Waves in Continuous Media PDF
Preview Waves in Continuous Media
Lecture Notes in Geosystems Mathematics and Computing S.L. Gavrilyuk N.I. Makarenko S.V. Sukhinin Waves in Continuous Media Lecture Notes in Geosystems Mathematics and Computing Serieseditors W.Freeden,Kaiserslautern Z.Nashed,Orlando O.Scherzer,Vienna Moreinformationaboutthisseriesathttp://www.springer.com/series/15481 S.L. Gavrilyuk • N.I. Makarenko (cid:129) S.V. Sukhinin Waves in Continuous Media S.L.Gavrilyuk N.I.Makarenko Aix-MarseilleUniversity RussianAcademyofSciences Marseille,France LavrentyevInstituteofHydrodynamics NovosibirskStateUniversity Novosibirsk,Russia S.V.Sukhinin RussianAcademyofSciences LavrentyevInstituteofHydrodynamics NovosibirskStateUniversity Novosibirsk,Russia LectureNotesinGeosystemsMathematicsandComputing ISBN978-3-319-49276-6 ISBN978-3-319-49277-3 (eBook) DOI10.1007/978-3-319-49277-3 LibraryofCongressControlNumber:2017930812 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Wave phenomena occur everywhere in nature and therefore are studied in many areasofscienceforalongtime. Themathematicalwavetheoryemergedasanindependentdisciplineinthemid- 1970sduetonumerousapplicationsinnaturalscienceandengineeringstimulating thefurtherdevelopmentofmathematicalmethods. The lecture course Waves in Continuous Media is one of the disciplines on continuum mechanics and mathematical modeling included into the education program at the Department of Mechanics and Mathematics, Novosibirsk State University.This course was first given by Professor L. V. Ovsyannikov,1 a distin- guishedscientistwhoobtainedanumberoffundamentalresultsinthefieldofwave hydrodynamics.Based on Ovsyannikov’s principles of selecting the material, the authorsdevelopednewvariantsofthecourseadaptedtogroupsofmaster’sstudents specializedinappliedmathematics,mechanics,andgeophysics. The textbook contains a rich collection of exercises and problems which have been carefully selected and tested at practical works and seminars of courses given by the authors at Novosibirsk State University (Russia) and Aix-Marseille University(France)formanyyears.Mostoftheproblemsandexercisesaresupplied withanswersandhints.Solutionsofsometypicalproblemsareexplainedindetail, and some theoretical background material is included in order to make the book self-containedand give students the necessary tools for self-education.More than 200problemsformulatedinthebookallowedustoproposetoeachmaster’sstudent anindividualsemestermini-projectconsistinginsolvinguptosixproblems.Most ofthemaresolvedbyapplyingthetheoreticalapproachesfromthecourse,butthe otheronesdemandadeeperunderstandingofthemethodsdiscussedinthecourse. Duringthesemester,the studentshavealsobeenworkingin researchlaboratories, so a set of problems specific to the research activity of the students was usually proposed. 1Ovsyannikov,L.V.:WaveMotionsofContinuousMedia.NovosibirskStateUniversity,Novosi- birsk(1985)[inRussian]. v vi Preface The textbook consists of three chapters. Chapters 1 and 2 present the basic notions and facts of the mathematical theory of waves illustrated by numerous examples and methods of solving typical problems. The reader learns how to recognizethe hyperbolicityproperty;find characteristics, Riemanninvariants,and conservationlawsforquasilinearsystemsofequations;constructandanalyzesolu- tionswithweakorstrongdiscontinuities;andinvestigateequationswithdispersion: analysis of dispersion relations, the study of large time asymptotic behavior of solutions, the construction of traveling wave solutions for models reducible to nonlinearevolutionequations,etc.Themajorityofproblemsareformulatedwithin the framework of wave models arising in gas dynamics, magnetohydrodynamics, elasticity and plasticity, linear and nonlinear acoustics, chemical adsorption, and otherapplications. Chapter3dealswithsurfaceandinternalwavesinanincompressiblefluid.The efficiencyofmathematicalmethodsisdemonstratedonahierarchyofapproximate submodelsgeneratedfromtheEulerequationsofhomogeneousandinhomogeneous fluids.Someproblemsillustratetheinfluenceofviscosityandvorticityonthewave processes. The list of references consists mainly of monographs and textbooks recom- mendedfor further reading.Three of them are generic [1–3], while others [4–33] are more specific for each chapter. These have been selected to allow readers to understand better mathematical statements whose proofs were skipped, and find solutions of relatively hard exercises. A separate bibliographyfor each chapter is maintained.ThereferencelistforChap.3alsocontainsfiveresearcharticlesonthe theory of water waves [16, 17, 21, 26, 31] we explicitly refer to. The books for furtherreadingarenotcitedinthetext. TheauthorsthanktheircolleaguesattheChairofHydrodynamics,Novosibirsk StateUniversity,fortheirhelpinthepreparationofthemanuscript. The authors would like to express their special gratitude to Professor V. M. Teshukov, who recently passed away, for his numerous useful advices and discussions. Marseille,France S.L.Gavrilyuk Novosibirsk,Russia N.I.Makarenko Novosibirsk,Russia S.V.Sukhinin October2016 Contents 1 HyperbolicWaves............................................................ 1 1.1 HyperbolicSystems ................................................... 1 1.2 PropagationofWeakDiscontinuities................................. 4 1.3 MotionwithStrongDiscontinuities .................................. 7 1.4 KinematicWaves ...................................................... 10 1.5 Multi-dimensionalWaveFronts....................................... 13 1.6 SymmetrizationofHyperbolicSystems ofConservationLaws ................................................. 16 1.7 Problems ............................................................... 19 2 DispersiveWaves............................................................. 43 2.1 DispersionRelation.................................................... 43 2.2 Multi-dimensionalWavePackets ..................................... 46 2.3 GroupVelocity......................................................... 49 2.4 StationaryPhaseMethod.............................................. 51 2.5 NonlinearDispersion.................................................. 56 2.6 Problems ............................................................... 60 3 WaterWaves.................................................................. 77 3.1 EquationsofMotion................................................... 77 3.2 LinearTheoryofSurfaceWaves...................................... 82 3.3 ShallowWaterTheory................................................. 85 3.4 ShearFlowsofShallowWater........................................ 88 3.5 NonlinearDispersiveEquations ...................................... 90 3.6 StationarySurfaceWaves ............................................. 94 3.7 WavesinTwo-LayerFluids........................................... 97 3.8 WavesinStratifiedFluids ............................................. 103 3.9 StabilityofStratifiedFlows........................................... 107 vii viii Contents 3.10 StationaryInternalWaves............................................. 109 3.11 Problems ............................................................... 112 References......................................................................... 137 Index............................................................................... 139 Chapter 1 Hyperbolic Waves 1.1 HyperbolicSystems Weconsiderthequasilinearsystemoffirstorderequations u CA.u;x;t/u Cb.u;x;t/D0; (1.1) t x where the n(cid:2)n-matrix A and vector b depend on x, t and u D .u1;:::;un/T. A directiondx=dt D c is called characteristic if there exists a linearcombinationof equations of the form (1.1) such that each unknown function u is differentiable i alongthisdirection.Thequantityc inthedefinitionofa characteristicdirectionis aneigenvalueofthematrixA,i.e., det.A(cid:3)cI/D0: (1.2) Foranyeigenvaluec andthecorrespondinglefteigenvectorl D .l1;:::;ln/ofthe matrix A (i.e., lA D cl) the system (1.1) implies the following condition on the characteristic(thecurvecorrespondingtoacharacteristicdirection): l(cid:4).duCb/D0; (1.3) t whered D@ Cc@ istheoperatorofdifferentiationalongthecharacteristic. t t x Thesystem(1.1)ishyperbolicifalleigenvaluesc ofthematrixAarereal(inthis i case,theycanbeordered:c1 6c2 6:::6cn)andthereexistnlinearlyindependent reallefteigenvectorsofthematrixA. Figure 1.1 shows the location of characteristics emanating from a given point M in the .x;t/-plane. A hyperbolic system of equations is equivalent to a system of n relations on characteristics. A system is hyperbolicif and only if the normal Jordan form of the matrix A is diagonal. We indicate the sufficient hyperbolicity ©SpringerInternationalPublishingAG2017 1 S.L.Gavrilyuketal.,WavesinContinuousMedia,LectureNotesinGeosystems MathematicsandComputing,DOI10.1007/978-3-319-49277-3_1
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