Shortwave grazing scattering of a plane wave on a smooth periodic boundary. II. Diffraction on an infinite periodic boundary

Scattering problem of a plane wave on a smooth periodic surface in the case of small grazing angle and shortwave approximation is considered (period and radius of curvature of the boundary are supposed to be large in compare with wave length). The solution of the problem is constructed as an infinite sum of multiple diffracted fields in a vicinity of the reflecting boundary (Fock's region and penumbra). Series of the multiple diffracted fields can be summed by means of a certain Winer–Hopf–Fock integral equation. Closed formulas for two leading terms of the shortwave aproximation of the solution is obtained.