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\ {\bf    EML 5060  \hfill Homework Set 4    \hfill     Fall 2002  }\\
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\begin{tabular}{|c|c|c|l|}\hline\hline
\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\
\   Page &  HW      & Class   & Topic                               \\\hline
\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\
\    33  & 3.22ade  &         & intro                               \\
\    35  & 3.38     &         & intro                               \\
\    35  & 3.39     &         & intro (see note 1 below)            \\
\    35  & 3.40     &         & intro (see note 2 below)            \\
\    35  & 3.41     &         & intro (see note 3 below)            \\
\    50  & 4.18     &         & intro                               \\
\    50  & 4.19     &         & intro                               \\
\    50  & 4.20     &         & intro (see note 4 below)            \\
\    17  & 2.19cfg  & 2.19e   & Classification                      \\
\    17  & 2.20     &         & Classification                      \\
\    17  & 2.21ac   & 2.21b   & Classification: assume $u=u(x,y,z[,t])$ \\
\    18  & 2.25     & 2.24    & Canonical form                      \\
\    18  & 2.26     &         & Canonical form                      \\
\    18  & 2.22bf   & 2.22d   & Characteristics                     \\
\    18  & 2.27b    & 2.27d   & 2D Canonical form                   \\
\    18  & 2.28egj  & 2.28nml & 2D Canonical form                   \\
\    98  & 7.20     & 7.19    & Unsteady heat conduction in a bar   \\
\    98  & 7.21     & 7.22    & Unsteady heat conduction in a bar   \\
\    98  & 7.25     & 7.24    & Unidirectional viscous flow         \\
\    99  & 7.27     & 7.28    & Acoustics in a pipe (use two methods)\\
\    99  & 7.35     & 7.36    & Steady supersonic flow              \\
\    99  & 7.37     & 7.37    & Steady heat conduction in a plate   \\
\    99  & 7.39     & 7.38    & Potential flow inside a cylinder    \\
\    00  &          &         & Unsteady heat conduction in a disk  \\\hline
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Note 1:  Guess the solution for $x^2+y^2<1$

Note 2: Use 3.37 with $f$ nonzero only in $\theta_1<\theta<\theta_2$
and see what part of the interior becomes nonzero.

Note 3: Assume $\nabla^2 u=0$ instead of 1.  Then make a physical
argument based on the physical interpretation of steady heat
conduction in a circle with a heat flux 2 entering through the
perimeter.

Note 4: Solutions must be of the form $u=\sin(\alpha\pi x)\sin(\alpha\pi t)$.
See when they satisfy the given conditions.

{\it Also solve the following problem:}

{\em NO working together on the problem below!  If you get stuck ask
  the instructor or TA.}


Solve the 2D unsteady vibrations $u(r,\vartheta,t)$ of a circular
membrane of radius $r_0$ if the membrane is fixed to a nonmoving drum at
its perimeter, i.e. $u(r_0,\vartheta,t)=0$.  Use the separation of
variables (eigenfunction expansion) method.

Initially, the membrane is at rest, but then at time $t=0$ it is hit
by a drum stick a distance $\frac13 r_0$ away from the center.  You may
assume that the initial displacement $u(r,\vartheta,0)$ of the membrane
is still zero, but that the initial velocity is a delta function
positioned at $r=\frac13 r_0$ and $\vartheta=0$:
$u_t(r,\vartheta,0)=\delta(r-\frac13 r_0) \delta(\vartheta)$.

To solve the problem, you are required to answer the following
questions in the order asked (list question number with your answer):
\begin{enumerate}
\item The governing P.D.E.~is the two-dimensional wave equation
  $u_{tt} = a^2 \nabla^2 u$ where $a$ is the given wave propagation speed.
  Write this equation out in polar coordinates.
\item Identify the spatial domain of the problem.
\item Identify the boundary conditions.
\item Identify the initial conditions.
\item You will need two eigenfunction expansions to reduce this P.D.E.
  in three variables, $r$, $\vartheta$, and $t$, into ordinary
  differential equations with respect to $t$ only.  These
  expansions are similar to those of the heat conduction problem
  covered in class, but {\em not} the same.
\item Write down the Sturm-Liouville problem for $\Theta(\vartheta)$
  completely, including the boundary conditions by substituting a term
  of the form $T(r,t)\Theta(\vartheta)$ into the homogeneous PDE.
  Solve it, or find a place in the lecture notes where that problem
  has been solved before (same problem with same boundary conditions.)
\item Find the transformation formulae that for an arbitrary function
  $F(r,\vartheta,t)$ produce the Fourier coefficients $F^i_n(r,t)$ of
  that function in the eigenfunction expansion:
  \begin{displaymath}
    F(r,\vartheta,t)=\sum_{n,i} F^i_n(r,t) \Theta^i_n(\vartheta)
  \end{displaymath}
\item Now that an eigenfunction expansion in the $\vartheta$
  coordinate has been found, find a subexpansion in the $r$ coordinate
  by substituting a single term of the form $T^i_n(t) R^i_n(r)$ into
  the equation for the $T^i_n(r,t)$.  Make $R^i_n$ satisfy the right
  boundary conditions.  Where is the Sturm Liouville problem different
  from the problem solved in class?  How does that affect the
  solution?
\item Find the transformation formulae that for an arbitrary function
  $F(r,\vartheta,t)$ with first Fourier coefficients $F^i_n(r,t)$ produces
  the second Fourier coefficients $F^i_{nm}(t)$ of the double eigenfunction
  expansion:
  \begin{displaymath}
    F(r,\vartheta,t)=\sum_{n,i} F^i_n(r,t) \Theta^i_n(\vartheta) =
    \sum_{n,i} \sum_m F^i_{nm}(t) R^i_{nm}(y) \Theta^i_n(\vartheta) =
  \end{displaymath}
\item Find the net transformation formulae that for an arbitrary
  function $F(r,\vartheta,t)$ produce the Fourier coefficients
  $F^i_{nm}(t)$ of that function in the eigenfunction expansion above
  directly.  (In terms of a multiple integral.)  Where is it different
  from the heat conduction problem?
\item Expand everything in the PDE in terms of the double
  eigenfunction expansions obtained and solve the resulting O.D.E.
  for the $u^i_{nm}$.
\item Expand everything in the IC in terms of the double eigenfunction
  expansions obtained.  While finding the Fourier coefficients of the
  given initial conditions involves double integrals, they can
  be found by noting that for Dirac delta functions the following is
  true:
  \begin{displaymath}
    \int\int \mbox{any-integrand}(r,\vartheta)
    \delta(r-{\textstyle \frac13 r_0})\delta(\vartheta)\;
    {\rm d}r {\rm d}\vartheta \equiv
    \mbox{any-integrand}({\textstyle \frac13 r_0},0)
  \end{displaymath}
\item Work everything out as far as possible.
\item What is the lowest frequency produced by the drum? What is its
  amplitude?
\end{enumerate}

\begin{rawhtml}
<a href="7q21/">7.21 solution</a>  
<a href="7q37/">7.37 solution</a>  
<a href="7q39/">7.39 solution</a>  
\end{rawhtml}

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