On the scattering vibrations of the plane periodic grating

The existence of non-trivial solutions of homogeneous boundary value problems of scattering theory for the Helmholtz equation is closely connected with the resonance phenomena. That is, if the frequency of time-harmonic forces belongs to some discrete set on the real axis, then the solution of corresponding non-stationary initial and boundary value problems for the wave equation with a time-harmonic righthand side is unbounded as t→∞. In this paper, a class of non-trivial solutions of the homogeneous Dirichlet´s boundary value problem for the Helmholtz equation in the exterior of the periodic grating of smooth obstacles in R² is researched. The essential feature of this solution is that there can be an unbounded energy in the stripe of one period of the grating. The theorem of uniqueness is proved under fulfilment of one condition for the boundary