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{\bf EGN 5456 \hfill Intro to Computational Mechanics \hfill 10/21/96 }\\
{\it Closed book \hfill Van Dommelen \hfill 12:55-2:10 pm }
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

\item
The velocity potential $\phi$ for compressible unsteady
potential flow in a pipe satisfies
\[ \phi_{tt} + 2 \phi_x \phi_{xt} + (\phi_x^2 - a^2) \phi_{xx} = 0\]
Classify this equation.  According to the classification,
what sort of additional conditions would you need to solve this?
For example, would it be appropriate to specify only the velocity
potential at a starting time?  Or the velocity potential at both
a starting and an ending time?  Would you expect singularities in the
solution at say the starting time to smooth out for later times?

\item
The weak formulation for an hypothetical conservation
law equation is:
\[ - \int\int [\phi_t u + \phi_x u^3 ] dx\/dt = 0 \]
for all infinitely smooth $\phi$ that vanish outside a bounded region.
Show that the following function $u(x,t)$ is a weak solution:
\[\begin{array}{ll}
u = 0           & \mbox{ for } x < 0, \\
u = \sqrt{x/3t} & \mbox{ for } 0 < x < 3t, \\
u = 1           & \mbox{ for } 3t < x < 1+t, \\
u = 0           & \mbox{ for } 1+t < x.
\end{array}\]
Is the entropy condition satisfied, i.e.
do the characteristics run into the shock?

\item
For a certain uniform compressible flow in a pipe, small perturbations
$p(x,t)$ in the uniform pressure, $u(x,t)$ in the uniform velocity, and
$s(x,t)$ in the uniform entropy satisfy:
\[
\left(\begin{array}{c} p \\ u \\ s \end{array}\right)_t +
\left(\begin{array}{ccc}1&{\textstyle\frac14}&0\\1&1&0\\0&0&1\end{array}\right)
\left(\begin{array}{c} p \\ u \\ s \end{array}\right)_x =
\left(\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right)\]
Find the perturbations for all times, assuming that the initial
conditions are:
\[ p(x,0)=\sin(x) \quad u(x,0) = \cos(x) \quad s(x,0)= \sin(x+1) .\]
Next suppose that we would have had a finite domain $0 \le x \le 1$, how many
boundary conditions would we have needed at $x=0$? And how many at $x=1$?

\end{enumerate}

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