Symbol of the Dirichlet-to-Neumann operator in 2D diffraction problems with large wavenumber

Abstract

Consider the Dirichlet-to-Neumann operator N in the exterior problem for the 2D Helmholtz equation outside a bounded domain with smooth boundary. Using parametrization of the boundary by normalized arclength, we treat N as a pseudodifferential operator on the unit circle. We study its discrete symbol. We put, forward a conjecture on the universal behaviour, independent of shape and curvature of the boundary, of the symbol as the wavenumber k /spl rarr/ /spl infin/. The conjecture is motivated by an explicit formula for circular boundary, and confirmed numerically for other shapes. It also agrees, on a physical level of rigor, with Kirchhoff's approximation. The conjecture, if true, opens new ways in numerical analysis of diffraction in the range of moderately high frequencies.