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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 4/23/92 }\\
{\it Closed book \hfill Van Dommelen \hfill 12:30-2:30pm}
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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}

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\item
The wave equation $w_{tt} = a^2 w_{xx}$ can be reduced to a first order
system by defining $u=w_t$ and $v=w_x$:
$$u_t = a^2 v_x$$
$$v_t = u_x$$
Solve this as a system and verify that the same solution is obtained
as we get from a canonical form of the wave equation.

\item
A semi-infinite bar with unit heat conduction coefficient is initially
at zero temperature, but at the end heat is being added at a unit rate:
$${\partial T \over \partial t} =
{\partial^2 T \over \partial x^2}$$
$$T(x,0)=0 \qquad {\partial T \over \partial x} (0,t)=1$$
Find the temperature in the bar for arbitrary times.

\item
A circular plate has an insulated circumference.
The heat conduction coefficient is unity:
$${\partial T \over \partial t} = \nabla^2 T$$
$${\partial T \over \partial r} = 0 \hbox{ at } r=1$$
The initial temperature has an arbitrary variation.
Approximately how long will it take until the initial temperature
variations decay by a factor 10?
Your estimate for this time should be exact under certain conditions.
Which ones?

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