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\title{WOOD'S ANOMALIES AND CONSTRUCTION OF RAYLEIGH-BLOCH SURFACE WAVES}
\author{Ilya V. Kamotski, Sergey A. Nazarov}
{Chair of Mathematical Physics \\
Department of Mathematics and Mechanics \\
St.Petersburg University \\
194295, St.-Petersburg, Russia, pr. Khudojnikov 18-1-100}


The problems of scattering of  elastic, electromagnetic, and acoustic
plane waves by periodical gratings are considered. It is well-known from
experimental works by Wood that for certain wave propagation angles the
small perturbation (of the angle leads to a considerable perturbation of
the scattered  field.  We derive the asymptotics  of the scattering
matrixes in power of ${\vert\varepsilon\vert}^{1/2}$ as
${\varepsilon\to\pm}0$.  This is the reason for the above
mentioned perturbation of all the scattered field characteristics.
This fact appears to be connected with the problem of existence of
the Rayleigh-Bloch surface waves .By using of these asymptotics, the
Rayleigh-Bloch surface waves can be constructed in a special cases. We
have found such waves in the plane with periodically families  of small
holes with the Neumann conditions (it is known from the works of Rellich,
Wilcox, Alber that if the boundary is defined globally as graph of smooth
function then there are no surface waves in the case of Dirichlet boundary
condition). Both the singularities of  derivatives in $\varepsilon$
of  the scattered
field characteristics and the existence of surface waves may be regarded
as a reason for Wood's anomalies.

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