A contribution to the mathematical theory of diffraction: a note on double fourier integrals. Issue 1 (5th December 2022)

• Record Type: Journal Article
• Authors: Assier, R C; Shanin, A V; Korolkov, A I
• Is Part Of: Quarterly journal of mechanics and applied mathematics. Volume 76:Issue 1(2023)
• Page Start: 1
• Page End: 47
• Publication Date: 2022-12-05
• Subjects: Mechanics -- Mathematics -- Periodicals; Applied mathematics -- Periodicals; 530.1
• DOI: 10.1093/qjmam/hbac017
• Languages: English
• ISSNs: 0033-5614

Abstract

Summary: 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 closed-form 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 far-field asymptotic component of $u$ . We conclude the article by validating this theory against full numerical computations for two specific examples.