DN-Scattering of a plane wave by wedges

Abstract

Communicated by M. CostabelWe continue our study of a nonstationary scattering by wedges. In this paper we consider nonstationary scattering of plane waves by a 'hard-soft' wedge. We prove the uniqueness and existence of a solution to the corresponding DN-Cauchy problem in appropriate functional spaces. We also give the explicit form of the solution and prove the Limiting Amplitude Principle.

Sommerfeld's solution as the limiting amplitude and asymptotics for narrow wedges

“…where u (x, t) is a unique solution from the classes introduced in previous studies. [15][16][17][18][19][20] Namely, in other studies, [16][17][18][19] the LAP was proved for DD, DN, and NN boundary conditions respectively for smooth incident waves and, in Komech et al, 20 for distributional incident waves.…”

“…The existence and uniqueness of solutions to the time-dependent diffraction problems (4) and (5) in suitable function classes were proved for the first time for smooth profiles f in recent works [15][16][17][18][19] for the DD, DN, and NN boundary conditions, respectively. In Komech et al, 20 these results were extended to more general class of profiles f including distributions.…”

“…In papers, [16][17][18][19][20] we have proved for the first time the limiting amplitude principle (LAP) for the scattering by wedges with angle ∈ (0, 2 ) for a broad class of the profile functions:…”

Wave diffraction by wedges having arbitrary aperture angle

“…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.…”

“…Also, the limiting amplitude principle in the two-dimensional scattering of an incident plane harmonic wave by a wedge has recently been successfully applied, cf. [13,49,56].…”