Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution

Citation

Kirpichnikova AS & Kirpichnikova NY (2019) Leontovich-Fock Parabolic Equation Method in the Neumann Diffraction Problem on a Prolate Body of Revolution. Journal of Mathematical Sciences, 238 (5), pp. 658-675. https://doi.org/10.1007/s10958-019-04265-2

Abstract
This paper continues a series of publications on the shortwave diffraction of the plane wave on prolate bodies of revolution with axial symmetry in the Neumann problem. The approach, which is based on the Leontovich–Fock parabolic equation method for the two parameter asymptotic expansion of the solution, is briefly described. Two correction terms are found for the Fock's main integral term of the solution expansion in the boundary layer. This solution can be continuously transformed into the ray solution in the illuminated zone and decays exponentially in the shadow zone. If the observation point is in the shadow zone near the scatterer, then the wave field can be obtained with the help of residue theory for the integrals of the reflected field, because the incident field does not reach the shadow zone. The obtained residues are necessary for the unique construction of the creeping waves in the boundary layer of the scatterer in the shadow zone. Bibliography: 16 titles. We consider a shortwave diffraction of a plane incident wave on the strictly convex, prolate body of revolution. The geometric characteristics of the scatterer (i.e., radii of curvatures of the surface of body of revolution) are assumed to be much larger than the incident wavelength. The incident wave propagates along the axis of revolution. The total wave field U is the sum of the incident U inc and reflected U ref waves, U = U inc + U ref. The field is constructed in the vicinity of the light-shadow border (i.e., in the penumbra of Fock's region, [1]), which is the " seed " zone for fields both in the vicinity of the limit rays and in the shadowed part of the body. The shortwave field in the illuminated area near the scatterer is described by means of the ray method. The field U satisfies the Helmholtz equation with Neumann or Dirichlet boundary conditions. Fock's boundary layer O(sk 1 3) = O(1), O(nk 2 3) = O(1) is introduced in a neighborhood of point s = 0, which belongs to the geometric border (Equator) of the shadow; here k is the wave number, n is the distance along the outer normal on the scatterer, and s is the arclength of the geodesic. The ray method does not work in the vicinity of the light-shadow border, i.e., in the Fock's boundary layer. The total wave field in the Fock's zone can be represented as U = e iks (W inc + W ref), where e iks is the oscillating factor of the wave field along the geodesic; the function W is called the attenuation function. Introducing dimensionless coordinates σ, ν instead of s and n, and rewriting e ikz in the new coordinates σ, ν, we obtain the first three terms of the expansion W inc in the form W (σ, ν) = W inc 0 + W inc 1 k 1 3 + W inc 2 k 2 3 + O(k −1), k 1, here W inc 0 is the main, W inc 1 is the first, and W inc 2 is the second terms of the asymptotic expansion. The functions W inc i , i = 0, 1, 2, have the form of integrals of linear combinations of the Airy function v(t) and its derivative v (t) with polynomials in the dimensionless normal coordinate ν in Fock's region. We apply the Leontovich–Fock parabolic equation method [1,4] to the function under investigation W ref (σ, ν) = W ref 0 + W ref 1 k 1 3 + W ref 2 k 2 3 + O(k −1), k 1.

Notes
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 461, 2017, pp. 148–173.

Journal
Journal of Mathematical Sciences: Volume 238, Issue 5