Leontovich–Fock parabolic equation method in the problems of short-wave diffraction by prolate bodies

Application of the parabolic equation to the problems of short-wave diffraction by prolate convex bodies in rotational symmetric case is considered. The wave field is constructed in the Fock's region and in the shaded part of the body where creeping waves appear. In the problems under consideration the following two large parameters arise: $\mathbf M=(k\rho/2)^{1/3}$ and $\mathbf\Lambda=\rho/f$, where $k$ is wave number, $\rho$ is radius of curvature along geodesic (meridians) and $f$ is radius of curvature in the transversal direction. The first one is so-called Fock parameter and the second one $\mathbf\Lambda$ characterizes prolateness of the body. Under condition $\mathbf\Lambda=\mathbf M^{2-\varepsilon}$, $0<\varepsilon<2$, the parabolic equation method in classical form is valid and describes the wave field in terms of Airy and integrals of it. In the case when $\varepsilon=0$ some coefficients in the corresponding recurrent equations become singular and question of solvability of the equations in terms of regular and smooth functions remains open.