Tangent ray diffraction and the Pekeris caret function

• Published in: Wave motion Vol. 57; pp. 257 - 267
• Main Author: Hewett, D.P.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2015
• Subjects: Boundaries; Equivalence; Fourier analysis; Grazing incidence; Integrals; Matched asymptotic analysis; Mathematical analysis; Parabolic wave equation; Penumbra field; Penumbras; Plane waves; Representations; Tangent rays; Wave diffraction; Matched asymptotic analysis; Wave diffraction; Tangent rays; Penumbra field; Grazing incidence; Parabolic wave equation
• ISSN: 0165-2125; 1878-433X
• DOI: 10.1016/j.wavemoti.2015.05.003

We study the classical problem of high frequency scattering of an incident plane wave by a smooth convex two-dimensional body. We present a new integral representation of the leading order solution in the “Fock region”, i.e. the neighbourhood of a point of tangency between the incident rays and the scatterer boundary, from which the penumbra (light–shadow boundary) effects originate. The new representation, which is equivalent to the classical Fourier integral representation and its well-studied “forked contour” regularisation, reveals that the Pekeris caret function (sometimes referred to as a “Fock-type integral” or a “Fock scattering function”), a special function already known to describe the field in the penumbra, is also an intrinsic part of the solution in the inner Fock region. We also provide the correct interpretation of a divergent integral arising in the analysis of Tew et al. (2000), enabling the results of that paper to be used for quantitative calculations. •We give a new contour integral solution for the wave field at a point of ray tangency.•The integrand involves the Pekeris caret function.•The solution can be matched to the outer wave field by the steepest descent method.