Waves in Complex Continua (Wavinar): Andrey Korolkov (Moscow State University)
From Liam Holligan
Tuesday 9th June, 2020
Andrey Korolkov (Moscow State University)
Title: Sommerfeld integral method for discrete diffraction problems
Abstract: Two diffraction problems on a 2D square lattice are studied: finding the Green’s function and scattering by a half-line. The governing equation in both cases is the Helmholtz equation with a standard discrete Laplacian. For the Green’s function of an entire plane, we build the field in the form of a plane wave decomposition. This decomposition is rewritten as a contour integral of an analytical differential 1-form on a complex manifold, which is the lattice dispersion diagram (the solution of the dispersion equation). The dispersion diagram of the lattice is, topologically, a torus, and this makes possible to develop an invariant representation of the field. For the diffraction problems, an analogue of the Sommerfeld integral is built. This is an integral of a Sommerfeld transformant (a two-valued function on the torus) along a system of appropriate integrals.
As a result, we obtain new representations of the discrete wave fields. Potentially, these methods can be applied to a wider class of problems.