The field of radiation of whispering gallery waves over the concave-convex boundary

The wave field over a reflex surface with a flex point in Kirchhoff approximation is investigated. The incident wave is the whispering gallery wave, propagating from the concave side of the boundary. The short wave asymptotics of Kirchhoff integral describing the radiated wave field is obtained in a neighbourhood of the flex point and in the vicinity of the tangent in this point. The asymptotics is expressed in terms of special functions, which are tabulated. The wave field behavior is illustrated by diagrams.