierencefrequencywave=1 2excitedinanonlinear mediumunder theinter-actionoftwodampednondiractinghighfrequencywaveswithclosefrequencies1and2:u(1)= f(y, z)ex/xd[A1ei1(tx/c0)+ A2ei2(tx/c0)] +c.c. . (4.39)Here xdis a characteristic damping length for 1and 2, the functionf(y, z)describesthetransversestructureof thesewaves beam, andc.c.meanscomplexconjugatedterms.SolutionUsing(4.39) as therst approximation, obtaintheSpherical and cylindrical waves and nonlinear beams 85followingequationfrom(4.22)tondthesecondapproximationu(2)1c202u(2)t2=

c30

2u(1)2t2. (4.40)The right-hand side of (4.40) which describes nonlinear sources at thedierence frequency by taking (4.39) into account, will take the formq(x, y, z)eit+ c.c. ;q=2c302A1A2f2(y, z) exp[2xxd+ ic0x] + c.c. . (4.41)Seeking a solution of (4.40) in the form u(2)= A exp(it) we obtaintheinhomogeneousHelmholtzequationA + K2A= q , K= /c0, (4.42)forthedierencefrequencywaveamplitudeA. ThesolutionofthisequationiswrittenasA(

R) = 14_vq( R1)eiKrrdV1, r = [

R

R1[ . (4.43)Here

R= x, y, zistheradiusvectoroftheobservationpoint, and

R1= x1, y1, z1 is the point inside the volume V1occupied by the 1and2intersectionregion.Inthefardiractionregionwhere [

R[ [

R1[ itcanbeassumedthatr =_R22

R R1 + R21 R(1

RR1/R2) = R (xx1 + yy1 + zz1)/R. (4.44)Substitutingexpressions(4.41), (4.44)intotheintegral (4.43)were-ducethelattertotheformA 22c30A1A2eiKRRDtDl. (4.45)86 Chapter 4OfspecialinterestisthestructureofexpressionsDt, DldeningtheradiationdirectivityassociatedwiththetransverseDt=_ _f2(y, z)eiKR(yy1+zz1)dy1dz1, (4.46)andthelongitudinal(alongthexaxis)Dl=_0exp[2x1xd+ iK(1 x/R)x1] dx1. (4.47)distributionof theprimaryeldu(1)(4.39). Theintegral (4.46)ap-pears as an expansion into the angular spectrum of the function f2(y, z).Ithasthe sameformasthatobservedin thecasewhenthe wave isradiateddirectlybyahighfrequencywavewitha1and2source.Theintegral (4.47)describesthedirectivity, withthewavebeingexcitedbyspatiallydistributednonlinearsourcesDl= [2/xdiK(1 x/R)]1,[Dl[ =xd2[1 + (Kxd)2sin4(/2)]1/2. (4.48)Informula(4.48)therelationship1 x/R=1 cos =2 sin2(/2)wasused; istheanglebetweenthebeamaxisxandthedirectionwithrespecttotheobservationpoint.For K xd 1 (many dierence frequency wave lengths are presentalongtheinteractionregion)theradiationproceedsatsmall angleswithrespecttotheaxis. Thecharacteristicangularwidthofthedi-rectivitydiagramis,asfollowsfrom(4.48),equaltol (Kxd)1/2 (/xd)1/2. (4.49)Thewidth(4.49)denedbythelongitudinal distributionof theeldu(1)is,asarule,muchlessthanthewidthdenedbythetrans-versedistribution(4.46), whichiswhyformula(4.49)isresponsibleforthehighdirectivityofthelow-frequencysignal.Spherical and cylindrical waves and nonlinear beams 87Problem4.16Evaluatetheangularwidthof thelow-frequencyradiationdirec-tivitydenedbyformula(4.49)ofthepreviousproblem. A100kHzdierence frequency signal is excited by two pump waves with frequen-cies1MHzand1.1MHzinwater. TheabsorptioncoecientvalueisgiveninProblem3.3.Answer l 2102istheangularwidthapproachingonedegree.Problem4.17Calculatealongitudinalaperturefactor(4.47)fortheinteractionrangeof nondecaying(xd ) pumpwaves whichis, for x=l,boundedbyalterabsorbinghighfrequenciescompletelyandonlytransmittinglowfrequency.SolutionDl=_0exp[iK(1 x/R)x1] dx1=exp[iKl(1 x/R)] 1iK(1 x/R),[Dl[ = lsinc[Kl sin2(/2)]) , (4.50)Herewemadeuseof thefollowingnotation: sinc(x) sin(x)/xandtookintoconsiderationthat 1 x/R 2 sin2(/2). It has tobe notedthat the directivitydiagramdescribedby(4.50) containslateral lobes (of course, it assumedthat Kl1i.e. manywavelengths t inside the interaction length). The presence of lateral lobesisassociatedwiththeabrupt(downtozero)jumpoftheinteractingwave amplitudes for x = l. In the case when the interaction range wasboundedbyanexponentialdampinglawnolobeswereobserved(see(4.48)).88 Chapter 5Chapter 5HIGHINTENSITYACOUSTICNOISEProblem5.1Neglecting frequency uctuations, nd the probability distributionand the average for the plane quasi-monochromatic wave discontinuity(shock)formationlengthassumingtheprobabilitydistributionoftheamplitudeWa(a)isknown.SolutionThe results of Problem 2.4 imply that the discontinu-ityformationlengthxSforaplanemonochromaticwaveisequal toxS= c20/(a), where is frequency and a is the wave amplitude. Thesameformulacanbeappliedforaquasimonochromaticwaveaswell,when the amplitude and the frequency change insignicantly over onewave period. Therefore, the problem is reduced to that of a nonlineartransformationxS= f(a) . (5.1)Providedtheinversefunctiona = f1(xS)issingle-valued,theprob-abilitydistributionWx(xS)isrelatedwithWa(a)throughWx(xS) = Wa[f1(xS)][df1(xS)/dxS[ . (5.2)ForthemomentsofthequantityxSthefollowingexpressionisvalidxnS) =_fn(a)W(a) da. (5.3)FortheprobabilitydistributionoftheshockformationlengthanditsaveragewegetWx(xS) = Wa(c20xS)(c20x2S) (5.4)xS) =c20_0Wa(a)a1da. (5.5)High intensity acoustic noise 89Problem5.2Through the results of Problem 5.1, analyze the two dierent con-ditions:(a)theamplitudeofthesignalisdistributeduniformlyinsidethein-terval[a1, a2];(b)theinputsignalhasGaussianstatisticswithdispersion2. TakeintoaccounttheprobabilitydistributionoftheamplitudeofaGaus-siansignalWa(a) =a2 exp(a222) . (5.6)whichisalsoknownasRayleighdistribution.AnswerUsing (5.5) calculate probablility distributions and av-eragesofshockformationdistance:a)Wx(xS) =_c20(a2a1)x2S, xS [c20a2,c20a1],0 xS ,= [c20a2,c20a1](5.7)xS) =c20(a2a1) ln a2a1. (5.8)b)Ws(xS) =c40

222x3Sexpc402222x2S . (5.9)xS) =_2c20. (5.10)Problem5.3Find the probability distribution of the quasi-monochromatic wavediscontinuity(shock)amplitudeu = a sin(+) , (5.11)90 Chapter 5assuming the input signal to be Gaussian. The frequency uctuationsaretobedisregarded.SolutionThediscontinuityamplitudeuSisdenedfromtheequation(seeProblem2.13)arcsin(uSa) =

c20uSx. (5.12)Usingformula(5.2) andconsideringthat for a V(5.57)whereV= c20/(x).Problem5.19Usingthe limitingsolutionof the Burgers equationinthe caseofinnitelysmallviscosity(Problem3.10),showthatstationaryandcontinuous input noiseis transformedintoasequenceof saw-toothpulseswiththesameslopeatsucientlylargedistances. Findthevelocitiesofthediscontinuities.Solution Let the input noise have dispersion2u=u20())andbecharacterizedbythescale0. Thenthecharacteristiccurva-tureofthefunctionS0()enteringthesolution(3.37),(3.38)equalsS

0() u/0. Theparabolacurvatureinthesamesolutionis1/x. For ux/0 1 parabola (t, , x) will be a smooth function oftinthescaleS0(t).Becauseofthis,thetouchingpointsbetweenS0(t)and(t, , x)willbeclosetosomemaximaS0(t)(seeFigure5.2).102 Chapter 5uttS (t)q qq0k-1kk+1kk+1k-1Figure 5.2: Transformationof stationary noise into a sequence ofsawtooth-shapedpulseshavingstraight-lineconnections-all withthesameinclination.The eld u(x, ) is completely dened by a system of critical parabo-las, i.e. theparabolashavingdoubletouchingpointswithfunctionS0(t). Therewith,thecriticalparabolacentrecoordinates denethepositionof discontinuities Kandthe critical parabolaintersectionpoints(coincidingwithsomemaximaS0(t))denethezerosKontheeldu(x, ). Indeed, withintheinterval betweendiscontinuitiesKand K+1, the parabola touches the function S0(t) practically inthesamepointKwhichmeansthattheeldu(x, )hasauniversalHigh intensity acoustic noise 103structureintheintervalsbetweendiscontinuitiesu(x, ) = (K)/(x) , K< < K+1. (5.58)The discontinuitypositionis determinedfromthe double touchingcondition between and S0and for the discontinuity coordinate onecanwriteK=K + K12xS0(K) S0(K1)KK1. (5.59)Besides,thediscontinuitytravelspeedisconstantVK=dKdx= S0(K) S0(K1)KK1. (5.60)Therefore, theeldu(x, ) proleat this stageappears as aset ofinclinedlines withthe same slope 1/xemergingfromthe zeros =K. Theselinesareconnectedbyvertical lines-discontinuitieshaving coordinates K. The distance between individual neighbouringdiscontinuities K=K+1 Kcanbothincrease or decrease. IfKdecreasesthediscontinuitiesmergetoformasingleonewithanamplitudeequaltothesumofthemergeddiscontinuityamplitudes.Problem5.20Bysupposingthatarandomeldu(x, )ischaracterizedbytheonlyscale(x),estimatethegrowthofthisscaleduetodiscontinuitycoalescence.SolutionInthecaseof randomperturbationsu0(), thedis-continuity velocities are random as well. Because of this, collision andstickingtogetherof thediscontinuitieswill occurwhichgivesrisetothe increase of the eld (x) characteristic time scale. The estimate ofthe(x)growthcanbeobtainedthroughwritinganequationfortheaveragefrequencyof thediscontinuity(perunittime)n(x)whichisrelated to the external scale by means of n(x) = 1/(x). A decrease ofn(x) due to collisions is proportional both to the discontinuity number104 Chapter 5n(x)andtotheratioofthecharacteristicspeedofthediscontinuityapproach v= VK+1VKto the characteristic distance between themdndx= nv= n2v. (5.61)Thediscontinuityapproachspeedvcanbeassumedtobeontheorderofcharacteristicscatteringofthediscontinuityvelocity v2K).Useof theexpressionforthediscontinuityvelocity(5.60)yieldsthefollowingestimatev2K) v2K) 2(S0( + ) S0())2)2. (5.62)Or, with B0() = u0( +)u0()) and B0(0) = 20being the speciedinputsignalcorrelationfunction,thenv2) = 2n_1/n0(1 n)B0() d=_n200, D ,= 0,n22020, D = 0.(5.63)HereD=_0B0() disthevalueof theinitial perturbationspec-trumat zerofrequency. (For D ,=0thereferencecorrelationtime0isdeterminedfromtheconditionD=200andforD=0fromthe condition_0B0() d= 2020). Substituting (5.63) into (5.61)yieldsthefollowingestimatesfortheexternalscalegrowth(x) _0(x/xS)2/3, D ,= 0,0(x/xS)1/2, D = 0,(5.64)wherexS= 0/0isthecharacteristiclengthofthenonlineareect.Problem5.21Assuming that statistical properties of the intense noise tend to beself-similar, write an expression for the wave power spectrum (a) and,byusing(5.64), evaluatetheeldenergyat thestageof developeddiscontinuities(b).High intensity acoustic noise 105Answer a)g(x, )=3(x)2x2 g((x)), where g()isauniversaldimensionlessfunction.b)