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\     EML 5060  \hfill Homework Set 4    \hfill     Fall 1998      }\\\hline
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\   Page &  HW      & Class   & Topic                               \\\hline
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\    17  & 2.19cdg  & 2.19e   & Classification                      \\
\    17  & 2.20     &         & Classification                      \\
\    17  & 2.21bc   & 2.21a   & Classification: take $p=p(x,y,z)$!  \\
\    18  & 2.24     & 2.26    & Canonical form                      \\
\    18  & 2.25     &         & Canonical form                      \\
\    18  & 2.22cgh  & 2.22d   & Characteristics                     \\
\    18  & 2.27ab   & 2.27d   & 2D Canonical form                   \\
\    18  & 2.28ajkp & 2.28nml & 2D Canonical form                   \\
\    98  & 7.19     & 7.20    & Unsteady heat conduction in a bar   \\
\    98  & 7.21     & 7.22    & Unsteady heat conduction in a bar   \\
\    98  & 7.24     & 7.25    & Unidirectional viscous flow         \\
\    98  & 7.26     &         & Unidirectional viscous flow         \\
\    99  & 7.28     & 7.27    & Acoustics in a pipe                 \\
\    99  & 7.31     & 7.29    & Vibrations of a string              \\
\    99  & 7.35     & 7.36    & Acoustics above a plate             \\
\    99  & 7.37     & 7.37    & Steady heat conduction in a plate   \\
\    99  & 7.38     & 7.39    & Steady heat conduction in a disk    \\\hline
\multicolumn{4}{|c|}{ Solve the problem below                      }\\\hline
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\begin{enumerate}

\item
Solve the unsteady vibrations of the membrane on a circular drum.
The membrane displacement $u(r,\theta,t)$ satisfies the wave equation
\[ u_{tt} = a^2 \nabla^2 u. \]
The boundary condition at the edge is
\[ u(2,\theta,t)= 0. \]
For initial condition, assume that the membrane is struck at
the point $r=1$ and $\theta=\pi/2$. In other words assume that
$u(r,\theta,0)=0$ and that $u_t(r,\theta,0)$ is a delta
function at $r=1$ and $\theta=\pi/2$. This means that
\[ \int_0^1 \int_0^{2\pi} u_t(r,\theta,0) f(r,\theta)\/
r \/{\rm d} r\/ {\rm d}\theta = f(1,\pi/2) \]
for any function $f(r,\theta)$.
What frequencies will be present in the vibrations, and with
what relative strength?

\end{enumerate}

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