Geodesic
Ungtationary modes in a~thin and curved waveguide of variable width
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 210-218

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The propagation of nonstationary modes in a thin waveguide with variable properties is considered and asymptotical formulas for these modes are obtained. It is noted, that for some initial data the overturn-effect of nonstationary waves is possible. 
@article{ZNSL_1979_89_a14,
     author = {I. A. Molotkov and A. B. Plachenov},
     title = {Ungtationary modes in a~thin and curved waveguide of variable width},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {210--218},
     publisher = {mathdoc},
     volume = {89},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_89_a14/}
}
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I. A. Molotkov; A. B. Plachenov. Ungtationary modes in a~thin and curved waveguide of variable width. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 10, Tome 89 (1979), pp. 210-218. http://geodesic.mathdoc.fr/item/ZNSL_1979_89_a14/