# What is the difference between the domain of influence and the domain of dependence?

When analysing the wave equation $$u_{tt} = c^2 u_{xx}$$ in my PDE's module, I understand the 'domain of dependence' which is where the value $u(x_0,t_0)$ is only depends on the initial value of $x$ (at $t=0$) in the closed interval $[x_0 - ct_0, x_0 + ct_0]$ and so it forms a triangle on the $x\, \, t$ plane.

Can someone please explain, intuitively, what the 'domain of influence' is? This is apparently the upside down triangle on the $x \, \, t$ plane starting at point $(x_0, 0)$?

Thanks!

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I think this terminology comes from special relativity. Points in $(x, t)$ plane are called "events" and, since the maximum propagation speed of interactions is finite (and equal $c$), an event can only influence events in that upside down triangle starting at it (which is called light cone). I think this is called principle of causality: try looking for it in Feynman's Lectures on Physics, vol. I. – Giuseppe Negro May 3 '12 at 17:47
As you said, you understand the "domain of dependence", then you can understand the "domain of influence" of point $P$ as the set $S$ of points in the $(x,t)$ plane such that for each $Q\in S$, $P$ is in the "domain of dependence" of $Q$. – Jack May 3 '12 at 18:04