Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics

476 Pages·2012·8.41 MB·English

by  S. N. Gurbatov

NONLINEAR PHYSICAL SCIENCE
NONLINEAR PHYSICAL SCIENCE
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S.N. Gurbatov
O.V. Rudenko
A.I. Saichev
Waves and Structures in
Nonlinear Nondispersive
Media
General Theory and Applications to Nonlinear
Acoustics
With 181 figures
Authors
S.N.Gurbatov O.V.Rudenko
RadiophysicsDepartment PhysicsDepartment
NizhnyNovgorodStateUniversity MoscowStateUniversity
603950NizhnyNovgorod,Russia 119991Moscow,Russia
Email:[email protected] Email:[email protected]
A.I.Saichev
RadiophysicsDepartment
NizhnyNovgorodStateUniversity
603950NizhnyNovgorod,Russia
Email:[email protected]
ISSN1867-8440 e-ISSN1867-8459
NonlinearPhysicalScience
ISBN978-7-04-031695-7
HigherEducationPress,Beijing
ISBN978-3-642-23616-7 e-ISBN978-3-642-23617-4
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011935427
(cid:2)c HigherEducationPress,BeijingandSpringer-VerlagBerlinHeidelberg2011
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Preface
Thebookisaimedatnaturalscienceundergraduates,aswellasatgraduateandpost-
graduatestudentsstudyingthetheoryofnonlinearwavesofvariousphysicalnature.
It may also beuseful as a handbookfor engineersandresearcherswhoencounter
thenecessityoftakingnonlinearwaveeffectsintoaccountintheirwork.
Evolutionofsufficientlyintensewavesisdeterminedbynonlinearprocesses,in
which the progress is substantially influencedby dispersion (a dependenceof the
phase velocity on its frequency). Media without dispersion, where the phase ve-
locitydoesnotdependonthefrequency,arethesimplestoneswithrespecttotheir
physicalpropertiesandarethemostcommoninnature.Butnonlinearinteractionsof
theFourierspectralcomponentsinsuchmediaareparticularlycomplexanddiverse.
Here,practicallyall“virtual”energy-exchangeprocessesbetweenwavesofdifferent
frequenciesbecomeresonantonesandoccurwithahighefficiency.Anavalanche-
likeincreaseofthenumberofspectralcomponentsofthefieldtakesplace,which,
within the space-time representation, corresponds to formation of structures with
stronglypronouncednonlinearproperties.Examplesofsuchstructuresarediscon-
tinuitiesofafunctiondescribingthewavefieldordiscontinuitiesofits derivative,
steepshockfrontsofvarioustypesandmultidimensionalcellularstructures.
Nonlinearstructurescanbestableonlyinstrongfields,undertheconditionsof
competitionwitheffectsofabsorption,dispersion,etc,whichcontributetothedecay
of such structures. These objects have properties of quasiparticles. For instance,
shockfrontsundergoinelasticcollisions.Thus,innondispersivemedia,nonlinearity
providesbotha possibilityofinteractionsbetweenstablestructuresandtheirvery
existence. Solitons are other well-known objects in nonlinear physics, which are,
generallyspeaking,stableonlyinidealizedconservativesystems.Atthesametime,
quasistabilityofshock-frontstructuresorsawtoothwavesoccursinrealdissipative
systems.
Structures of different physical nature are described by similar mathematical
models. These models are used not only in the wave theory, but also to describe
variousnon-waveobjects,viz.:forest-firefronts,densityofaflowofnon-interacting
particles,etc.Becauseoftheuniversalityofsuchnonlinearmodels,itisnecessaryto
vi Preface
analyzethemonthebasisofgeneralprinciplesofmathematicalphysics,irrespective
ofthenatureofthedescribedphenomena.
Ontheotherhand,nondispersivewavesandstructuresarewidelyusedinscience
andtechnology.Areviewoftheseapplications,fromtheauthors’viewpoint,iswhat
“brightensup”thetheoryandmaybeofinteresttomanyreaders.
Thetheoryofnonlinearwavesandstructuresisaveryextensiveandconstantde-
velopingfieldofphysics(especiallyradiophysicsandmathematicalphysics).Ithas
manyspecificapplications.Amongthemthereareboththewell-knownproblemsof
acoustics,electrodynamicsandplasmaphysics(see,e.g.,[1–5]),andtheless-known
problems,such as surface-growthdescription[6,7], dynamicsof turbulence[8,9]
anddevelopmentofagravitationalinstabilityofthelarge-scaledistributionofmat-
terintheUniverse[10–14].Awiderangeofphenomenaarisingherehaveledtothe
developmentofavarietyofmathematicalmethods,whichareeffectiveinaddress-
ingvariouskindsofnonlinearfieldsandwaves(see,e.g.,[15–17]).Itis clearthat
within a single monograph,it is not possible to give an exhaustivelycomprehen-
siveoverviewofthewholeproblem.Forthisreason,theauthorslimitedthemselves
to a discussion of the “hydrodynamic” type of nonlinear waves in nondispersive
medium. First of all, the properties of solutions to such standard nonlinear wave
equationsinnondispersivemediaasthesimplewaveequation,theBurgersequation
and the Kardar-Parisi-Zhangequation have been studied in detail. Apart fromthe
importanceoftheseequationsforthetheoryandapplications,ananalysisofthese
solutions allows us to trace stages of development of typical nonlinear processes
and,aboveall, nonlineardistortionofprofiles,the gradientcatastropheandemer-
genceofshockwaves.Inorderforthetheoryofnonlinearwavesinnondispersive
media not to look too abstract, the presentation is based on illustrative geometric
interpretationsofboththeequationsthemselvesandtheirsolutions,aswellasona
comprehensivediscussionofthephysicalmeaningofthesesolutionsandthemeth-
odsusedtoobtainthem.
Themonographconsistsoftwoparts.Thefirstpartisdevotedtoadetailedde-
scriptionoftheconceptsandanalysismethodsofnonlinearwavesandstructuresin
nondispersivemedia.Thesecondpartfocusesonanin-depthdescriptionofthenon-
lineartheoryasappliedonlytoonetypeofwaves—highintensityacousticwaves.
Thisobject,ontheonehand,isthemoststraightforwardand,ontheotherhand,has
importantpracticalapplications.
Theauthorshaveattemptedtocommunicateallmateriallsatthefollowing“two
levels” of complexity. The first level is intended to introduce beginning investi-
gators (above all undergraduate, graduate and PhD students) to the concepts and
methods of the theory of nonlinear waves and structures in nondispersive media.
Inordertoachieveadeeperunderstandingofthefoundations,itisusefultosolve
the problems given in the end of the chapters in Part I. The second, higher, level
is meant for researchers, who already have experience in this field of study and
are interested in the state of the art or in specific results. Naturally, it is impossi-
ble to reflect the entire diversity of approachesused to study nonlinearfields and
waves in a single monograph. This is why the material is presented at a simple,
“physical”levelofrigor,wherepossible.Those,whoareinterestedinamorerigor-
References vii
ousmathematicalfoundationoftheproblemsdiscussedhere,areadvisedtoturnto
monographs[15,17],wheremathematicalfoundationsofmanytopicstouchedupon
in this bookare thoroughlydiscussed.An in-depthreviewofthemethodsusedto
solvenonlinearproblems,alongwithprofoundresultsofthenonlinearfieldtheory,
canbefoundinbook[16].Inmonograph[18],andalsointextbook[19],thetheory
ofgeneralizedfunctionsnecessaryforconstructionofgeneralizedsolutionsofnon-
linear equations is comprehensivelyelucidated. We recommendthose who intend
deepertodelveintothenonlinearfieldtheory,withoutburyingthemselvesinmath-
ematicalsubtleties,thefollowingthoroughmonographsandtextbooks:[1,2,4,5],
whicharewrittenbyphysicistsforphysicists.Basicconceptsofthenonlinearwave
theory, along with illustrative physical examples, can be found in the remarkable
textbook[14].To those who aregoingprofessionallyto engagethemselvesin the
fieldofnonlinearacoustics,werecommendmonograph[3]andthebooksofprob-
lems[20,21],whereasetofproblemsaidinginmasteringvariousaspectsofnon-
linear acoustics is given. If one is interested in statistical properties of nonlinear
randomwavesasappliedtononlinearacoustics,astrophysicsandturbulence,heor
shecanpickupnecessaryinformationfrommonograph[10].Wealsoadvisetoturn
tomonograph[8],whichcoversthefoundationsofthetheoryofstrongturbulence
anditsinherentphenomena,suchasintermittencyandmultifractality.
Wearegratefultotherenownedscientists,fruitfulinteractionswithwhomover
the years have formed our vision of the problems and methods of the nonlinear
science. First of all, they are: academicians A.V. Gaponov-Grekhov, Ya.B. Zel-
dovich, R.V. Khokhlov, V.I. Arnold and Ya.G. Sinai; corresponding members of
the Russian Academy of Sciences M.I. Rabinovich and D.I. Trubetskov; Profes-
sors A.N.Malakhov,L.A.Ostrovsky,S.A.Rybak,S.I.Soluyan,A.P. Sukhorukov,
A.S. Chirkin and S.F. Shandarin.We are delighted to rememberthe years of col-
laborationwithinternationalcolleagues,amongwhomare:D.Crighton,U.Frisch,
B.Enflo,D.Blackstock,M.Hamilton,L.Cram,E.Aurell,A.Noullez,W.A.Woy-
czynskiandmanyothers.
Wewouldlikealsotothankourtranslators,O.SimdyankinaandS.Simdyankin,
notonlyforthespeedyproductionofanEnglishtranslationofthisbook,butalso
forthelucidclarityoftheirliteraryrepresentationoftheoriginaltext.
NizhnyNovgorod,Moscow, SergeyN.Gurbatov
July2011 OlegV.Rudenko
AlexanderI.Saichev
References
1. M.J.Lighthill,WavesinFluids,2ndedn.(CambridgeUniversityPress,2002)
2. M.I.Rabinovich,D.I.Trubetskov,OscillationsandWavesinLinearandNonlinearSystems.
(Springer,Berlin,1989)
3. O.V.Rudenko,S.I.Soluyan,TheoreticalFoundationsofNonlinearAcoustics(Plenum,New
York,1977)
viii Preface
4. M.B. Vinogradova, O.V. Rudenko, A.P. Sukhorukov, Theory of Waves (Nauka, Moscow,
1979). InRussian
5. G.B.Whitham,LinearandNonlinearWaves(Wiley,NewYork,1974)
6. A.L. Baraba´si, H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University
Press,1995)
7. M.Kardar,G.Parisi,Y.C.Zhang,Dynamicalscalingofgrowinginterfaces,Phys.Rev.Lett.
56,889–892(1986)
8. U.Frisch,Turbulence:TheLegacyofA.N.Kolmogorov(CambridgeUniversityPress,1995)
9. A.S.Monin,A.M.Yaglom,StatisticalFluidMechanics,vol.2(MITPress,Cambridge,Mass,
1975)
10. S.N.Gurbatov, A.N.Malakhov, A.I.Saichev, Nonlinear Random Waves and Turbulence in
NondispersiveMedia:Waves,RaysandParticles.(ManchesterUniversityPress,1991)
11. P.J.E.Peebles,Large-ScaleStructureoftheUniverse(PrincetonUniversityPress,1980)
12. S.F.Shandarin,Y.B.Zeldovich,Thelarge-scalestructureoftheuniverse:turbulence,intermit-
tency,structuresinaself-gravitatingmedium,Rev.Mod.Phys.61,185–220(1989)
13. S.Weinberg,GravitationandCosmology(Wiley,NewYork,1972)
14. Y.B.Zeldovich,ElementsofMathematicalPhysics(Nauka,Moscow,1973). InRussian
15. V.I.Arnold,OrdinaryDifferentialEquations(MITPress,Cambridge,Mass,1978)
16. R.Richtmyer,PrinciplesofAdvancedMathematicalPhysics,vol.1(Springer,Berlin,1978)
17. B.L. Rozhdestvenskii, N.N. Yanenko, Systems of Quasilinear Equations (Nauka, Moscow,
1978). InRussian
18. A.I. Saichev, W.A. Woyczyn´ski, Distributions in the Physical and Engineering Sciences
(Birkha¨user,Boston,1997)
19. S.A.Lapinova,A.I.Saichev,V.A.Filimonov,GeneralizedFunctionsandAsymptoticMethods
(NizhnyNovgorodUniversityPress,2006). InRussian
20. S.N.Gurbatov, O.V.Rudenko (eds.), Acoustics inProblems(Fizmatlit,Moscow, 2009). In
Russian
21. O.V. Rudenko, S.N. Gurbatov, C.M. Hedberg, Nonlinear Acoustics through Problems and
Examples(Trafford,2010)
Contents
PartI FoundationsoftheTheoryofWavesinNondispersiveMedia...... 1
1 NonlinearEquationsoftheFirstOrder........................... 3
1.1 Simplewaveequation....................................... 3
1.1.1 Thecanonicalformoftheequation ..................... 3
1.1.2 Particleflow ........................................ 4
1.1.3 DiscussionoftheRiemannsolution..................... 5
1.1.4 Compressionsandexpansionsoftheparticleflow ......... 6
1.1.5 Continuityequation .................................. 8
1.1.6 Constructionofthedensityfield........................ 9
1.1.7 Momentum-conservationlaw .......................... 10
1.1.8 Fouriertransformsofdensityandvelocity ............... 11
1.2 Line-growthequation ....................................... 13
1.2.1 Forest-firepropagation................................ 13
1.2.2 Anisotropicsurfacegrowth............................ 16
1.2.3 Solutionofthesurface-growthequation ................. 18
1.3 One-dimensionallawsofgravitation........................... 20
1.3.1 Lagrangiandescriptionofone-dimensionalgravitation..... 20
1.3.2 Euleriandescriptionofone-dimensionalgravitation ....... 22
1.3.3 Collapseofaone-dimensionalUniverse ................. 24
1.4 ProblemstoChapter1....................................... 25
References..................................................... 36
2 GeneralizedSolutionsofNonlinearEquations..................... 39
2.1 Standardequations ......................................... 39
2.1.1 Particle-flowequations................................ 40
2.1.2 Linegrowthinthesmallangleapproximation ............ 40
2.1.3 Nonlinearacousticsequation .......................... 41
2.2 Multistreamsolutions ....................................... 42
2.2.1 Intervalofsingle-streammotion........................ 42
x Contents
2.2.2 Appearanceofmultistreamness ........................ 42
2.2.3 Gradientcatastrophe ................................. 44
2.3 Sumofstreams ............................................ 46
2.3.1 Totalparticleflow.................................... 46
2.3.2 SummationofstreamsbyinverseFouriertransform ....... 47
2.3.3 Algebraicsumofthevelocityfield...................... 47
2.3.4 Densityofa“warm”particleflow ...................... 48
2.4 Weaksolutionsofnonlinearequationsofthefirstorder........... 50
2.4.1 Forestfire .......................................... 50
2.4.2 TheLax-Oleinikabsoluteminimumprinciple ............ 52
2.4.3 Geometricconstructionofweaksolutions................ 53
2.4.4 Convexhull......................................... 54
2.4.5 Maxwell’srule...................................... 56
2.5 TheE-Rykov-Sinaiglobalprinciple ........................... 59
2.5.1 Flowofinelasticallycoalescingparticles................. 59
2.5.2 Inelasticcollisionsofparticles ......................... 60
2.5.3 Formulationoftheglobalprinciple ..................... 61
2.5.4 Mechanicalmeaningoftheglobalprinciple .............. 62
2.5.5 Conditionofphysicalrealizability ...................... 63
2.5.6 Geometryoftheglobalprinciple ....................... 66
2.5.7 Solutionsofthecontinuityequation..................... 69
2.6 Line-growthgeometry ...................................... 70
2.6.1 Parametricequationsofaline.......................... 71
2.6.2 Contourinpolarcoordinates........................... 72
2.6.3 Contourenvelopes ................................... 74
2.7 ProblemstoChapter2....................................... 77
References..................................................... 82
3 NonlinearEquationsoftheSecondOrder......................... 83
3.1 Regularizationofnonlinearequations.......................... 83
3.1.1 TheKardar-Parisi-Zhangequation...................... 84
3.1.2 TheBurgersequation................................. 85
3.2 PropertiesoftheBurgersequation ............................ 86
3.2.1 Galileaninvariance................................... 86
3.2.2 Reynoldsnumber .................................... 87
3.2.3 Hubbleexpansion.................................... 89
3.2.4 Stationarywave ..................................... 92
3.2.5 Khokhlov’ssolution.................................. 93
3.2.6 Rudenko’ssolution................................... 95
3.3 GeneralsolutionoftheBurgersequation .......................100
3.3.1 TheHopf-Colesubstitution............................101
3.3.2 GeneralsolutionoftheBurgersequation.................102
3.3.3 AveragedLagrangiancoordinate .......................103
3.3.4 SolutionoftheBurgersequationwithvanishing
viscosity............................................104