A Canonical Diffraction Problem With Two Media

Abstract

A Wiener–Hopf equation in L2 being equivalent [5] to a boundary value problem (of the first kind) for a wave-scattering Sommerfeld half-plane Σ=ℝ+×{0} which faces two different media Ω-: x2<0, Ω+: x2>0, as a special configuration in [3], is solved by canonical Weiner–Hopf factorization of its L2-regular scalar symbol γo=γo- γo+. The factors are calculated by solving a Riemann–Hilbert boundary value problem on the semi-infinite branch cuts of tj(ξ):=(ξ2−k2j)1/2, kj∈ℂ++ for j=1,2: taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener–Hopf–Hilbert(–Hurd) method [2] and requires the evaluation of elliptic-type integrals. Formula (3.7) seems not to be contained in tables of integrals.