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A Contribution to the Mathematical Theory of Diffraction: A Note on Double Fourier Integralsстатья
- Авторы: Assier R.C., Shanin A.V., Korolkov A.I.
- Журнал: Quarterly Journal of Mechanics and Applied Mathematics
- Том: 76
- Номер: 1
- Год издания: 2022
- Издательство: Oxford University Press
- Местоположение издательства: United Kingdom
- Первая страница: 1
- Последняя страница: 47
- DOI: 10.1093/qjmam/hbac017
- Аннотация: 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.
- Добавил в систему: Шанин Андрей Владимирович