Summatory and integral equations in periodic problems of the diffraction of elastic waves by defects in layered media

In this paper we consider the diffraction problem for an elastic wave at a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a paired summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the paired equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.