Solutions of the helmholtz equation, concentrated near a plane periodic boundary

The existence of solutions of the Helmholtz equation, exponentially decreasing with distance from a periodic boundary in the upper half-plane, is proved. These solutions exist for a special form of the boundary under the Dirichlet or Neumann boundary conditions. In either case, the boundary has the form of a chain of resonators joined with the upper half-plane by narrow splits. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 83–96. Translated by V. Yu. Gotlib