11 HW 11

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 four separate graphs, draw $u(x,0)$, $u(x,0.25)$, $u(x,0.5)$, and $u(x,1.25)$. For all but the first graph, also include the separate terms $\frac12F(x-at)$, $\frac12F(x+at)$, and $\int_{x-at}^{x+at}G(\xi) {\rm d}\xi$. Use graph or raster paper or a plotting package. Use the D'Alembert solution only to plot, do not use a separation of variables solution in your software package. 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.

   Make sure to include your source code if any.

3. Using the D'Alembert solution of the previous problems, find $u(0.1,3)$. Be sure to show the value of each term in the expression.