Matching of local asymptotics in the illuminated part of Fock domain

Exploration of shortwave diffraction by elongated bodies of revolution requires detail consideration of matching of the local asymptotics in the illuminated part of Fock domain. In the paper we solve that task by means of the direct construction of the reflected wave with the help of the ray method. The main problem on that way, which was estimated by V. A. Fock as rather complicated, is the calculation of the eikonal and geometrical spreading in curvilinear coordinates used in the boundary layer method in the vicinity of light-shadow zone.