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Preface
7
veryslenderregion,andthenumericalsolutionshouldtakeintoaccount
allthesephenomena.Forthisreasontheasymptoticapproachseems
tobepromising.Elasticwavemeasurementinvolvesusingvariouswave
sources(steelballimpact,pencilbreak,laserheatingetc.)andreceivers
(piezo-ceramicssensors,laserinterferometryetc.).
Theinversescatteringproblemistoreconstructtheshapeorthe
Lameconstantsoftheunknownscattererfromthescatteringdatamea-
suredoutside.However,therearetwomajortheoreticaldifficultiesin
solvingthisproblem:uniquenessandstability(i.e.,illposedness).Asthe
measureddataisnotsufficientoriscontaminatedbynoise,reconstruc-
tionisquitedifficult.Therefore,morepriorknowledgeabouttheobject
ishelpfulforreconstruction.
Inthework,thedirectandinverseelastodynamicscatteringproblems
forthininhomogeneitieswithinaninfiniteisotropicandhomogeneous
elasticmediumareconsidered,bymeansofelasticwavesfields,assum-
ingthatthegeneraltopologyofscatteringobject,typeofboundaryor
contactconditionsandsoundingwavesareknown.Theamplitude-phase
characteristics(diagrams)ofscatteringorenergeticcharacteristics-the
scatteringcross-sectionsaretakenasinputdata,andtheplanewaves
oflongitudinaltypearetakenassoundingwaves.Theproblembeing
considered,istypicalinmanybranchesofknowledge,whenphysicsof
processisknown,anditisnecessarytodeterminethegeometryand
materialstructureofascatterer.
Effectivenessoftheapproachestotheinverseproblemsresearchof-
feredintheworkisbasedontheuseofalong-andshort-waveranges
ofsoundingfrequencies.Applicationofsuchrangesgivesapossibility,in
anumberofcases,todescribeanalyticallythescatteringcharacteristics
inawavezone,andalsoallowstodisplayandtointerpretthephysicsof
diffractionphenomena,thatinmanycasesprovidesasuccessfulstudyof
theinverseproblems.Atthat,theknownconclusionisconfirmed,that
thelocalscattererformidentificationproblemcanbereliablysolvedonly
byusingofsoundingfieldfrequencyranges.
Asitwasalreadyremarked,atypicalpeculiarityoftheinverseprob-
lemsistheirillposednessintheHadamardsenseinmostnatural(from
thepointofviewoftheapplications)functionalspaces.Thefactisthat
experimentallyobservedwavefields,scatteredbyobjectswithdifferent
materialorgeometricalproperties,byreasonofintegralcharacterof
thesefieldsinfarzone,canbesimilar.Aquestionaboutuniquenessofa
solutionoftheinverseproblemarises.Fromtheotherside,becausethe
experimentisperformedwithsomeerror,alsoaquestionarisesabout
