Geodesic
Diffraction of a high-frequency wave by a grating with a complicated period in the case of grazing incidence
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 212-239 
A. V. Shanin. Diffraction of a high-frequency wave by a grating with a complicated period in the case of grazing incidence. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 212-239. http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a12/
@article{ZNSL_2012_409_a12,
     author = {A. V. Shanin},
     title = {Diffraction of a~high-frequency wave by a~grating with a~complicated period in the case of grazing incidence},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {212--239},
     year = {2012},
     volume = {409},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a12/}
}
TY  - JOUR
AU  - A. V. Shanin
TI  - Diffraction of a high-frequency wave by a grating with a complicated period in the case of grazing incidence
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2012
SP  - 212
EP  - 239
VL  - 409
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a12/
LA  - ru
ID  - ZNSL_2012_409_a12
ER  - 
%0 Journal Article
%A A. V. Shanin
%T Diffraction of a high-frequency wave by a grating with a complicated period in the case of grazing incidence
%J Zapiski Nauchnykh Seminarov POMI
%D 2012
%P 212-239
%V 409
%U http://geodesic.mathdoc.fr/item/ZNSL_2012_409_a12/
%G ru
%F ZNSL_2012_409_a12

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A 2D problem of diffraction of a plane wave on a branched surface is studied. The configuration of branch points of the surface is periodic; these branch points play the role if a diffraction grating. The period of the grating is composed of two branch points. The incident wave travels at a grazing incidence angle. The consideration is held in the parabolic approximation; the axis of the parabolic coordinates is the edge of the grating. Edge Green's functions of the problem, i.e. the the fields generated by point sources placed near the branch points, are introduced. The embedding formula is proven. It expresses the coefficients of generation of diffraction orders in terms of the directivities of the edge Green's functions. A spetral equation is derived for the directivities of the edge Green's functions. This is an ordinary differential eqiation, the coefficient of which is unknown. Finally, for finding of this coefficient, an OE-equation is derived.

[1] E. D. Shabalina, N. V. Shirgina, A. V. Shanin, “High frequency modes in a two dimensional rectangular room with windows”, Acoust. Phys., 56 (2010), 525–536 | DOI

[2] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[3] L. A. Vainshtein, Teoriya difraktsii i metod faktorizatsii, Sov. radio, M., 1966

[4] B. Nobl, Primenenie metoda Vinera–Khopfa dlya resheniya differentsialnykh uravnenii v chastnykh proizvodnykh, Nauka, M., 1962 | MR

[5] A. V. Shanin, “Weinstein's diffraction problem: embedding formula and spectral equation in parabolic approximation”, SIAM J. Appl. Math., 70 (2009), 1201–1218 | DOI | MR | Zbl

[6] V. V. Zalipaev, M. M. Popov, “Korotkovolnovoe skolzyaschee rasseyanie ploskoi volny na gladkoi periodicheskoi granitse. I. Difraktsiya polutenevogo polya na gladkom vypuklom konture”, Zap. nauchn. semin. LOMI, 165, 1987, 59–90 | MR | Zbl

[7] V. V. Zalipaev, M. M. Popov, “Korotkovolnovoe skolzyaschee rasseyanie ploskoi volny na gladkoi periodicheskoi granitse. II. Difraktsiya na beskonechnoi periodicheskoi granitse”, Zap. nauchn. semin. LOMI, 173, 1988, 60–86 | MR | Zbl

[8] V. V. Zalipaev, “Shortwave scattering by a diffraction echelette-grating”, J. Math. Sci., 102 (2000), 4203–4219 | DOI | MR

[9] A. A. Anis, E. H. Lloyd, “On the range of partial sums of a finite number of independent normal variates”, Biometrika, 40 (1953), 35–42 | DOI | MR | Zbl

[10] J. Boersma, “On certain multiple integrals occuring in a waveguide scattering problem”, SIAM J. Math. Anal., 9 (1978), 377–393 | DOI | MR | Zbl