Tangent ray diffraction and the Pekeris caret function

TL;DR

This paper introduces a new integral representation for high-frequency scattering near a convex body, highlighting the intrinsic role of the Pekeris caret function in the Fock region and clarifying previous divergent integral issues.

Contribution

It presents a novel integral representation of the solution in the Fock region, emphasizing the Pekeris caret function's role and resolving earlier divergence problems for practical use.

Findings

New integral representation of the Fock region solution

Pekeris caret function is intrinsic to the scattering field

Clarification of divergent integral interpretation

Abstract

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. (Wave Motion 32, 2000), enabling the results of that paper to be used for quantitative calculations.