Geodesic
Asymptotics of the radiating plane wave coefficients in the scattering problem on a smooth periodic boundary
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 20, Tome 186 (1990), pp. 71-86

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Scattering problem of a plane wave on a smooth periodic curve in the case of small grazing angle and shortwave approximation is considered (period and radius of curvature of the curve are supposed to be large in compare with wave length). Using the solution previously obtained as an infinite sum of multiple diffracted fields in the vicinity of reflecting boundary the shortwave asymptotics of the coefficients of radiating plane waves is derived. 
@article{ZNSL_1990_186_a6,
     author = {V. V. Zalipaev},
     title = {Asymptotics of the radiating plane wave coefficients in the scattering problem on a smooth periodic boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {71--86},
     publisher = {mathdoc},
     volume = {186},
     year = {1990},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1990_186_a6/}
}
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V. V. Zalipaev. Asymptotics of the radiating plane wave coefficients in the scattering problem on a smooth periodic boundary. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 20, Tome 186 (1990), pp. 71-86. http://geodesic.mathdoc.fr/item/ZNSL_1990_186_a6/