Initial Dirichlet Problem for Half-Plane Diffraction: Global Formulae for its Generalized Eigenfunctions, Explicit Solution by the Cagniard-de Hoop Method

Abstract

This paper deals with time-dependent plane wave diffraction by a soft/soft Sommerfeld half-plane $\sum :x>0,y=\pm 0$. The explicit solution is obtained as a time-convolution in two ways: The first is directly applying the Cagniard de Hoop method to the generalized Wiener-Hopf solution of the corresponding stationary problem due to Meister & Speck (1989). The second way makes use of the Laplace integral representation of the generalized eigenfunctions with respect to the spatial (Cartesian) variables derived by Ali Mehmeti in his habilitation thesis (1995) from formulae of Meister (1983). After deforming the path of.integration into the semi-infinte branch cut lines of the characteristic square root $\sqrt{{\xi }^2-k^2}$ of the Helmholtzian, he obtains representations where real wave numbers may appear. But for convergence of the integrals one has to distinguish the cases $x\ge 0$ and $x<0$, where the obstacle is present or not. We set the time Laplace variable $s=–ik$ and recover the time domain functions for the diffracted field from the eigenfunctions of the stationary problem. There follows a global formula representation with polar coordinates having the diffracting edge of $\sum$ as its center. The solution of the initial boundary value problem is seen to coincide with that obtained in the first way, indeed.