the domain of dependence
Suppose you want to know the height of a wave at a particular spot x at a particular future time t. Which part of the starting data could possibly matter? Common sense says: only the part close enough that a signal travelling at speed c had time to arrive. The exact set of starting points that can affect (x,t) is called its domain of dependence.
For the one-dimensional wave equation, d'Alembert's formula makes this precise and even a little surprising. The value u(x,t) depends ONLY on the initial data on the interval from x - ct to x + ct — and on nothing else. Draw the point (x,t) and run two lines down to t = 0 with slopes corresponding to speed c; they cut out a triangle whose base is exactly that interval. Data outside the base simply has not had time to reach (x,t): a disturbance starting a kilometre away, with c = 343 m/s, cannot be heard for about three seconds. The interval is the base of the backward characteristic cone.
This is the concrete face of finite propagation speed, and it is one of the sharpest contrasts between equation types. The wave equation has a tight, finite domain of dependence, so information is local. The heat equation is the opposite: its domain of dependence is the WHOLE line — the temperature anywhere at any t > 0 depends, however faintly, on the initial data everywhere, because heat spreads at infinite speed. Knowing the domain of dependence also tells a numerical analyst how to choose a stable time step (the CFL condition): the numerical scheme's domain of dependence must contain the true one.
On a string with c = 1, the value at the point x = 0, t = 2 depends only on the initial data between x = -2 and x = +2. Change the string's starting shape anywhere outside [-2, 2] and u(0,2) does not move one bit — that information simply has not arrived.
Only data in [x - ct, x + ct] can influence (x,t); the rest is too far to have arrived.
Domain of dependence (what the present point depends ON) is the mirror image of range of influence (what a given starting point reaches). Do not confuse the two: one looks backward in time, the other forward.