Diffraction of a High-Frequency Grazing Wave by a Grating With a Complicated Period

A 2D problem of propagation of a plane wave on a branched surface with a periodic set of branch points is studied. The periodic system of branch points plays the role of a diffraction grating. The period of the grating is composed of two branch points. The incident wave falls at a grazing angle with respect to the edge of the grating. The consideration is held in the parabolic approximation. The axis of propagation coincides with the edge of the grating. Edge Green's functions of the problem are introduced. They are wave fields generated by point sources located near the branch points. An embedding formula is proved representing the unknown scattering coefficients in terms of the directivities of the edge Green's functions. A spectral equation is derived for the directivities of the edge Green's functions. This equation is an ordinary differential equation with unknown coefficient. An OE-equation is proposed to find this coefficient. Bibliography: 10 titles. © 2013 Springer Science+Business Media New York.