2 7.28, §2 P.D.E. Model

Always draw the depend-variables picture first.

**Figure 2:** Dependent variables.
$\begin{figure}\begin{center} \leavevmode \epsffile{figures/2.ps} \end{center}\end{figure}$

Observations:

Unknown pressure .
Constant speed of sound .
Finite domain $\bar \Omega$ : $0\le x \le \ell$ .
Hyperbolic PDE (waves, singularities, evolution).
Two initial conditions.
One homogeneous Neumann boundary condition at and one homogeneous Dirichlet condition at $x=\ell$ .

We will try to find a solution of this problem in the form

$\begin{displaymath} u = \sum_n u_n(t) X_n(x) \end{displaymath}$

i.e. a sum, in which each term is a product of a function of

only times a function of

only.