Geodesic
The field of radiation of whispering gallery waves over the concave-convex boundary
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 11, Tome 104 (1981), pp. 49-65

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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. 
@article{ZNSL_1981_104_a5,
     author = {V. S. Buldyrev and A. I. Lanin},
     title = {The field of radiation of whispering gallery waves over the concave-convex boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {49--65},
     publisher = {mathdoc},
     volume = {104},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_104_a5/}
}
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V. S. Buldyrev; A. I. Lanin. The field of radiation of whispering gallery waves over the concave-convex boundary. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 11, Tome 104 (1981), pp. 49-65. http://geodesic.mathdoc.fr/item/ZNSL_1981_104_a5/