Plane-wave diffraction by a rational wedge

Rawlins, A. D.

doi:10.1098/rspa.1987.0066

Cited by 12 publications

(4 citation statements)

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“…Note that the papers and concern not only right angles. The exact solutions to the stationary diffracted problems by the different types of wedges were obtained in . To analyze problem we use a general method which we call the Method of Complex Characteristics (MCC) .…”

Section: Introductionmentioning

confidence: 99%

DN-Scattering of a plane wave by wedges

Merzon¹,

Méndez²

2011

Math. Meth. Appl. Sci.

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Section: Introductionmentioning

confidence: 99%

DN-Scattering of a plane wave by wedges

Merzon¹,

Méndez²

2011

Math. Meth. Appl. Sci.

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“…For example, classical methods allow us to obtain analytic solutions for particular incoming waves by constructing resolvent operators (see e.g. the papers by A. D. Rawlins in the 1980s [29]). …”

Section: Introduction and Formulation Of The Problemsmentioning

confidence: 99%

Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors

Duduchava

Tsaava

2013

Georgian Mathematical Journal

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The purpose of the present research is to investigate the mixed Dirichlet-Neumann boundary value problems for the Helmholtz equation in a 2D domain R with finite number of non-cuspidal angular points on the boundary. Using localization, the problem is reduced in [Proc. A. Razmadze Math. Inst. 162 (2013), 37-44] to model problems in plane angles of magnitude˛j 2 OE0; 2 , j D 1; : : : ; m.In the present paper, we apply the potential method and reduce the model mixed BVP (with Dirichlet-Neumann conditions on the boundary) to an equivalent boundary integral equation (BIE) of Mellin convolution type. Applying the recent results on Mellin convolution equations with meromorphic kernels in Bessel potential and Sobolev-Slobodeckij (Besov) spaces obtained by V. Didenko and R. Duduchava and by R. Duduchava, the unique solvability criteria (Fredholm criteria) of the above mentioned mixed BVP are obtained in classical finite energy space H 1 .@ ˛/ D W 1 .@ ˛/ and also in non-classical Bessel potential spaces H 1 p .@ ˛/ when 1 < p < 1. The problem has been tackled before only for angular domains of magnitude˛for rational angles˛D m=n.

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“…In this review (apart from Macdonald's approach discussed in A), we will focus primarily on plane wave incidence rather than line sources. It has to be noted, however, that a broad range of work (Bromwich, 1915;Oberhettinger, 1954;Rawlins, 1987Rawlins, , 1989 has also been carried out for both acoustic and electromagnetic sources. For other reviews of some of the methods used for various types of incident waves (plane, cylindrical, spherical, dipole and pulse), see Oberhettinger (1958) and Bowman et al (1987).…”

Section: Introduction and Formulationmentioning

confidence: 99%

Analytical methods for perfect wedge diffraction: A review

2020

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The subject of diffraction of waves by sharp boundaries has been studied intensively for well over a century, initiated by groundbreaking mathematicians and physicists including Sommerfeld, Macdonald and Poincaré. The significance of such canonical diffraction models, and their analytical solutions, was recognised much more broadly thanks to Keller, who introduced a geometrical theory of diffraction (GTD) in the middle of the last century, and other important mathematicians such as Fock and Babich. This has led to a very wide variety of approaches to be developed in order to tackle such two and three dimensional diffraction problems, with the purpose of obtaining elegant and compact analytic solutions capable of easy numerical evaluation.The purpose of this review article is to showcase the disparate mathematical techniques that have been proposed. For ease of exposition, mathematical brevity, and for the broadest interest to the reader, all approaches are aimed at one canonical model, namely diffraction of a monochromatic scalar plane wave by a two-dimensional wedge with perfect Dirichlet or Neumann boundaries. The first three approaches offered are those most commonly used today in diffraction theory, although not necessarily in the context of wedge diffraction. These are the Sommerfeld-Malyuzhinets method, the Wiener-Hopf technique, and the Kontorovich-Lebedev transform approach. Then follows three less well-known and somewhat novel methods, which would be of interest even to specialists in the field, namely the embedding method, a random walk approach, and the technique of functionally-invariant solutions.Having offered the exact solution of this problem in a variety of forms, a numerical comparison between the exact solution and several powerful approximations such as GTD is performed and critically assessed.

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Plane-wave diffraction by a rational wedge

Cited by 12 publications

References 10 publications

DN-Scattering of a plane wave by wedges

DN-Scattering of a plane wave by wedges

Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors

Analytical methods for perfect wedge diffraction: A review

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