Fourier and PDEs: Problem Set 7

1. Domain of dependence of the wave equation

Consider the wave equation $y_{tt} = c^2 y_{xx}$ with initial conditions $y(x,0) = 0$ and $y_t(x,0)$ on the entire real line.

a) Show that the solution is \[ y(x, t) = \frac{1}{2c} \int_{x-ct}^{x+ct} v(s) \, \de{s}. \]

b) If \[ v(x) = \begin{cases} cx/a & a < x < 2a \\ -cx/a & -2a < x < -a \\ \text{undefined} & \text{otherwise} \end{cases} \]

give a clear sketch of the part (or parts) of the $(x, t)$ plane where $y(x, t)$ can be determined, and obtain an expression for $y(x, t)$ in each such region.


2. Axisymmetric solutions of the wave equation

Let $r$ be the distance from the origin in $(x,y,z)$-space, and let $u(r,t)$ be the solution of the spherically symmetric wave equation \[ u_{tt} = c^2 u_{rr} + \frac{2}{r} u_r, \qquad r > 0, \]

which satisfies the initial conditions $u(r,0) = 0$ and $u_t(r,0) = e^{-r^2}$ for $r > 0$.

a) Let $w(r,t) = ru(r,t)$. Show that $w$ is a solution of the wave equation \[ w_{tt} = c^2 w_{rr}, \qquad r > 0 \]

for which $w(r,0) = 0$ and $w_t(r,0) = re^{-r^2}$ for $r > 0$ and $w(0, t) = 0$ for $t \geq 0$.

b) Find $w(r,t)$ and show that \[ u(r,t) = \frac{\text{sinh}(2crt)}{2cr} e^{-(r^2 + c^2 t^2)}. \]

c) What is the value of the limit $\lim_{r\to 0} u(r,t)$?


3. Laplace's equation in a square

Let $T(x,y)$ satisfy Laplace's equation in the square $0 < x < \pi$ and $0 < y < \pi$. Let $T = 1$ on the side $y = \pi$, $0 < x < \pi$, and let $T = 0$ on the other three sides.

a) Show that \[ T(x, t) = \frac{4}{\pi} \sum_{n=0}^\infty \frac{\text{sinh}[(2n + 1)y]}{\text{sinh}[(2n + 1)\pi]} \frac{\sin[(2n +1)x]}{(2n+1)}. \]


b) By considering three similar problems, show that $T = 1/4$ at the centre of the square.


4. Axisymmetric Laplace's equation

a) (i) If $u(x, y)$ is a twice-differentiable function of the Cartesian coordinates $x$ and $y$, and if $v(r, \theta) = u(r\cos\theta, r\sin\theta)$, where $r$ and $\theta$ are plane polar coordinates, show that \[ u_{xx} + u_{yy} = v_{rr} + \frac{1}{r} v_r + \frac{1}{r^2} v_{\theta\theta} \]


(ii) Hence show that the functions \[ 1 \qquad \log r \qquad r^n\cos(n\theta) \qquad r^{-n}\cos(n\theta), \]

where $n$ is a positive integer, are all solutions of Laplace's equation.


b) A conductor occupies the region $r \geq a$, i.e. exterior to the disk whose centre is the origin and whose radius is $a$. If the steady state temperature, $T(r, \theta)$, satisfies the boundary condition $T(a, \theta) = \cos^2 \theta$ for $0 \leq \theta \leq 2\pi$ and $T$ remains bounded as $r \to \infty$, show that \[ T(r,\theta) = \frac{1}{2} + \frac{a^2}{2r^2} \cos(2\theta). \]