An Explicit Formula for the Nonstationary Diffracted Wave Scattered on a NN-Wedge

Abstract

We consider two dimensional nonstationary scattering of plane waves by a NN-wedge. We prove the existence and uniqueness of a solution to the corresponding mixed problem and we give an explicit formula for the solution. Also the Limiting Amplitude Principle is proved and a rate of convergence to the limiting amplitude is obtained.

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

“…In the present paper, we analyze the Sommerfeld solution to the stationary diffraction by a half-plane, which corresponds to the case = 0, 21 (see also Sommerfeld 22,23 ). In this case, our results [15][16][17][18][19][20] are not applicable directly. Nevertheless, we prove for the first time that even in this case, the LAP holds, ie,…”

“…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:…”

“…Our proofs essentially rely on our methods and results in other studies. [15][16][17][18][19][20] In particular, we use the key formula (27) for the diffracted wave (see Komech et al 20, formula (1.7)) .…”