10 03/28 F

1. In 7.27, acoustics in a pipe with closed ends, graphically identify the extensions $F(x)$ and $G(x)$ of the given $f(x)$ and $g(x)$ to all $x$ that allow the solution $u$ to be written in terms of the infinite pipe D'Alembert solution.

2. Using the solution of the previous problem, and taking $\ell=1$, $a=1$, $f(x)=x$, and $g(x)=1$, draw $u(x,0)$ as a function of $x$ between say $x=-6$ and 6. Use raster paper or equivalent and a ruler, with 4 raster cells per unit length. Then draw $u(x,0.5)$, by first drawing ${\textstyle\frac{1}{2}}F(x-at)$, ${\textstyle\frac{1}{2}}F(x+at)$, and $\int_{x-at}^{x+at}G(\xi){\rm {d}}\xi$, and then $u$ from that. If you are unable to draw a neat and clear graph, use a plotting package to do it. Note that the boundary conditions are now satisfied, though the initial condition did not.

3. Repeat, but now draw $u(x,0.25)$. Are the boundary conditions still satisfied? Does that mean they will be satisfied for all nonzero times?

4. Using the solution of the previous problems, find $u(0.25,3)$.

5. Write the complete (Sturm-Liouville) eigenvalue problem for the eigenfunctions of 7.27.

6. Find the eigenfunctions of that problem. Make very sure you do not miss one. Write a single symbolic expression for the eigenfunctions in terms of an index, and identify all the values that index takes.

7. Write $f=x$ and $g=1$ in terms of these eigenfunctions for the case $\ell=1$. Be very careful with one particular eigenfunction.

8. Substitute $u(x,t)=\sum_nu_n(t)X_n(x)$ into the PDE to convert it into an ordinary differential for each separate coefficient $u_n(t)$.