Correctness of the problem of whispering gallery waves in a vicinity of the-points where boundary's curvature vanishes

M. M. Popov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 261-269

Résumé

Two problems of propagation of whispering gallery waves are considered. The first one arises when in some point $s=0$ boundary's curvature $K(s)$ equal zero, but $\frac{dK}{ds}|_{s=0}\ne0$; the second – when $K(0)=\frac{dK}{ds}|_{s=0}=0$, but $\frac{d^2K}{ds^2}|_{s=0}\ne0$ in a point $s=0$. Using technique of the wave operators of the scattering theory it is proved that each problem has one and only one solution.