Helmholtz problem for plane periodical structures

The plane Helmholtz problem of the periodical disc structures, with the phase shifts conditions of the solutions along the basis lattice vectors and the Dirichlet conditions on the disc boundaries, is considered. The Green function satisfying the quasi-periodical conditions on the lattice is constructed. The Helmholtz problem is reduced to the boundary integral equations for the simple layer potentials of this Green function. The methods of the discretization of the arising integral equations are proposed. The procedures of calculation of the matrix elements are discussed. The reality of the spectral parameter of the nonlinear continuous and discretized problems is shown. Some numerical results are presented.