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\begin{document}



\title{ON NATURAL FREQUENCIES OF THE RECTANGULAR ACOUSTIC RESONATOR
WITH PIECEWISE-HOMOGENEOUS ELASTIC WALL}
\author{Yu.A.Lavrov}
{St.Petersburg Institute of Economics and Commerce,\\
Novorossiyskaya, 50, St.Petersburg, 194018, Russia}
\coauthor{V.D.Lukyanov}
{St.Petersburg Military Engineering Building Institute\\
Zacharievskaya, 22, St.Petersburg, 191194, Russia}

\noindent
The flat rectangular resonator
is assumed to be filled with an inviscid, compressible,
and nonrotational fluid.
Acoustic pressure in a fluid obey the homogeneous Helmholtz equation.
One wall consists of two conjoint thin elastic walls with different
mechanical properties.
The bending plates displacements satisfy Kirchhoff equation.
The edges of plates are clamped.
Three other walls of acoustic resonator are absolutely rigid.
The Neumann condition should be fulfilled on this part of the bound.

     Acoustic pressure in the resonator is searched in terms of the
pulse forms for the resonator with three rigid walls.
For the coefficients of this decomposition
one can obtain an infinite linear set of algebraic equations.
The matrix form of this system is
$$(I+V)X=0\>,$$
where $I$ is identity operator,
$V$ is completely continuous operator.
Condition of existence of nontrivial solution of the system
$${\rm det}(I+V)=0$$
is an equation for natural frequencies.
This equation was investigated numerically.
The dependencies of natural frequencies
on geometrical and mechanical parameters were studied.

\end{document}
