Application of the parabolic equation to problems of short-wave diffraction by prolate convex bodies in the rotational symmetry case is considered. The wave field is constructed in the Fock domain and in the shaded part of the body, where creeping waves appear. In the problems under consideration, the following two large parameters arise: M = (kρ/2)1/3 and Λ = ρ / f, where k is the wave number, ρ is the radius of curvature along geodesics (meridians) and f is the radius of curvature in the transversal direction. The first one is the so-called Fock parameter, and the second one Λ characterizes the prolateness of the body. Under the condition Λ = M 2—ε, 0 < ε < 2, the parabolic equation method in classical form is valid and describes the wave field in terms of the Airy function and integrals of it. In the case of ε = 0, some coefficients in the corresponding recurrent equations become singular and the question on the solvability of the equations in terms of regular and smooth functions remains open. Bibliography: 9 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 409, 2012, pp. 55–79.
Translated by P. M. Popov.
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Kirpichnikova, N.Y., Popov, M.M. The Leontovich−Fock Parabolic Equation Method in Problems of Short-Wave Diffraction by Prolate Bodies. J Math Sci 194, 30–43 (2013). https://doi.org/10.1007/s10958-013-1504-5
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DOI: https://doi.org/10.1007/s10958-013-1504-5