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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 12/11/98}\\
{\it Closed book \hfill Van Dommelen \hfill 10:00-12:00am}
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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, as well as a standard size handwritten formula sheet.

\begin{enumerate}

\item (25\%)
Classify the following Partial Differential Equation:
\begin{displaymath}
    u_{xx} + 4 u_{xy} + 3 u_{yy} = 0
\end{displaymath}
Put it in characteristic coordinates and solve it.  Rewrite the
solution in terms of the original coordinates $x$ and $y$.
\begin{rawhtml}
<A HREF="figures/1.gif">Solution.</A>
\end{rawhtml}

\item (25\%)
A pipe that is closed at both ends contains air.  Initially the air
and pipe are at rest. Then, at time zero, the pipe is impulsively
given a unit velocity along its axis.  Find the velocity $u(x,t)$ of
the air in the pipe.  From fluid mechanics, it may be derived that
this velocity satisfies the following Partial Differential Equation:
\begin{displaymath}
    u_{tt} = a^2 u_{xx},
\end{displaymath}
where $a$ is a known constant (the speed of sound.)
For this problem, the initial and boundary conditions are:
\begin{displaymath}
    u(x,0)=u_t(x,0)=0, \qquad u(0,t)=u(\ell,t)=1,
\end{displaymath}
where $\ell$ is the length of the pipe.  Give a set of suitable
functions to expand the solution $u(x,t)$ in.  Then give the solution
$u(x,t)$ in terms of these functions.
\begin{rawhtml}
<A HREF="figures/2a.gif">Solution part 1.</A>
<A HREF="figures/2b.gif">Solution part 2.</A>
\end{rawhtml}

\item (25\%)
One end of a very long bar is suddenly, at time zero, raised
to a unit temperature.  The further evolution of the temperature
distribution $u(x,t)$ in the bar is described by the following
Partial Differential Equation:
\begin{displaymath}
    u_t = \kappa u_{xx}, \quad (x\ge 0;\ t\ge 0),
\end{displaymath}
where $\kappa$ is the conduction coefficient.  The initial and
boundary conditions are:
\begin{displaymath}
    u(x,0)= 0, \qquad u(0,t) = 1.
\end{displaymath}
Find the solution $u(x,t)$ using a Laplace transform.  From it, show that
the temperature is constant on points that move away from the end
as $x =$ (constant) $\sqrt t$.
\begin{rawhtml}
<A HREF="figures/3.gif">Solution.</A>
\end{rawhtml}

\item (25\%)
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?
\begin{rawhtml}
<A HREF="figures/4.gif">Solution.</A>
\end{rawhtml}

\end{enumerate}

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