The impedance boundary value problem for the Helmholtz equation in a half-plane

Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 (1997) The impedance boundary value problem for the Helmholtz equation in a half-plane. Mathematical Methods in the Applied Sciences, 20 (10). pp. 813-840. ISSN 0170-4214 doi: 10.1002/(SICI)1099-1476(19970710)20:10<813::AID-MMA883>3.0.CO;2-R

Abstract/Summary

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].