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Plane wave decompositions: general notes

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WHTW02 - WHT Follow on: the applications, generalisation and implementation of the Wiener-Hopf Method

A starting point of solving a diffraction problem is choosing an Ansatz for the field, and such an Ansatz is usually a plane wave decomposition. Some of plane waves decompositions follow directly from the Fourier analysis (this is the case of reprepresentation of Green’s functions), and some are the matter of guess (the Sommerfeld integrals). The talk describes common properties of the plane wave decompositions in various physical situations (2D and 3D Helmholtz equations, 2D discrete lattice equation, WFEM equation for a waveguide,  Laplace-Beltrami equation on a sphere). A starting point of the consideration is introducing of a dispersion manifold of the space, that is a set of all plane waves possibly admitted by the equation. For planar geometries, the plane waves are usual complex plane waves, and for the the sphere the sitiation is a bit more complicated.  In all cases studied in the talk, we assume that the dispersion manifold posseses a structure or complex manifold. Thus, one can study a “wave field” that is an integral over some contour (or, more generally, a cycle) on the diffraction manifold. The integrand is a product of the plave wave, a transformant and a holomorphic differental form. The transformant is assumed to be a meromorphic function of the dispersion manifold. The complex structure on the dispersion manifold enables one to use the Cauchy’s theorem and deform the integration contour if necessary. The next step is introducing of “sliding” contours of integration on the dispersion manifold. Usually, it is impossible to describe the field in the whole domain of interest by a single integral. Thus, the plane wave decomposition comprises a family of contours, by means of which the field is described in overlapping domains covering the whole domain of interest. For consistency, the contours should be deformed into each other for the areas of overlapping.  The representations of Green’s function and of solutions of plane wave diffraction problems are different by the choice of the families of contours. The latter can be described using the Sommerfeld contours that can be localized near infinity, while the former necessarily pass through the finite parts of the dispersion manifold.  The talk is based on common works with O.I.Makarov and K.S.Kniazeva from Moscow State University, and with R.C.Assier, A.I.Korolkov, and V.Kunz from the University of Manchester.

This talk is part of the Isaac Newton Institute Seminar Series series.

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