11 11/13 F

FOLLOW CLASS PROCEDURE IN ALL QUESTIONS.

1. 5.25. Also: (c) Assume that
   
   $$f(x) = e^{-\vert x-2\vert}$$
   
   
   In a single very neat plot, draw $u(x,1)$, $u(x,2)$, and $u(x,3)$ versus $x$. Make sure you draw a complete covering of characteristics in the $x,y$-plane. And show the path of the singularity as a fattened characteristic in the $x,y$-plane.

2. 5.26b. IGNORE THE HINT. Include a very neat sketch of the complete set of characteristic lines. Fatten the asked characteristic in the $x,y$-plane. Simplify your answer as much as possible.

3. 5.27(a). Include a very neat sketch of the complete set of characteristic lines. Is the solution you get valid everywhere?

4. 5.27(b). Do not try to use an initial condition written in terms of two different, related, variables. Get rid of either $x$ or $y$ in the condition. Then call the argument of your undetermined function $s$ and rewrite its expression in terms of $s$. Include a sketch of the complete set of characteristic lines and the initial condition line.

5. 5.29 Explain why there is no solution.

6. 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.

7. 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.

8. 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.