There's probably something really obvious I should be getting, but I haven't yet developed the intuition for working with the wave equation.
Suppose we're given the wave equation $u_{tt} = c^{2} u_{xx}$ with $(x,t) \in \mathbb{R} \times \mathbb{R}$, and $c > 0$ constant, subject to the initial conditions that $u_{t}(x,0) = 0$ and $u(x,0) = h$ provided $|x| < a$ and $0$ otherwise.
The solution to this initial value problem for the wave equation is given by d'Alembert's formula, $$u(x,t) = \frac{1}{2}\left(u(x+ct,0) + u(x-ct,0) \right)$$
So there is a left-moving and a right-moving part. Apparently, these waves separate when $t = \frac{a}{c}$, but I don't see how to think about this to be able to deduce when the waves separate. I want to have a bit more intuition so that I can know how to think about this for other examples.
The initial value $h$ is supported on the interval $[-a,a]$. It splits into two equal parts: $h_l=h_r=h/2$. $h_l$ starts traveling towards the left with speed $c$, while $h_r$ travels towards the right with speed also $c$. They separate when the rightmost extreme of the support of the left traveling wave reaches $0$, what happens at the same time that the leftmost extreme of the support of the right traveling wave reaches $0$. The distance travelled by each wave is $a$ and the speed is $c$, so the time is $a/c$.
Another way to see this is to calculate the support of both waves. The support (as a function of $x$) of $h_l(x+c\,t)$ is $[-a-c\,t,a-c\,t]$, and the support (as a function of $x$) of $h_r(x+c\,t)$ is $[-a+c\,t,a+c\,t]$. They become disjoint when $a-c\,t=-a+c\,t$.