Shortwave scattering of a plane wave on a smooth periodic boundary

Diffraction problems of a plane wave by gratings form now a separate branch of the diffraction theory. There are many papers devoted to the scattering of a plane wave on smooth gratings and usually these investigations are based on direct numerical methods. But these numerical methods face serious difficulties in the case of a sufficiently short wavelength, especially when illuminated and shadow parts of the grating arise. However, one may expect the short wave asymptotic methods to be useful in such problems. Using the main ideas of the shortwave approximation the authors develop an asymptotic theory of smooth gratings in special case of grazing incidence