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SUMMARY:Plane wave decomposition for discrete diffraction problems - Andre
 y Korolkov (University of Manchester)
DTSTART:20240702T130000Z
DTEND:20240702T133000Z
UID:TALK214897@talks.cam.ac.uk
DESCRIPTION:One way to solve a diffraction problem is to look for a soluti
 on in the form of a plane wave decomposition integral. Over a century ago 
 this approach was successfully applied by A. Sommerfeld to the problem of 
 diffraction by a half-plane and later extended by Maliuzinets for the wedg
 e problem. In this talk\, we show that this method can also be applied to 
 diffraction problems on discrete lattices. Particularly\, we show that dis
 persion surface for the discrete Helmholtz equation on a grid is topologic
 ally a torus. The plane wave integral is built as an integral over a canon
 ical dissection of the torus with the integrand being a product of a plane
  wave\, a transformant and Abel differential of the first kind. Depending 
 on the point of observation contours of integration slide along the torus.
  &nbsp\;The transformant is supposed to be a meromorphic function over a t
 orus. Then three discrete diffraction problems are considered: (1) the pro
 blem with a point source on an entire plane\; (2) the problem of diffracti
 on by a half-plane\; (3) the problem of diffraction by a right-angled wedg
 e. It is shown that for the first problem the transformant is trivial\, an
 d for the rest two it is built using the theory of algebraic fields of fun
 ctions on Riemann surfaces. An analogy with continuous case and relation t
 o Wiener-Hopf method is discussed.\nThe work is being done in collaboratio
 n with A. V. Shanin.\n&nbsp\;\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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