Creating Materials with a Desired Refraction Coefficient

102 Pages·2021·3.72 MB·English

by  Alexander G. Ramm

Creating Materials with a
Desired Refraction Coefficient
(Second Edition)

Creating Materials with a
Desired Refraction Coefficient
(Second Edition)
Alexander G Ramm
IOP Publishing, Bristol, UK
ªIOPPublishingLtd2020
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ISBN 978-0-7503-3391-7(ebook)
ISBN 978-0-7503-3389-4(print)
ISBN 978-0-7503-3392-4(myPrint)
ISBN 978-0-7503-3390-0(mobi)
DOI 10.1088/978-0-7503-3391-7
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Contents
Preface ix
Author biography xi
1 Introduction 1-1
References 1-4
2 Wave scattering by many small impedance particles 2-1
2.1 Scalar wave scattering by one small body of an arbitrary shape 2-1
2.1.1 Impedance bodies 2-1
2.1.2 Acoustically soft bodies (the Dirichlet boundary condition) 2-8
2.1.3 Acoustically hard bodies (the Neumann boundary condition) 2-10
2.1.4 The interface (transmission) boundary condition 2-13
2.1.5 Summary of the results 2-19
2.2 Scalar wave scattering by many small bodies of an arbitrary shape 2-20
2.2.1 Impedance bodies 2-20
2.2.2 The Dirichlet boundary condition 2-26
2.2.3 The Neumann boundary condition 2-28
2.2.4 The transmission boundary condition 2-31
2.2.5 Wave scattering in an inhomogeneous medium 2-33
2.2.6 Summary of the results 2-36
References 2-37
3 Creating materials with a desired refraction coefficient 3-1
3.1 Scalar wave scattering. Formula for the refraction coefficient 3-1
3.2 A recipe for creating materials with a desired refraction coefficient 3-2
3.3 A discussion of the practical implementation of the recipe 3-3
3.4 Summary of the results 3-4
References 3-4
4 Wave-focusing materials 4-1
4.1 What is a wave-focusing material? 4-1
4.2 Creating wave-focusing materials 4-3
4.3 Computational aspects of the problem 4-11
4.4 Open problems 4-14
vii
CreatingMaterialswithaDesiredRefractionCoefficient(SecondEdition)
4.5 Summary of the results 4-14
References 4-14
5 On non-over-determined inverse problems 5-1
5.1 Introduction 5-1
5.2 Proof of theorem 5.1.1 5-2
5.3 A numerical method 5-4
5.4 Summary of the results 5-8
References 5-8
6 Experimental verification of the method for creating materials 6-1
6.1 Moving the refraction coefficient in the desired direction 6-1
6.2 The case of a bounded region 6-3
6.3 Embedding acoustically soft particles 6-6
6.4 Summary of the results 6-7
References 6-7
7 A symmetry property in harmonic analysis 7-1
7.1 Summary of the results 7-7
References 7-7
8 Inverse scattering problem 8-1
8.1 Summary of the results 8-5
References 8-5
Bibliography 9-1
viii
Preface
The goals of this work are:
1. to give a recipe for creating materials with a desired refraction coefficient;
2. to draw the interest in solving practically (technologically) the problem of
producing small particles with a prescribed boundary impedance;
3. to give a method for solving many-body scattering problem for small
scatterers;
4. to prove uniqueness theorems for inverse scattering problems with non-over-
determined scattering data;
5. to formulate a symmetry property for PDE;
6. toformulateandapplyPropertyCforaproofoftheuniquenesstheoremsfor
inverse scattering with fixed-energy potential scattering;
7. to give a method for creating materials with a wave-focusing property.
Creating materials with a prescribed boundary impedance is a problem of high
interestboththeoreticallyandpractically.Suchmaterialsmaybemeta-materials,for
example.Therefore,problem2isimportantbothpracticallyandtheoretically.Thisis
why it is discussed in detail in chapters 3 and 6. Problem 3 is basic for solving
problem 1. The authorʼs solution to this problem is given in chapter 2. In chapter 4
creating of materials with a wave-focusing property is discussed. In chapter 5 the
inverse scattering theory with non-over-determined scattering data is developed for
obstacle scattering. In chapter 6 creating materials with a desired refraction
coefficient in bounded domains is considered for the first time. In chapter 7 a
symmetry property for PDE is presented. This result allows the author to prove the
SchifferʼsconjectureandtosolvethePompeiuproblem,see[4,9].Inchapter7anew
result, a symmetry property for harmonic analysis is formulated, probably, for the
first time. Its proof is given. In chapter 8 the author’s old result, the uniqueness
theoremforinversescatteringwithfixed-energydataispresented.Theproofisbased
on the Property C introduced by the author for a study of inverse problems.
Inthissmallbooktheresultsoftheauthorarepresented.Theywerepublishedin
detail in [1–9], and in the authorʼs papers cited in the bibliography.
Inthisworktheauthorpresentsthebasicideasandresultsinaself-containedway
anddealswiththescalarwavescattering.Theemphasisisontheformulationofthe
results and on the importance of solving problem 2 practically.
The author thanks Momentum Press for permission to use parts of his earlier
monograph [1], specifically chapters 2–4 of [1]. In chapter 5 some recent papers of
the author are used [5, 6].
Thisbookisanexpandedversionofthebook[7],Chapters6,7and8areadded.
References
[1] Ramm A G 2013 Scattering of Acoustic and Electromagnetic Waves by Small Bodies
of Arbitrary Shapes. Applications to Creating New Engineered Materials (New York:
Momentum)
ix