Sommerfeld-type integrals for discrete diffraction problems

Three problems for a discrete analog of the Helmholtz equation are studied analytically using the plane wave decomposition and the Sommerfeld integral approach. They are: (1) the problem with a point source on an entire plane; (2) the problem of diffraction by a Dirichlet half-line; (3) the problem of diffraction by a Dirichlet right angle. It is shown that the total field can be represented as an integral of an algebraic function over a contour drawn on some manifold. The latter is a torus. As a result, explicit solutions are obtained in terms of recursive relations (for the Green’s function), algebraic functions (for the half-line problem), or elliptic functions (for the right angle problem). •Three problems for a discrete analog of the Helmholtz equation are studied.•The problem with a point source on an entire plane is studied.•The problem of diffraction by a Dirichlet half-line is studied.•The problem of diffraction by a Dirichlet right angle is studied.•The plane wave decomposition and the Sommerfeld integral approach are applied.•The solution is obtained in terms of the algebraic functions.