Wood's anomalies and surface waves in the problem of scattering by a periodic boundary. I

The solution of the problem of diffraction of an acoustic plane wave by a periodic boundary for frequencies close to threshold values is studied. Wood's well-known experiments show that deviations from the threshold frequency values by a small quantity bring about drastic changes in the diffraction pattern. The asymptotic formula with respect to the small parameter $\varepsilon$ is obtained for the corresponding scattering matrix.