9 03/21 F

1. 4.17a. (Question (b) was done in class, and the stated condition that $F$ only needs to be continuous is not sufficient, but integrable and continuous would do it.)

2. 4.18. (This is the basic solution for the temperature $u$ in a bar of length 1 where the ends of the bar are kept at zero temperature. Of course, the values of the constants $C_n$ will normally follow from some given initial temperature, and the number of terms in the sum $N$ will normally be infinite.)

3. 4.19. Plane wave solutions are solutions that take the form (2) in solved problem 4.12, with $\vec\alpha$ a constant vector and $\mu$ a constant. This sort of solutions are a multi-dimensional generalization of the $f(x-at)$ moving “wave” solution of the one-dimensional wave equation. In fact, if you take $\vec\alpha$ to be a unit vector, it gives the oblique direction of propagation of the wave and $\mu$ gives the wave propagation speed. However, in this case you will see that the function $F$ cannot be an arbitrary function unless $b=0$.

4. 4.20. A number $T$ is rational if it can be written as the ratio of a pair of integers, e.g. 1.5 = 3/2 = 6/4 = 9/6 = .... It is irrational if it cannot, like $\sqrt{2}$. Near any rational number, irrational numbers can be found infinitely closely nearby, and vice versa. For example, the value $\pi$ to one billion digits, as found on the internet, is the rational number $31415927\ldots/10000000\ldots$; $\pi$ itself is not rational. The wave equation problem when $T=\pi$ has no nonzero solutions, but when $T=\pi$ to 1 billion digits has infinitely many of them. Obviously, in physics it is impossible to determine the final time to infinitely many digits, so there is no physically meaningful solution to the stated problem.

   For nonzero solutions, try $u=\sin(n\pi x)\sin(n\pi t)$, which satisfies the wave equation and the boundary conditions at $x=0$ and $x=1$ and the initial condition at $t=0$. See when it satisfies the end condition at $t=T$.

   This is the boundary value problem for the wave equation, and would be perfectly OK for if it would have been the Laplace equation. (For the Laplace equation, the $\sin(n\pi t)$ becomes $\sinh(n\pi t)$ and only a unique, zero, solution is possible.) The wave equation needs two initial conditions at $t=0$, not one condition at $t=0$ and one at $t=T$.

5. 5.27. In (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! In both problems, include a sketch of the characteristic lines.

6. 5.29. Making a picture of the characteristics and where the “initial” condition is given may help clarify the problem.