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\begin{document}



\title{DIFFRACTION OF SCALAR WAVES IN THE DOMAIN
WITH THIN MATERIAL COATINGS}
\author{M.A.Lyalinov}{St.Petersburg University, Russia}

The problem of diffraction in an arbitrary two dimensional
compact domain $Q\,=\,Q_0\,\bigcup \,Q_1\,$ $\,\bigcup\,Q_2$ is
studied. The boundary $S_0$ of the domain is coated by a thin
material layer $Q_1\,\bigcup\,Q_2$ which has discontinuity of its
properties on the boundaries $S_0\,$, $S_1\,$, $S_2\,$. It is
assumed that the Dirichlet condition is satisfied on $S_0\,$. The
continuity of the wave  field $U$ on $S_1\,$, $S_2\,$, $q^1\,$,
$q^2\,$ is supposed to be valid. The value ${1\over
\mu_i}\,{\partial U\over \partial n}\,$ is also continuous on
these boundaries, where $\mu_i$ is the material constant of the
medium in $Q_i$, $i\,=\,1,\,2,\,3$, $n$ is the normal unit
vector. The wave field, excited by a compact source $F(\vec{x})$,
is described by the Helmholtz equation with discontinuous
coefficient $\chi$ on  $S_i\,$, $q^i\,$, $i\,=\,1,\,2$.

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The solution of the problem is developed in the form of
asymptotic expansion by the powers of
the small parameter $"h"$ which is the
thickness of the coating. The conception of limit problems is
exploited. The wave field is sought as a sum of the appropriate
solutions of the limit problems. One of these problems determines
the boundary - layer functions in the narrow vicinity of
discontinuities of $q^1$ and $q^2$. The asymptotics of the wave
field is matched by use of the solutions of  limit problems.
The developed asymptotics enables us to compare it with the
solution of  so called boundary - contact problem of
diffraction in the domain $Q_0$. The boundary - contact problem
is the problem for the Helmholtz equation in $Q_0$ and
Generalized Boundary - Contact Conditions (GIBCC) on $\overline
{S_1\,\bigcup\,S_2}\,$. The GIBCCs simulate the thin material
coatings with discontinuities in their coefficients at the points
$A$ and $B$. Some preliminary results of such a comparison will be
discussed.


\end{document}
