Wood's anomalies in the scattering problem of a plane wave on a smooth periodic boundary

The scattering problem of a plane wave on a smooth periodic boundary in short wave approximation and for small grazing angle is considered. In previous our papers the asymptotics of wave field of the problem has been constructed in terms of infinite series of multiple scattering fields and this series has been summed arising an integral equation of Wiener–Hopf type if the latter equation has unique solution. It turns out that uniqueness of the solution of this equation coincides with absence of Wood's anomalies in the scattering problem. The main result of this paper consists in summation of multiple scattering fields when Wood's anomalies arise. To this end we had to choose appropriate solution of the integral equation when spectral parameter $\Omega$ belonged to the spectrum of corresponding operator. This was achieved by a detailed analysis of transition from penumbral wave field to asymptotics of wave field in Fock's region.