Asymptotic solutions of the wave equation localized in a tabular vicinity of the geodesics and the Fock problem in $3D$

M. M. Popov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 52, Tome 516 (2022), pp. 253-266

Résumé

The $3D$ problem of the short wave diffraction by a compact body with the smooth strongly convex boundary surface is considered. At that the particular interest presents the diffraction wave field in a vicinity of the light-shadow curve on the scatterer boundary surface. This curve is formed by the geometrical locus of tangent points of the incident rays. In $2D$ case this problem was the first time studied by academician V. A. Fock long ago and we call it now the Fock problem. But in $3D$ case new additional difficulties emerge in the problem: i) in the shadowed part of the boundary surface the diffraction wave field slides along geodesics which form caustics and ii) the geodesics are not plane curves due to the torsion. In the paper we propose a solution of $3D$ Fock problem in terms of superposition (integral) of asymptotic solutions of the wave equation localized in a vicinity of each geodesics from corresponding geodesic flow. The solution gets over the difficulties.