the range of influence
Turn the previous question around. Instead of asking which starting data affects a future point, ask: if I poke the string right here, right now, who will ever feel it, and when? The set of all future places and times that a single starting point can reach is its range of influence.
For the wave equation a disturbance at a point x0 at time 0 can only reach places within distance ct of x0 by time t — it travels outward at speed c and no faster. So the range of influence of x0 is the wedge (in the x-t picture) bounded by the two lines x = x0 - ct and x = x0 + ct opening upward: the forward characteristic cone. A pluck at one spot of an infinite string sends a signal left and right at speed c; far away, nothing happens until the wave physically arrives, and then it does. This is the exact partner of the domain of dependence — that triangle pointed down into the past, this wedge opening up into the future.
Range of influence is finite propagation speed seen from the cause's side, and it is why the world is causal and orderly: a clap here is not heard a mile away for several seconds, a light switched on does not brighten a distant room instantly. The heat equation has no such courtesy — its range of influence is everywhere at once (infinite speed), so in the idealized model a match struck in one room raises the temperature, infinitesimally, in every room on Earth the very next instant. That is a real and honest difference between the wave (hyperbolic) and heat (parabolic) worlds, not a mathematical quibble.
Drop a stone in a still pond. A ring of ripples expands outward at the wave speed; a leaf floating beyond the ring bobs not at all until the ring reaches it, and only then. The leaf is inside the stone's range of influence only after the wavefront arrives.
A point disturbance reaches only a growing cone — causality at speed c.
In three dimensions Huygens' principle sharpens this further: the influence is concentrated ON the expanding sphere of radius ct, not throughout the filled ball — after the sphere passes, a point returns to rest. In two dimensions it does not, leaving a lingering wake.