Short-wave grazing scattering of a planar wave at a smooth periodic boundary. I: Diffraction of a half-shadow field of a smooth convex contour

Scattering problem of a plane wave on a smooth periodical surface in the case of small grazing angle and short wave approximation is concidered (period and radius of curvature are supposed to be small in compare with wave length). In the first part of the paper the short wave penumbral asymptotics of two-fold scattering field is constructed both in the vicinity of a point of diffraction and in the vicinity of the limiting ray. Basing on these results asymptotics of many-fold scattering penumbral fields is derived.