12 04/11 F

1. In 7.27, acoustics in a pipe with closed ends, assume $\ell=1$, $a=1$, $f(x)=x$, and $g(x)=1$. 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. Continuing the previous problem, in three separate graphs, draw $u(x,0)$, $u(x,0.25)$, and $u(x,0.5)$. For the latter two graphs, also include the separate terms $\frac12F(x-at)$, $\frac12F(x+at)$, and $\int_{x-at}^{x+at}G(\xi) {\rm d}\xi$. Use raster paper or a plotting package. Use the D'Alembert solution only to plot, do not use a separation of variables solution. Comment on the boundary conditions. At which times are they satisfied? At which times are they not meaningful? Consider all times $0\le t<\infty$ and do not approximate.

   Include your code if any.

3. Using the D'Alembert solution of the previous problems, find $u(0.1,3)$.

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

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

6. Continuing the previous homework, write $f=x$ and $g=1$ in terms of the eigenfunctions you found for the case $\ell=1$. Be very careful with one particular eigenfunction. Note that sometimes you need to write a term in a sum or sequence out separately from the others.