Abstract

One way to solve a diffraction problem is to look for a solution in the form of a plane wave decomposition integral. Over a century ago this approach was successfully applied by A. Sommerfeld to the problem of diffraction by a half-plane and later extended by Maliuzinets for the wedge problem. In this talk, we show that this method can also be applied to diffraction problems on discrete lattices. Particularly, we show that dispersion surface for the discrete Helmholtz equation on a grid is topologically a torus. The plane wave integral is built as an integral over a canonical dissection of the torus with the integrand being a product of a plane wave, a transformant and Abel differential of the first kind. Depending on the point of observation contours of integration slide along the torus.  The transformant is supposed to be a meromorphic function over a torus. Then three discrete diffraction problems are considered: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a half-plane; (3) the problem of diffraction by a right-angled wedge. It is shown that for the first problem the transformant is trivial, and for the rest two it is built using the theory of algebraic fields of functions on Riemann surfaces. An analogy with continuous case and relation to Wiener-Hopf method is discussed. The work is being done in collaboration with A. V. Shanin.