Time-dependent approach to the uniqueness of the Sommerfeld solution of the diffraction problem by a half-plane: https://arxiv.org/abs/1908.01663
A. Merzon, P. Zhevandrov, J.E. De la Paz Méndez, T.J. Villalba Vega
We consider the Sommerfeld problem of diffraction by an opaque half-plane with a real wavenumber
interpreting it as the limiting case, as time tends to infinity, of the corresponding time-dependent diffraction problem. We prove that the Sommerfeld formula for the solution is the limiting amplitude of the solution of this time-dependent problem which belongs to a certain functional class and is unique in it. For the proof of uniqueness of solution to the time-dependent problem we reduce it, after the Fourier-Laplace transform in t, to a stationary diffraction problem with a complex wavenumber. This permits us to use the proof of uniqueness in the Sobolev space H1. Thus we avoid imposing the radiation and regularity conditions on the edge from the beginning and instead obtain it in a natural way.