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\def\p{\varphi}

\title{ON APLICABILITY OF POINT MODELS OF NON-HOMOGENEITIES IN
FLUID LOADED THIN ELASTIC PLATES}
\author{Ivan V. Andronov}{University of St.Petersburg, Russia}
\cont

\paragraph{Introduction}

The problems of acoustic waves scattering by thin elastic plates or shells with
inhomogeneities are of continuous interest. One of the simpliest problems of
that kind is that of scattering by a fluid loaded plate with point-wise
inhomogeneity (crack, stiffener, etc.) The solution of the corresponding
boundary-contact problem can be found explicitely (see  for examle
\cite{Kouzov1}, \cite{Konov}, \cite{Crighton}) and thus convenient for the
physical analysis.

The flexure vibrations of thin plates are described according to the
Kirchhoff model by the boundary condition containing fourth order tangential
derivatives
\begin{equation}
\left( D \frac{d^4}{dx^4} - \rho \omega^2 h \right) \xi + U = 0 , \qquad
\xi = \frac{1}{\rho_0\omega^2}\ \frac{\partial U}{\partial y}|_{y=0}
\label{andr1}
\end{equation}
Here $\xi$ is the flexural displacement of the plate, $U$ is acoustic
pressure on its surface, $\omega$ is frequency, $D$, $\rho$, $\rho_0$ and $h$
are the parameters of the system: stiffness, densities of plate matherial and
liquid and thickness of the plate. For the uniqueness of the solution special
contact
conditions should be fixed in the point where the uniformity of the plate
is violated. These conditions describe the mechanical behaviour of the
system and are formulated when limit values of $\xi$ and its derivatives are
interpretted as displacements, angles, momentums and forces. However the
reduction of the Lame equations to the condition (\ref{andr1}) is approved
only for infinite homogeneous elastic layer, i.e. in a vicinity of an
edge of the plate the Kirchoff approximation generally saying is not 
applicable.

The work presents an example showing that a crack of finite width
is not described by the point model. That is point model 
reproduce the properties of narrow crack only for cracks of exponentially 
small width with respect to $kh$.

\paragraph{The problem formulation and results}

Consider the two-dimensional system consisting of liquid half-space
$\{y>0\}$ bounded by a plate $\{y=0\}$ with a crack on segment $|x|<a$. Let
a plane wave is incident at angle $\varphi_0$. The solution of the Helmholts
equation in $\{y>0\}$ with the boundary condition (\ref{andr1}) for $|x|>a$ and
\begin{equation}
u(x,0) = 0 , \qquad |x|<a
\label{andr2}
\end{equation}
can be reduced to integral equation of the convolution on an interval $|x|<a$.
This equation and the scattered field are analysed then asymptotically. At 
large distances the scattered field presents a
cylindrical wave with pattern $\Psi(\varphi, \varphi_0)$ and two
surface waves travelling to the left and to the right from the crack.

The main object of analysis is the pattern $\Psi$ \cite{andr}
$$
\Psi(\p,\p_0) = \frac{i}{\pi} \frac{k^2 \sin\p \sin\p_0}{L(\p) L(\p_0)}
 \left\{ \frac{k^4}{D_0} \cos^2\p \cos^2\p_0
   + \frac{k^6}{D_2} \cos^3\p \cos^3\p_0 \right. 
$$
\begin{equation}
\left.
   + \frac{1}{A} \left( \pi L_1(\p)   - \frac{k^2 B}{D_0} \cos^2\p \right)
                 \left( \pi L_1(\p_0) - \frac{k^2 B}{D_0} \cos^2\p_0 \right)
   + \ldots \right\},
\label{andr3}
\end{equation}
where
$$
L_1(\p) = k^4\cos^4\p - k_0^4, \qquad
L(\p) = i k \sin\p L_1(\p) + \nu , \qquad
A = -\pi \left( \log\left(\frac{ka}{2}\right) + \gamma -
\frac{\pi i}{2} - \frac{\nu}{2} \int\limits_{-\infty}^{+\infty}
\frac{d\lambda}{l(\lambda)\sqrt{\lambda^2-k^2}}
 \right) ,
$$
$$
B = \int\limits_0^{+\infty}
    \frac{\lambda^2 d\lambda}{l(\lambda)} , \qquad
D_j = \frac{1}{2\pi} \int\limits_{-\infty}^{+\infty}
\left\{ \frac{k_0^4}{\nu} \sqrt{\lambda^2-k^2} + 1
\right\}
\frac{(i\lambda)^j d\lambda}{l(\lambda)} , \quad j=0,2,
$$
and $\gamma$ is the Euler constant.

The main in $ka$ term of 
the asymptotic expansion \ref{andr3} coincides with the result of scattering 
by the point model \cite{Kouzov1}. The next order term is logarithmically 
smaller. However the coefficients of the main order term and first correction
depend on the other small parameter $kh$. The Kirchhoff model is applicable 
only for thin plates and the coefficients of the main order terms are 
$(kh)^{8/5}$ times smaller than that of correction. Thus the terms reproduced 
in the point model dominate only for exponentially narrow cracks when
\begin{equation}
ka < \exp\left(-(kh)^{-8/5}\right) .
\label{andr4}
\end{equation}

\paragraph{Conclusion}

In the asymptotics (\ref{andr3}) the first two terms of order $(kh)^0$ are
reproduced by point model of crack. However these terms are small in $kh$ 
parameter. The asymptotic analysis for thin plates shows that the point model
is valid only for exponentially narrow cracks when (\ref{andr4}) holds.
Numerical computations show that for ordinary ranges of the acoustic-elastic
system parameters the main effect is also given by the correcting terms in
(\ref{andr3}).

Analogous tests can be done for the other point models. For the case of a
narrow heavy stiffener \cite{andr1} for example it was discovered that the 
point model is valid. The case of a stiffener of finite mass is expected to 
present some intermediate situation. 

\vskip -1cm
\begin{thebibliography}{99}
\bibitem{Kouzov1} {\sc D.P.Kouzov} Diffraction of plane hydroacoustic wave
by a crack in elastic plate. {\it J. Appl. Math. Mech} 1963, v.27, no 6. 
(in Russian)

\bibitem{Konov} {\sc I.P.Konovalyuk, V.N.Krasil'nikov} Influence of
stiffener on the reflection of plane sound wave by thin elastic plate.
{\it Scattering and radiation of waves 4}, Leningrad Univ. Press, 1965.
(in Russian)

\bibitem{Crighton} {\sc D.G.Crighton, D.Innes} Low frequency acoustic radiation
	and vibration response of locally excited fluid-loaded structures
	{J. Sound Vibration} 1983, v.91.

\bibitem{andr} {I.V.Andronov, B.P.Belinskiy, J.P.Dauer} 
	Acoustic Scattering by a narrow crack in elastic plate.
	{\it Wave Motion} 1996. (to appear)

\bibitem{andr1} {I.V.Andronov, B.P.Belinskiy} 
	Acoustic Scattering by a narrow stiffener in elastic plate.
	{\it Wave Motion}. (submitted)
\end{thebibliography}
\end{document}
