Time-dependent plane wave diffraction by a half-plane: Explicit solution for Rawlins' mixed initial boundary value problem

摘要

This paper deals with the diffraction of a time-dependent plane wave field G(t-x cos theta - ysin theta) governed by the two-dimensional wave equation and striking the edge of the half-plane Sigma: x > 0, y = 0 at time t = 0 with some incident angle theta. The explicit solution formula for Me total wave field is derived as a convolution with respect to time for homogeneous initial data and homogeneous boundary conditions: Dirichlet on the upper Neumann on the lower bank of Sigma. The Cagniard de Hoop method [1] is seen to be applicable due to the Wiener-Hopf solution of the corresponding stationary Rawlins problem [13] obtained in [16] by generalized L-2-factorization of its piece-wise continuous Fourier matrix symbol relative to the real line on the basis of [17]. This approach is also inspired by the attempt to solve transient half-plane problems via spectral theory ([2, 16]). The method to prove the limiting absorption principle for the pure Dirichlet problem in [2] (by deforming integral paths [9]) has intrinsic analogies to the Cagniard de Hoop method used here. [References: 18]