Equations of Mathematical Diffraction Theory

291 Pages·2005·1.762 MB·English

by  Mezhlum A. Sumbatyan, Antonio Scalia

EQUATIONS OF
MATHEMATICAL
DIFFRACTION THEORY
© 2005 by CRC Press LLC
Differential and Integral Equations and Their Applications
A series edited by:
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Volume 1
Handbook of First Order Partial Differential Equations
A.D. Polyanin, V.F. Zaitsev and A. Moussiaux
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Group-Theoretic Methods in Mechanics and Applied Mathematics
D.M. Klimov and V. Ph. Zhuravlev
Volume 3
Quantization Methods in the Theory of Differential Equations
V.E. Nazaikinskii, B.-W. Schulze and B. Yu. Sternin
Volume 4
Hypersingular Integral Equations and Their Applications
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Volume 5
Equations of Mathematical Diffraction Theory
Mezhlum A. Sumbatyan and Antonio Scalia
© 2005 by CRC Press LLC
EQUATIONS OF
MATHEMATICAL
DIFFRACTION THEORY
Mezhlum A. Sumbatyan
Antonio Scalia
CHAPMAN & HALL/CRC
A CRC Press Company
Boca Raton London New York Washington, D.C.
© 2005 by CRC Press LLC
TF1668 disclaimer.fm Page 1 Thursday, July 15, 2004 2:40 PM
Library of Congress Cataloging-in-Publication Data
Sumbatyan, Mezhlum A.
Equations of mathematical diffraction theory / Mezhlum A. Sumbatyan, Antonio Scalia.
p. cm.
Includes bibliographical references and index.
ISBN 0-415-30849-6 (alk. paper)
1. Diffraction—Mathematics. I. Scalia, A. II. Title.
QC415.S95 2004
535'.42'0151—dc22 2004051957
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© 2005 by CRC Press LLC
PREFACE
The connection between heuristic and strictly formal methods is seemingly one of
the most interesting and debatable questions in modern mathematics. Each of these two
differentapproaches,whosefoundationswerelaidbySocratesandAristotle,respectively,
andinthenewhistoryarere(cid:3)ectedindiscussionsandwrittenpapersbyDescartes,Leibnitz
and Bacon, has its own intrinsic merits and restrictions. Moreover, a large number of
discoveriesin science were made owing to a combination of strict and heuristic methods
ofinvestigation.
Apparently,oneofthebrightestexamplesinmodernmathematicalphysicsisdiffraction
theory, where the combination of the two approaches would lead so ef(cid:2)ciently to such
impressive results. Many important and interesting solutions and even some classical
theories appeared from heuristic ideas, and the most impressive example was given by
Kirchhoff’s physical diffraction theory, which is based upon a clear (cid:147)light and shadow(cid:148)
concept for diffracted wave (cid:2)elds. Subsequently, many of these heuristic results were
rigorouslysubstantiatedandprovedastheorems. Ontheotherhand,unsuccessfulattempts
to prove some other heuristic ideas caused signi(cid:2)cant progress in the development of
formal methods that yielded correct solutions, different sometimes from those prompted
bysomeone’sintuition.
The above speci(cid:2)c features have affected the style of presentation of the book. Each
section deals with a discussion of heuristic ideas, which as a rule are substantiated (or
disproved)with the use of rigorous mathematical methods. Due to limited volume of the
book,atsomeplaceswegiveonlyabriefsketchofthesubstantiation,referringthereader
totheoriginalliteratureformoredetails.
Onemorespeci(cid:2)cfeatureofthepresentedmaterialisconnectedwiththerapidprogress
in computertechnologyover the last 20 years, whichhas signi(cid:2)cantlychangedour view›
point on what could be accepted as ef(cid:2)cient methods of investigation. Only recently,
expansionofunknownfunctionsintoseriesintermsofspecialfunctions,whenaproblem
reduced to in(cid:2)nite system of linear algebraic equationswith respect to coef(cid:2)cientsof the
expansion, was regarded as a standard method. Such a (cid:147)semi›analytical(cid:148) approach was
ef(cid:2)cient15(cid:150)20yearsago,whentheevaluationofregularityoftheobtainedin(cid:2)nitesystems
seemedtobeveryimportant,sincethiscouldguaranteeaccuracyofasolutionbyretaining
onlyfew(cid:2)rstequations,whichwasacceptablefor(cid:2)rst›generationscomputers. Nowadays,
whenthereisnotmuchdifferencebetween10·10and500·500systemsevenfor home
personalcomputers,suchaviewpointlooksarchaic,sincethetimerequiredtoconvertthe
system to a form appropriate for (cid:147)fast computations(cid:148) is much greater than that for (cid:147)slow
computations(cid:148)basedonmoderndirect numericalmethodslike boundaryelementmethod
and (cid:2)nite element method. Apparently, it should be agreed that in the caseswhere direct
numerical techniques provide reliable results in an acceptable computational time, they
should be regarded as most ef(cid:2)cient for the problem in question. It is very important to
recognizethe caseswhere one has a priori to reject direct numerical methods. Theseare
listedbelow.
(cid:14)
1 . Problems where an exact analytical solution or a good approximation to it can be
obtained. Diffractiontheoryshowsmanyexamplesofthiskind.
(cid:14)
2 . Studying dynamic processes with high frequencies. Here, one has to take at least
© 2005 by CRC Press LLC
Pagev
10 nodes per wavelength to obtain reliable results by any direct numerical method. As
the wavelength decreases (i.e., the frequency increases) within a given frequency range,
the total number of nodes increases very rapidly, which results in too large algebraic
systems. An impressive example is given by room acoustics. Suppose a sound wave of
frequencyf = 2kHz,whosewavelengthin air is 17cm,propagatesin a 17›m long room.
For reasonable numerical accuracy, one should hence take at least 1000 nodes along the
room length. If the room has a width of 8m and a height of 5.1m, one has to consider
1000·500·300»108 (cid:2)nite›elementnodesandperformcomplex›valuedarithmetic. This
cannot be implemented even on the most powerful super computers. Here, a reasonable
criterionforacceptabilityofanumericalapproachisitsimplementabilityonaPCorsimilar
computer. So, obtaining solutions to such high›frequency problems in exact formulation
bydirectnumericaltechniquesdoesnotseemtobefeasibleinthevisiblefuture.
(cid:14)
3 . Studying phenomena of complex qualitative nature. Since direct numerical methods
provideonlynumbers,whichare usuallytabulatedandplotted, it is often verydif(cid:2)cult to
extract such complex qualitative effects from numerous tables and graphs. Instead, it is
preferable to construct an approximate analytical solution, from which qualitative effects
maybeextractedexplicitly.
(cid:14)
4 . Cases where an exact analytical solution has been obtained but its representation is
inapplicabletopracticefor speci(cid:2)ccalculations. Anexampleofthiskindisconsideredin
Section6.1. In suchinteresting cases,oneshouldlookfor analternativeapproach,which
isoftentheconstructionofanapproximatesolutionthatwouldbemoreappropriateforfast
computationsthantheexactanalyticalsolutionobtained.
The above situations are not widespread but, when met, are very dif(cid:2)cult to cope
withef(cid:2)ciently,especiallyiftheresearcherdoesnothavesuf(cid:2)cientexperienceintackling
them. Thispromptedustoconcludeeachsectionwithaspecialsubsectiontitled(cid:147)Helpful
Remarks,(cid:148)whichmayhelpthereadertobuilduphisorherownlessformalconceptionand
allowthecreationofamorecompletepictureoftheissueunderconsideration.
Notethattheapplicationofnumericalmethodsinregularcasesiswelldescribedinthe
classicalliterature. Forthisreason,weonlyconsidernumericalmethodsforsomeirregular
operatorproblems;seeChapter9.
To summarize, the main purpose of the present book is to show the close connection
between heuristic and rigorous methods in mathematical diffraction theory. We focus on
differentialandintegralequationsthatcaneasilybeutilizedinpracticalapplications.
Such an approach is accounted for by the choice of our potential readers. The book
presentsclearandelegantmethodsandisaimedatgraduateandpost›graduatestudents,so
that they could quickly examine the state of the art in a speci(cid:2)c (cid:2)eld of interest. At the
same time, researcherswith considerableexpertise in dealing with diffraction theory will
hopefully discoverthat the time of clear explicit solutions in unsolvedcomplexproblems
has not passed yet(cid:151)this is demonstrated by the authors’ original results in Sections 4.5,
4.6, 5.4(cid:150)5.7, and 6.3(cid:150)6.6 as well as in many sections of Chapters 7(cid:150)9. Furthermore, we
hope that an experienced reader will be able to discover for him› or herself new helpful
methods,bothanalyticalandnumerical.
The reader will seein what follows that we prefer to rely upon classicalresults of the
foundersof modernscienceunlikearatherwidespread(mistaken)pointof viewthatonly
very complicated recent (cid:147)abstract(cid:148) theories can provide further progressin contemporary
science. Westronglyrecommendtheyoungerreadertooperatewithclassicalmathematical
theories, and the present book will demonstrate that the fruitful ideas of Hilbert, Cauchy,
Fourier,Abel,Poisson,Weyl,Riemann,Green,Kirchhoff,Rayleigh,Helmholtz,Neumann,
andotherscanguidethereaderveryef(cid:2)cientlyaroundpresent›dayproblemsindiffraction
theory. It should also be stressedthat we tried to avoid too formal presentation, since we
© 2005 by CRC Press LLC
Pagevi
believethat wielding thoroughknowledgein anymathematical theoryimplies applyingit
effectivelyandsuccessfullyto practicerather than operatingwith the formal apparatusof
thetheory.
Duetoitslimitedvolume,anymonographcannotcoverallimportantquestions,andthe
presentbookisnoexception. Forexample,thereaderwillnot(cid:2)ndheretransientproblems
at all. The presentation is con(cid:2)ned to boundary problems for elliptic operators only, and
only those with constant coef(cid:2)cients (except Section 3.6). Moreover, the main focus is
on methods that provide solutions without too cumbersome mathematical manipulations.
For example,the readerwill not (cid:2)ndthe structure of the wave(cid:2)eldin the (cid:147)semi›shadow(cid:148)
zonein diffraction by convexobstacles,andin the method of (cid:147)edgewaves(cid:148)in diffraction
fromlinearsegments,thereaderwillonly(cid:2)ndtheleadinghigh›frequencyasymptoticterm,
whichisconstructedbyasimpleandeleganttechnique.
The sections, displayed formulas, and (cid:2)gures are enumerated independently within
eachchapterwiththechapternumberinfront.
Thebookisintendedforthereaderfamiliarwithfundamentalsofreal,complex›valued,
andfunctionalanalysiswithin a standardcourseoncalculusin the(cid:2)rst three yearsof any
universityprogramofmathematical,physical,orengineeringdepartments.
Thestyleandcontentofthisbookhavebeenin(cid:3)uencedbytheauthors’friends,teachers,
and colleagues, Alexander Vatulyan (Rostov State University, Russia), Mauro Fabrizio
(UniversityofBologna,Italy),andDorinIesan(UniversityofIasi,Romania).
The authors are grateful to Alexander Manzhirov and Alexei Zhurov for their helpful
discussionsandcomments.
The(cid:2)rstauthoristhankfultohiswife,AngelaSumbatyan,andtohisdaughters,Laura,
Carina,andAngelica,whoassistedandinspiredhiminthewritingofthebook.
M.A.Sumbatyan
A.Scalia
© 2005 by CRC Press LLC
Pagevii
AUTHORS
MezhlumA.Sumbatyan,Ph.D.,D.Sc.,isanotedscientistinthe(cid:2)eldofwavedynam›
ics,diffractiontheory,andill›posedproblems.
MezhlumSumbatyangraduatedfromtheFacultyofMechanicsandMathematics,Ros›
tovStateUniversity,Russia,in1969andreceivedhisPh.D.degreein1980attheInstitute
forProblemsinMechanics,RussianAcademyofSciences,Moscow. HisPh.D.thesiswas
devotedtoasymptoticmethodsfor solvingintegralequationsarisingin someproblemsof
mechanicsandacousticswith mixedboundaryconditions. In 1995,ProfessorSumbatyan
receivedhisDoctorofSciencesdegree;hisD.Sc.thesiswasdedicatedtodirectandinverse
problemsofdiffractiontheory,withapplicationstoreconstructionofobstaclesbyultrasonic
techniques.
In1985(cid:150)2000,MezhlumSumbatyanworkedintheResearchInstituteofMechanicsand
Applied Mathematics of the Rostov State University. Since 2001, Professor Sumbatyan
hasbeenamemberofthestaffoftheFacultyofMechanicsandMathematics,RostovState
University.
Professor Sumbatyan has made important contributions to new methods in the theory
andanalyticalmethodsappliedtodirectandinversediffractionproblems. Heisanauthor
ofmorethan120scienti(cid:2)cpublications,amemberoftheRussianAcousticalSociety,and
amemberoftheAcousticalSocietyofAmerica.
Address: FacultyofMechanicsandMathematics E›mail: [email protected]
ZorgeStreet5
344090Rostov›on›Don,Russia
Antonio Scalia, Ph.D., D.Sc., is a prominent scientist in the (cid:2)eld of mathematical
physics.
Antonio Scalia graduated from the Faculty of Mathematics, Physics and Natural Sci›
ences, University of Catania, Italy, in 1972 and received his Ph.D. degree in 1980 at the
UniversityofCatania. Histhesiswasdevotedtosolvabilityanduniquenessinmathematical
formulationofmechanicalandacousticalproblemsformediawithmicro›structure.
Since 1989, Professor Scalia has been a member of the staff of the Department of
Mathematics and Informatics at the University of Catania as an Associate Professor, and
since2001,asaFullProfessor.
ProfessorScaliahasgreatexpertisein thetheoryof viscoelasticity,electromagnetism,
and acoustics. He has signi(cid:2)cant achievements in the mixture theory in solids, with
applications to wave processesin elastic and acoustic media. Recently, within the theory
ofporousmedia,hedevisedanewapproachtotheanalysisofwavepropagationinelastic
solids with pores. He explicitly studieda characteristic equationof the porouscomposite
mediaappliedtothelayeredgeometry. ProfessorScaliaisanauthorofnearly100scienti(cid:2)c
publications.
Address: DipartimentodiMatematicaeInformatica E›mail: [email protected]
Universita· diCatania,VialeA.Dorian.6
95125Catania,Italy
© 2005 by CRC Press LLC
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CONTENTS
Preface
Authors
Contents
1. SomePreliminariesfromAnalysisandtheTheoryofWaveProcesses
1.1. FourierTransform,LineIntegralsofComplex›ValuedIntegrands,andSeriesin
Residues
1.2. ConvolutionIntegralEquationsandtheWiener(cid:150)HopfMethod
1.3. SummationofDivergentSeriesandIntegrals
1.4. AsymptoticEstimatesofIntegrals
1.5. FredholmTheoryforIntegralEquationsoftheSecondKind
1.6. FredholmIntegralEquationsoftheFirstKind
1.7. SingularIntegralEquationswithaCauchy›TypeSingularityintheKernel
1.8. Hyper›SingularIntegralsandIntegralEquations
1.9. Governing Equations of Hydroaeroacoustics, Electromagnetic Theory, and
DynamicElasticity
2. Integral Equations of Diffraction Theory for Obstacles in Unbounded
Medium
2.1. PropertiesofthePotentialsofSingleandDoubleLayers
2.2. BasicIntegralEquationsoftheDiffractionTheory
2.3. Propertiesof IntegralOperatorsof Diffraction Theory: GeneralCaseandLow
Frequencies
2.4. FullLow›FrequencySolutionforSphericalObstacle
2.5. Application: ScatteringDiagramforObstaclesofCanonicalShape
2.6. AsymptoticCharacteroftheKirchhoffPhysicalDiffractionTheory
3. WaveFieldsinaLayerofConstantThickness
3.1. Wave Operator in Acoustic Layer: Mode Expansion, Homogeneous and
InhomogeneousWaves
3.2. PrinciplesofSelectionofUniqueSolutioninUnboundedDomain
3.3. WavesinElasticLayer
3.4. Generalized Riemann’s Zeta Function and Summation of Some Oscillating
Series
3.5. Application: Ef(cid:2)cient Calculation of Wave Fields in a Layer of Constant
Thickness
3.6. WavesintheStrati(cid:2)edHalf›Plane
© 2005 by CRC Press LLC
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4. AnalyticalMethodsforSimplyConnectedBoundedDomains
4.1. GeneralSpectralPropertiesoftheInteriorProblemforLaplacian
4.2. ExplicitFormulasforEigenfrequenciesofRoundDisc
4.3. SomeVariationalPrinciplesforEigenvalues
4.4. Weyl(cid:150)CarlemanTheoryofAsymptoticDistributionofLargeEigenvalues
4.5. ExactExplicitResultsforSomePolygons
4.6. ExplicitAnalyticalResultsforSomePolyhedra
5. IntegralEquationsinDiffractionbyLinearObstacles
5.1. IntegralOperatorsinDiffractionbyLinearScreenandbyaGapintheScreen
5.2. Operator Equation in Diffraction Problem on a Crack in Unbounded Elastic
Medium
5.3. High›FrequencyAsymptoticsinDiffractionbyLinearObstaclesinUnbounded
Medium
5.4. High›Frequency Asymptotics for Diffraction by Linear Obstacles in Open
Waveguides
5.5. High›FrequencyDiffractionbyaLinearDiscontinuityintheWaveguide
5.6. WavesinElasticHalf›Space. FactorizationoftheRayleighFunction
5.7. IntegralEquationoftheMixedBoundaryValueProblemforElasticLayer
6. Short›WaveAsymptoticMethodsontheBasisofMultipleIntegrals
6.1. Schoch’sMethod: ExactRepresentationof3DWaveFieldsbyOne›Dimensional
Quadratures
6.2. High›FrequencyWaveFieldsinElasticHalf›Space
6.3. AsymptoticNatureoftheGeometricalDiffractionTheory
6.4. High›FrequencyDiffractionwithRe›Re(cid:3)ections
6.5. Application: ExamplesofHigh›FrequencyMultipleDiffraction
6.6. Application: PhysicalDiffractionTheoryforNonconvexObstacles
6.7. Short›WaveIntegralOperatorinDiffractionbyaFlawinElasticMedium
6.8. High›Frequency Asymptotics of Integral Operator in a Three›Dimensional
DiffractionTheory
7. InverseProblemsoftheShort›WaveDiffraction
7.1. Some Basic Results in a Local Differential Geometry of Smooth Convex
Surfaces
7.2. ReducingInverseProblemoftheShort›WaveDiffractiontoMinkowskiProblem
7.3. ExplicitResultsforaDifferentialOperatorofthe2DInverseProblem
7.4. ExactExplicitInversionoftheBasicOperatorintheCaseofAxialSymmetry
7.5. NonlinearDifferentialOperatoroftheThree›DimensionalInverseProblem
7.6. Reconstruction of Nonconvex Obstacles in the High›Frequency Range: 2D
Case
7.7. Reconstruction of Nonconvex Obstacles in the High›Frequency Range: 3D
Case
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