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\Large \bf
Numerieke methoden in de natuurwetenschappen\\
\large 
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\noindent 
{\bf Ex.~6: The vibrating string: solution with finite differences and finite elements. }  
Points a) and b) are compulsory, point c) is facultative.

Consider a vibrating piano string fixed at both ends ($x=0$ and $x=L=$1~m) 
and under
a  uniform tension $T=10^3$~N. We wish to find the allowed frequency of 
vibrations 
$\omega$ and the corresponding displacements by solving the equation:
\begin{equation}
{d^2y(x) \over dx^2} + {\omega^2 \mu(x) \over T} y(x) = 0
\end{equation}
where $\mu(x)$ is the mass of the string per unit length. We take first
a constant mass density $\mu_0=0.954$~g/m, and then 
$\mu(x)=\mu_0+(x-{L \over 2})\Delta$ with $\Delta=$0.5~g/m$^2$. In the first 
case the solutions are known and correspond to a sinusoidal motion with 
frequency
\begin{equation}
\omega={n\pi \over L} \sqrt{T/\mu}~~~~~~~~n=1,2,...
\end{equation}
Note that this problem could be also solved by the shooting technique.
\begin{description}
\item {a)} Solve the problem by the finite difference method to find the first 
two eigenvalues and eigenvectors for both constant and variable mass density.
Write the equations onto a grid of 11 and 21 points from 0 to L and compare the results. Find the eigenvalues by searching for the zeroes of the determinant
of the resulting tridiagonal matrix and the eigenvectors by backsubstitution.

\item {b)} Find the lowest eigenvalue (the fundamental note of the piano)
by the inverse power method. 

\item {c)} Solve the same problem by the finite elements method. This method 
implys finding the eigenvalues of a generalized eigenvalue problem of the type 
{\bf A}$x=\lambda${\bf B}$x$ which can be solved by use of the NAG libraries.
\end{description}
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