A Contribution to the Mathematical Theory of Diffraction: A Note on Double Fourier Integrals

Raphael C. Assier, A. V. Shanin, A. I. KOROLKOV

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a large class of physical fields u written as double inverse Fourier transforms of some functions F of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closedform far-field asymptotic expansion of u. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form farfield asymptotic component of u. We conclude the article by validating this theory against full numerical computations for two specific examples.
Original languageEnglish
Article numberhbac017
JournalThe Quarterly Journal of Mechanics and Applied Mathematics
DOIs
Publication statusPublished - 5 Dec 2022

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