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SUMMARY:Whispering gallery waves near boundary inflection - Shiza Naqvi (U
 niversity of Cambridge)
DTSTART:20220214T150000Z
DTEND:20220214T160000Z
UID:TALK169901@talks.cam.ac.uk
CONTACT:Alistair Hales
DESCRIPTION:The Helmholtz equation typically admits two types of solutions
  of interest depending on the geometry of the domain: modal solutions or s
 cattering solutions. However\, in analogy with the Airy function facilitat
 ing the transition between sinusoidal and oscillatory asymptotic behaviour
 s\, there is not yet an equivalent object for wave solutions transitioning
  from having a discrete to a continuous spectrum. Based on work by Babich\
 , V.M. and Popov\, M.M.\, a boundary with an inflection point models this 
 fundamental problem\, where the concave part of the boundary exhibits whis
 pering gallery modal solutions\, and the convex part exhibiting scattered 
 rays. The big question lies in a neighbourhood of the inflection point\, w
 here asymptotic analysis and Green’s functions methods are used in attem
 pt to construct a uniformly valid expansion on the entire boundary. The bo
 undary value problem in the inflection region is reduced to two Volterra i
 ntegral equations with scope of solution in the form of a convergent Neuma
 nn series. A rigorous review of the whispering gallery asymptotics is pres
 ented as well as a plan for future work.
LOCATION:MR11
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