Speed of Propogation in PDE?

Question

Let $u(x,t)$ be a solution to some PDE, then how do we calculate the speed of propagation (I am not even sure what does that term mean). I have some intuitive idea for example if $u(x,t) = f(x-ct)$, then the graph of $u(x,t)$ is same as graph of $u(x-ct,0)$ and hence we have a vague notion of graph moving at a speed of $c$ in a particular direction. But for more complicated cases where the graph not only travels but also changes is shapes (sorry I couldn't think of an example), how does one go about calculating the speed of propagation.