On completeness of asymptotic formulae for amplitudes of the plane waves diffracted by a smooth periodic boundary

V. V. Zalipaev   ; M. M. Popov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 21, Tome 195 (1991), pp. 40-47

Résumé

It is almost evident that in the shortwave approximation various parts of a smooth periodic boundary are responsible for the amplitudes of appropriate plane waves diffracted by the boundary. (It follows from the Green's formula and the stationary phase method.) It is clear also that under the grazing incidence only top part of the boundary hills is illuminated while the main part of it turns out to be in a deep shadow. In the previous paper [3] two different types of asymptotic formulae for the diffracted wave amplitudes have been obtained. One of it is originated by the Fock's region near the top of boundary hill. The other type is originated by illuminated part of the boundary arc adjoining to the Fock's region. Different mathematics has been used to derive the formulae. It is shown in this paper that both type of the formulae are overlaped in some intermediate region of the diffraction agles. Thus these formulae describe complete asymptotics of diffracted wave amplitudes except those which are originated by shadow part of the boundary and therefore are small in the shortwave approximation.