# Paper 2, Section II, A

By means of the change of variables $\eta=x-t$ and $\xi=x+t$, show that the wave equation for $u=u(x, t)$

$\frac{\partial^{2} u}{\partial x^{2}}-\frac{\partial^{2} u}{\partial t^{2}}=0$

is equivalent to the equation

$\frac{\partial^{2} U}{\partial \eta \partial \xi}=0$

where $U(\eta, \xi)=u(x, t)$. Hence show that the solution to $(*)$ on $x \in \mathbf{R}$ and $t>0$, subject to the initial conditions

$u(x, 0)=f(x), \quad \frac{\partial u}{\partial t}(x, 0)=g(x)$

$u(x, t)=\frac{1}{2}[f(x-t)+f(x+t)]+\frac{1}{2} \int_{x-t}^{x+t} g(y) \mathrm{d} y$

Deduce that if $f(x)=0$ and $g(x)=0$ on the interval $\left|x-x_{0}\right|>r$ then $u(x, t)=0$ on $\left|x-x_{0}\right|>r+t$.

Suppose now that $y=y(x, t)$ is a solution to the wave equation $(*)$ on the finite interval $0 and obeys the boundary conditions

$y(0, t)=y(L, t)=0$

for all $t$. The energy is defined by

$E(t)=\frac{1}{2} \int_{0}^{L}\left[\left(\frac{\partial y}{\partial x}\right)^{2}+\left(\frac{\partial y}{\partial t}\right)^{2}\right] \mathrm{d} x$

By considering $\mathrm{d} E / \mathrm{d} t$, or otherwise, show that the energy remains constant in time.