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CATEGORIES:Waves Group (DAMTP)
SUMMARY:Whispering gallery waves near boundary inflection
- Shiza Naqvi (University of Cambridge)
DTSTART;TZID=Europe/London:20220131T150000
DTEND;TZID=Europe/London:20220131T160000
UID:TALK168392AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/168392
DESCRIPTION:The Helmholtz equation typically admits two types
of solutions of interest depending on the geometry
of the domain: modal solutions or scattering solu
tions. However\, in analogy with the Airy function
facilitating the transition between sinusoidal an
d oscillatory asymptotic behaviours\, there is not
yet an equivalent object for wave solutions trans
itioning from having a discrete to a continuous sp
ectrum. 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 whispering gallery modal
solutions\, and the convex part exhibiting scatte
red rays. The big question lies in a neighbourhood
of the inflection point\, where asymptotic analys
is and Greenâ€™s functions methods are used in attem
pt to construct a uniformly valid expansion on the
entire boundary. The boundary value problem in th
e inflection region is reduced to two Volterra int
egral equations with scope of solution in the form
of a convergent Neumann series. A rigorous review
of the whispering gallery asymptotics is presente
d as well as a plan for future work.
LOCATION:CMS\, MR11
CONTACT:Alistair Hales
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