On Sommerfeld representation and uniqueness in scattering by wedges

Komech, Alexander; Mauser, Norbert J.; Merzon, Anatoli

doi:10.1002/mma.553

Cited by 37 publications

(69 citation statements)

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“…In Reference [1], we have proved the Sommerfeld-Malyuzhinets-type integral representation and uniqueness of the solution of a nonstationary scattering problem by a wedge using the method of the complex characteristics [2].…”

Section: Introductionmentioning

confidence: 99%

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Limiting Amplitude principle in the scattering by wedges

Komech¹,

Merzon²

2006

Math Methods in App Sciences

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Section: Introductionmentioning

confidence: 99%

“…Here we give the next steps in our program of mathematical foundation of scattering by wedges [1]. Namely, we prove the existence of the solution and the Limiting Amplitude principle.…”

Section: Introductionmentioning

confidence: 99%

Limiting Amplitude principle in the scattering by wedges

Komech¹,

Merzon²

2006

Math Methods in App Sciences

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“…More precisely, here we use a potential theory approach combined with the use of concrete extension operators, and a corresponding integral description of the problems. Therefore, this is also different from the most classical approach due to the Malyuzhinets method [13] and corresponding representations of solutions by using the so-called Sommerfeld integral transform (cf., e.g., [11,23]). In addition, the present wedge diffraction problems present an increase of the difficulties when compared with the corresponding problems of wave diffraction by a half-plane; cf.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems

Castro

Kapanadze

2008

Proc. Amer. Math. Soc.

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Abstract. Boundary value problems originated by the diffraction of an electromagnetic (or acoustic) wave by a rectangular wedge with faces of possible different kinds are analyzed in a Sobolev space framework. The boundary value problems satisfy the Helmholtz equation in the interior (Lipschitz) wedge domain, and are also subject to different combinations of boundary conditions on the faces of the wedge. Namely, the following types of boundary conditions will be under study: Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, Impedance-Dirichlet, and Impedance-Neumann. Potential theory (combined with an appropriate use of extension operators) leads to the reduction of the boundary value problems to integral equations of Fredholm type. Thus, the consideration of single and double layer potentials together with certain reflection operators originate pseudo-differential operators which allow the proof of existence and uniqueness results for the boundary value problems initially posed. Furthermore, explicit solutions are given for all the problems under consideration, and regularity results are obtained for these solutions.

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“…Consequently, we can identify a significant number of publications where such analysis was taken for particular cases of wedge amplitudes and/or boundary conditions (cf., e.g., [2,7,8,9,10,11,12,13,18,21,22,33,34,35,42,48,44,45,46,49,50,55,56,58,60,66,72]). However, none of these listed papers contain final solvability results for the general problems in a rigourous mathematical Sobolev space setting as is done in the present paper.…”

Section: Introductionmentioning

confidence: 99%

“…The relevant work of Komech, Merzon and their collaborators [32,33,34,35] must also be referred, where the so-called method of complex characteristics for elliptic equations in nonconvex angles is used. Typically, the crucial part of the method is played by the connection equation on the Riemann surface of complex characteristics of the given elliptic operator.…”

Section: Introductionmentioning

confidence: 99%

Wave diffraction by wedges having arbitrary aperture angle

Castro

Kapanadze

2015

Journal of Mathematical Analysis and Applications

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The problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background. In fact, we may recognize significant research on this topic for more than one century. Despite this fact, up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate Sobolev space setting. In the present paper, we are considering the corresponding boundary value problems for the Helmholtz equation, with complex wave number, admitting combinations of Dirichlet and Neumann boundary conditions. The main ideas are based on a convenient combination of potential representation formulas associated with (weighted) Mellin pseudo-differential operators in appropriate Sobolev spaces, and a detailed Fredholm analysis. Thus, we prove that the problems have unique solutions (with continuous dependence on the data), which are represented by the single and double layer potentials, where the densities are solutions of derived pseudo-differential equations on the half-line.

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On Sommerfeld representation and uniqueness in scattering by wedges

Abstract: SUMMARYWe consider a non-stationary scattering of plane waves by a wedge. We prove the Sommerfeld-type representation and uniqueness of solution to the Cauchy problem in appropriate functional spaces developing the general method of complex characteristics (Math.

Cited by 37 publications

References 20 publications

Limiting Amplitude principle in the scattering by wedges

Limiting Amplitude principle in the scattering by wedges

Dirichlet-Neumann-impedance boundary value problems arising in rectangular wedge diffraction problems

Wave diffraction by wedges having arbitrary aperture angle

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