Geodesic
Wave field in the caustic region in a~vicinity of a~boundary flex point
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 246-260

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Propagation of whispering gallery waves in a vicinity of a boundary flex point is studied. It is shown, that shadow region arises on some distance from the boundary. Asymptotics of the wave field in this region is obtained. 
@article{ZNSL_1979_89_a17,
     author = {M. M. Popov},
     title = {Wave field in the caustic region in a~vicinity of a~boundary flex point},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {246--260},
     publisher = {mathdoc},
     volume = {89},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_89_a17/}
}
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M. M. Popov. Wave field in the caustic region in a~vicinity of a~boundary flex point. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 246-260. http://geodesic.mathdoc.fr/item/ZNSL_1979_89_a17/