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

Abstract

We analyze the Sommerfeld solution to the stationary diffraction by a half‐plane. We prove that this solution is the limiting amplitude for time‐dependent scattering of incident plane waves with a broad class of the profile functions. We also show that this solution is the asymptotics of the limiting amplitudes of solutions to time‐dependent scattering problem with narrow wedges when the angle of the wedge tends to zero.

Automorphic Functions on the Riemann Surface

“…Remark 8.1. i) In all the works, cited above, the limiting amplitude principle was not established, and the choice of suitable solution of the ill-posed problem (7.55) was not rigorously clarified. Nevertheless, as shown in [28,29,46], all the obtained solutions coincide with the limiting amplitudes calculated in [26] and admit the Sommerfeld representation.…”

“…The main results of the collaboration were the limiting absorption principle [44,45], proof of the completeness of Ursell's trapping modes [31] , the extension to the nonconvex angles [25,30], and the Sommerfeld representation [24]. Moreover, our general methods [30] allowed us to reproduce the formulas obtained by Sommerfeld, Sobolev and Keller [28,29,46]. The identifications justify these formulas as the limiting amplitudes in diffraction.…”

Time-dependent approach to the uniqueness of the Sommerfeld solution of the diffraction problem by a half-plane

“…In other words, all the classical known formulas are the limiting amplitudes of solutions to nonstationary problems as t → ∞. For the Sommerfeld problem, this was proven in [11], except for the proof of the uniqueness of the solution to the nonstationary problem in an appropriate class. This paper makes up for this omission.…”

“…The literature devoted to diffraction by wedges including the Sommerfeld problem is enormous (see e.g. reviews in [10] and [11]), and we will only indicate some papers where the uniqueness is treated. In paper [4] a uniqueness theorem was proven for the Helmholtz equation (∆ + 1)u = 0 in two-dimensional regions D of the semiplane type.…”

“…The detailed description of these papers is given in [25]. In [11], [25]- [31], the diffraction by a wedge of magnitude φ (which can be a half-plane in the case φ = 0 as in [11]) with a real wavenumber was considered as a stationary problem which is the "limiting case" of a nonstationary one. More precisely, we seeked the solutions of the classical diffraction problems as limiting amplitudes of solutions to corresponding nonstationary problems, which are unique in some appropriate functional class.…”