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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Plane wave decomposition for discrete diffraction
problems - Andrey Korolkov (University of Manchest
er)
DTSTART;TZID=Europe/London:20240702T140000
DTEND;TZID=Europe/London:20240702T143000
UID:TALK214870AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/214870
DESCRIPTION:One way to solve a diffraction problem is to look
for a solution in the form of a plane wave decompo
sition integral. Over a century ago this approach
was successfully applied by A. Sommerfeld to the p
roblem of diffraction by a half-plane and later ex
tended by Maliuzinets for the wedge problem. In th
is talk\, we show that this method can also be app
lied to diffraction problems on discrete lattices.
Particularly\, we show that dispersion surface fo
r the discrete Helmholtz equation on a grid is top
ologically a torus. The plane wave integral is bui
lt as an integral over a canonical dissection of t
he torus with the integrand being a product of a p
lane wave\, a transformant and Abel differential o
f the first kind. Depending on the point of observ
ation contours of integration slide along the toru
s. \;The transformant is supposed to be a mer
omorphic function over a torus. Then three discret
e diffraction problems are considered: (1) the pro
blem with a point source on an entire plane\; (2)
the problem of diffraction by a half-plane\; (3) t
he problem of diffraction by a right-angled wedge.
It is shown that for the first problem the transf
ormant is trivial\, and for the rest two it is bui
lt using the theory of algebraic fields of functio
ns on Riemann surfaces. An analogy with continuous
case and relation to Wiener-Hopf method is discus
sed.\nThe work is being done in collaboration with
A. V. Shanin.\n \;\n \;
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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