Solve the equation $u_{tt} = u_{xx}$ where the initial displacement is $u(0,x) = 1$ for $1<x<2$ and 0 otherwise. Also the initial velocity is 0.
I am not too sure how to go about this. I know d’Alembert’s formula, which tells us that $u(t,x) = \frac{f(x-t) + f(x+t)}{2}$, so the wave separates into two left and right travelling waves that superimpose (I think?).
However I cannot find an explicit formula for $u(t,x)$
Any help is appreciated!
Using the initial condition at $t=0$,
$u(0,x)$ = $\theta(x-1)-\theta(x-2)$
and defining $b(x)=\theta(x-1)-\theta(x-2)$
where $\theta(x)$ is the Heaviside step function,
then $u(t,x)=b(x-t)/2+b(x+t)/2$
which are two 'box' functions traveling right and left.