= Finite propagation speed
For the unforced, constant-$c$ <one-dimensional wave equation>, <finite propagation speed> says that changing <Cauchy data> outside $[x_*-ct_*,x_*+ct_*]$ cannot change the solution at $(x_*,t_*)$, where $c>0$ and $t_*\geq0$. This interval is the initial-data <domain of dependence>. Its endpoints are found by following the two <characteristic lines for the one-dimensional wave equation> backwards from the observation. <D'Alembert's formula> makes the dependence sharper: initial <displacement> is sampled at the endpoints, while initial <velocity> is integrated over the entire interval.<Math 220A: Partial Differential Equations of Applied Mathematics/5 Wave Equation in R>{ref}
\Image[/wave-equation-characteristic-causal-cones.png]
{title=Synthetic constant-speed spacetime geometry: backward dependence and widening support}
For the synthetic choice $c=1$, $(x_*,t_*)=(0.5,1.5)$ has initial-data <domain of dependence> $[-1,2]$. A change confined near $x=3$ is irrelevant to that observation, however large it is. If both <Cauchy data> vanish outside $[-a,a]$, their <compact support> can influence only $[-a-ct,a+ct]$ at $t\geq0$. Thus an observer at $x>a$ sees exactly zero <displacement> for $0\leq t<(x-a)/c$. There is no nonzero precursor beyond this <causal cone for the constant-speed wave equation> in the ideal model. This is an exact statement about the <partial differential equation>, stronger than an experimental failure to detect a small <signal>.
<Support widening for the wave equation> is a bound on possible influence, not a claim that every point inside the expanding interval moves. A <bump function> released with zero initial <velocity> splits into two separated <travelling waves>; the middle can become empty. An initial kick can instead leave an interior tail, as in the <D'Alembert formula with initial velocity>. Consequently <finite propagation speed> does not imply the <Strong Huygens principle>. A finite string needs <boundary conditions>, and the whole-line formula applies locally only until its backward <causal cone for the constant-speed wave equation> encounters an endpoint.
Back to article page
