Geodesic
Matching of local asymptotics in the illuminated part of Fock domain
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 49-63

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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. 
@article{ZNSL_2014_426_a5,
     author = {N. Ya. Kirpichnikova and M. M. Popov},
     title = {Matching of local asymptotics in the illuminated part of {Fock} domain},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {49--63},
     publisher = {mathdoc},
     volume = {426},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a5/}
}
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N. Ya. Kirpichnikova; M. M. Popov. Matching of local asymptotics in the illuminated part of Fock domain. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 44, Tome 426 (2014), pp. 49-63. http://geodesic.mathdoc.fr/item/ZNSL_2014_426_a5/