4.2 Causality: domains of dependence and influence

Formula (4.2) shows that \(u(x_0, t_0)\) depends only on the data in the interval \([x_0 - ct_0, x_0 + ct_0]\), the domain of dependence of the point, cut out by the two characteristics through it (Figure 6). Conversely, data in an interval \([a, b]\) influence only the wedge \(a - ct \leq x \leq b + ct\), the range of influence. This is finite speed of propagation: a signal cannot travel faster than \(c\).

There is a difference between the two kinds of data. Initial displacement \(f\) affects only the edges of the range of influence: a pulse in \(f\) passes a point and is gone. Initial velocity \(g\) enters through an integral, so its effect fills the whole wedge: a string struck by a hammer is left permanently displaced behind the waves (Problem 4.7.2). In one dimension, the sharp form of Huygens’ principle fails; whether it holds depends on the dimension, as the next section shows.

x(xxdxabrax00dea−0−+opn,tmegc0ccanet)t0t0id o≤nefx on ifcn≤eflube+nccte — Figure 6: Causality for the wave equation. Left: the value at \((x_0, t_0)\) depends only on the data between the two characteristics \(x \pm ct = x_0 \pm ct_0\) through the point. Right: data on \([a, b]\) affect only the region between the outgoing characteristics; outside it, the solution does not know they exist.

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