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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\, \
;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. \;\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. \;\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. \;\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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