Table Of ContentLecture 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
