Geodesic
On short-wave diffraction by strongly prolate body of revolution
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 46, Tome 451 (2016), pp. 156-177
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In the paper a short-wave diffraction problem by strongly elongated body of revolution (axisymmetric case) is considered. In that case the classical method of Leontovich–Fock parabolic equation (actually Schrödinger type equation) turns out to be inapplicable due to corresponding recurrent system of equations loses asymptotic character and, moreover, each equation gets singularity in coefficients, including the main parabolic equation. In the work, we introduce a new boundary layer defined by the new scaling of the internal coordinates of the layer differs from the Fock case. Unfortunately, the variables cannot be separated in the main parabolic equation and therefore it is hardly possible to construct the solution of the problem in closed analytic form. Instead, we formulated a non-stationary scattering problem for the Schrödinger type equation, where role of the time plays the arc length along the meridians, and solved it by numerical methods. 
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M. M. Popov; N. M. Semtchenok; N. Ya. Kirpichnikova. On short-wave diffraction by strongly prolate body of revolution. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 46, Tome 451 (2016), pp. 156-177. http://geodesic.mathdoc.fr/item/ZNSL_2016_451_a9/

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