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).

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sommerfeld integral halfline problem or elliptic functions discrete analog of the helmholtz equation plane wave decomposition problems total field dirichlet halfline function algebraic functions diffraction point source

A.V. Shanin, A.I. Korolkov,.Sommerfeld-type integrals for discrete diffraction problems. 97 (),102606.

A.V. Shanin, A.I. Korolkov,"Sommerfeld-type integrals for discrete diffraction problems" 97

A.V. Shanin, A.I. Korolkov,"Sommerfeld-type integrals for discrete diffraction problems"