4 7.28, §4 Solution

$\begin{displaymath} \hbox{\epsffile{figures/8.ps}} \end{displaymath}$

$\begin{displaymath} u(x,t) = \frac{\bar f(x-at) + \bar f(x+at)}{2} + \frac{1}{2a} \int_{x-at}^{x+at} \bar g(\xi) \d\xi \end{displaymath}$

Probably pretty easy to evaluate.

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In the range $0\le x\le\ell$ , the found solution is exactly the same as for the finite pipe!

Note that if and/or does not satisfy the given boundary conditions, $\bar f$ and $\bar g$ may have kinks or jumps.