11 04/03 F

1. 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! Include a sketch of the characteristic lines.

2. Solve the Burgers’ equation, as given in 5.34, using the method of characteristics. Give two different forms of the solution, one in which you set $C_1=C_1(C_2)$ and another in which you set $C_2=C_2(C_1)$ and then re-express that function argument in terms of the physical variables. Check that the solutions $v$ and $w$ are in each of the seven regions given by at least one of these two forms, and identify the corresponding function $C_1$ or $C_2$. If a function $C_1$ or $C_2$ does not exist, show why not.

3. Continuing 5.34, in two very neat $xt$-planes on raster paper, draw the given two solutions $v$ and $w$ up to time $t=3$. In particular, draw the two shocks (jump discontinuities) $x=\frac12t$ and $x=1+\frac12t$ in the $xt$ plane for $v$ as fat lines. Similarly draw the single shock in the $xt$-plane for $w$; this shock is the straight line $x=1+\frac12t$ below $t=2$, but starts curving upward a bit above $t=2$ as the expansion fan hits it. Draw the characteristic lines as thinner lines. Make sure you draw enough characteristics, especially inside the expansion fan of $w$.

4. Show that if the Burgers’ equation as written gives the physically correct conservation law, then the conserved quantity is $\int{u}\;{\rm d}x$ and its flux is $f=\frac12u^2$. Check that the characteristic speed is indeed $f'$. Then check (for $t\le2$) that all three shocks satisfy the jump relations:
   
   $$\frac{{\rm d}x_s}{{\rm d}t} = \frac{f_2-f_1}{u_2-u_1}$$
   
   
   where $x_s$ is the location of the shock and 2 stands for the point immediately behind the shock at a given time and 1 to the point immediately before it. Check (for $t\le2$) that two shocks satisfy the entropy condition
   
   $$f'_1 > \frac{f_2-f_1}{u_2-u_1} > f'_2$$
   
   
   but that the first shock of $v$ does not. Check from your graph that indeed the characteristics come out of this shock, rather than run into it. Conclude that solution $v$ has an unphysical expansion shock and that the correct solution is $w$.

5. 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.

6. 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. 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.

7. Using the solution of the previous problems, find $u(0.1,3)$.