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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/13/95 }\\
{\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.

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\item
(33 points).
Consider steady supersonic flow at a Mach number 3.33 under a 45 degree
angle with the $x$-axis.  Small changes in the horizontal velocity
component $u(x,y)$ satisfy
$$4 u_{xx} + 10 u_{xy} + 4 u_{yy} = 0.$$
Find new coordinates $\xi$ and $\eta$ such that the equation satisfied
by $u(\xi,\eta)$ takes the form
$$u_{\xi\eta} = 0.$$
Use this result to show that the solution for $u$ must be of the form
$$u(x,y) = f(2x-y)+g(2y-x),$$
where $f$ and $g$ are arbitrary functions.
\begin{rawhtml}
  <a href="figures/1.gif">Solution</a>
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\item
(33 points).
Solve heat conduction in a semi-infinite bar which is initially
at zero temperature. Heat is removed through the end at a
rate which increases linearly in time:
$$T_t = T_{xx} \qquad T(x,0)=0 \qquad T_x(0,t) = t.$$
Note: you will probably not succeed in simplifying the
final result for $u$.
\begin{rawhtml}
  <a href="figures/2.gif">Solution</a>
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\item
(34 points).
Solve steady heat conduction in a quarter circle.
The straight sides are insulated:
$$ \nabla^2 T = 0 \qquad T_\theta(r,0) = T_\theta(r,\pi/2) = 0
\qquad T(1,\theta)= f(\theta).$$
Write the solution in terms of $f(\theta)$.
Also, if $f(\theta)=\cos(4\theta)$, what is $u$?
\begin{rawhtml}
  <a href="figures/3.gif">Solution a</a>
  <a href="figures/3b.gif">Solution b</a>
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