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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Plane wave decompositions: general notes - Andrey 
 Shanin (Moscow State University)
DTSTART;TZID=Europe/London:20240701T143000
DTEND;TZID=Europe/London:20240701T150000
UID:TALK214855AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/214855
DESCRIPTION: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 fro
 m the Fourier analysis (this is the case of reprep
 resentation of Green's functions)\, and some are t
 he matter of guess (the Sommerfeld integrals). The
  talk describes common properties of the plane wav
 e decompositions in various physical situations (2
 D and 3D Helmholtz equations\, 2D discrete lattice
  equation\, WFEM equation for a waveguide\, &nbsp\
 ;Laplace-Beltrami equation on a sphere).\nA starti
 ng point of the consideration is introducing of a 
 dispersion manifold of the space\, that is a set o
 f all plane waves possibly admitted by the equatio
 n. For planar geometries\, the plane waves are usu
 al complex plane waves\, and for the the sphere th
 e sitiation is a bit more complicated.&nbsp\;\nIn 
 all cases studied in the talk\, we assume that the
  dispersion manifold posseses a structure or compl
 ex manifold. Thus\, one can study a "wave field" t
 hat is an integral over some contour (or\, more ge
 nerally\, a cycle) on the diffraction manifold. Th
 e integrand is a product of the plave wave\, a tra
 nsformant and a holomorphic differental form. The 
 transformant is assumed to be a meromorphic functi
 on of the dispersion manifold. The complex structu
 re on the dispersion manifold enables one to use t
 he Cauchy's theorem and deform the integration con
 tour if necessary.\nThe next step is introducing o
 f "sliding" contours of integration on the dispers
 ion manifold. Usually\, it is impossible to descri
 be the field in the whole domain of interest by a 
 single integral. Thus\, the plane wave decompositi
 on comprises a family of contours\, by means of wh
 ich the field is described in overlapping domains 
 covering the whole domain of interest. For consist
 ency\, the contours should be deformed into each o
 ther for the areas of overlapping.&nbsp\;\nThe rep
 resentations of Green's function and of solutions 
 of plane wave diffraction problems are different b
 y the choice of the families of contours. The latt
 er can be described using the Sommerfeld contours 
 that can be localized near infinity\, while the fo
 rmer necessarily pass through the finite parts of 
 the dispersion manifold.&nbsp\;\nThe talk is based
  on common works with O.I.Makarov and K.S.Kniazeva
  from Moscow State University\, and with R.C.Assie
 r\, A.I.Korolkov\, and V.Kunz from the University 
 of Manchester.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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