On the asymptotic properties of a canonical diffraction integral. (28th October 2020)

• Record Type: Journal Article
• Authors: Assier, Raphaël C.; Abrahams, I. David
• Is Part Of: Proceedings. Volume 476:Number 2242(2020)
• Publication Date: 2020-10-28
• Subjects: Wiener–Hopf -- Cauchy integrals -- diffraction -- quarter-plane; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2020.0150
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : We introduce and study a new canonical integral, denoted I + − ε, depending on two complex parameters α 1 and α 2 . It arises from the problem of wave diffraction by a quarter-plane and is heuristically constructed to capture the complex field near the tip and edges. We establish some region of analyticity of this integral in C 2, and derive its rich asymptotic behaviour as | α 1 | and | α 2 | tend to infinity. We also study the decay properties of the function obtained from applying a specific double Cauchy integral operator to this integral. These results allow us to show that this integral shares all of the asymptotic properties expected from the key unknown function G +− arising when the quarter-plane diffraction problem is studied via a two-complex-variables Wiener–Hopf technique (see Assier & Abrahams, SIAM J. Appl. Math., in press). As a result, the integral I + − ε can be used to mimic the unknown function G +− and to build an efficient 'educated' approximation to the quarter-plane problem.