Finite propagation speed

For the unforced, constant-$c$ one-dimensional wave equation, finite propagation speed says that changing Cauchy data outside $[x_{\ast }-c{t}_{\ast },x_{\ast }+c{t}_{\ast }]$ cannot change the solution at $(x_{\ast },t_{\ast })$, where $c>0$ and $t_{\ast }\ge 0$. 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.^[ref]

For the synthetic choice $c=1$, $(x_{\ast },t_{\ast })=(0.5,1.5)$ 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 $t\ge 0$. Thus an observer at $x>a$ sees exactly zero displacement for $0\le t<(x-a)/c$. 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.

For $c>0$, the backward causal cone at $(x_{\ast },t_{\ast })$ isIts boundaries are characteristic lines for the one-dimensional wave equation; its base on $t=0$ is the initial-data domain of dependence. The forward cone from $(s,0)$ is $t\ge 0$, $|x-s|\le ct$. Here the cone is set by the model's wave speed, which may be acoustic or a string wave speed; it does not assert propagation at the speed of light.

For whole-line Cauchy data with support contained in $[-a,a]$, the displayed inclusion holds at $t\ge 0$. A short check uses D'Alembert's formula: if $x>a+ct$, both sampled endpoints and the whole interval of integration lie to the right of $a$, so all contributions vanish; the argument for $x<-a-ct$ is identical. The inclusion can be strict, and a nonzero response at the first permitted arrival is not guaranteed.

The weak Huygens principle is the domain-of-dependence statement that initial data outside a backward light cone cannot influence its tip. It permits an interior tail and is weaker than the Strong Huygens principle. For $u_{tt}-\Delta u+V(x)u=0$ with $V\ge 0$, integrating the local energy over shrinking balls proves this statement: the boundary contribution is $u_t{\partial }_{\nu }u-\frac{1}{2}(u_t^2+|\nabla u{|}^2+V{u}^2)\le 0$.

For the homogeneous speed-one wave equation, integrate its wave energy density over $B(x_{\ast },T-t)$. The moving boundary contributes $-e$, while the usual flux contributes $u_t{\partial }_{n}u$. Their sum is $-\frac{1}{2}(u_t-{\partial }_{n}u{)}^2-\frac{1}{2}|{\nabla }_{\tan }u{|}^2\le 0$. Zero Cauchy data in the initial ball therefore force zero solution in its backward light cone.

For Cauchy data prescribed on $x=0$ over $|y|<a<1$, put $S=\mathrm{artanh}a$. The maximal region determined solely by those data isIn the travel-time coordinate for a one-dimensional variable-speed wave equation $(x,s)$ this is the ordinary characteristic diamond $|x|+|s|<S$, as follows from finite propagation speed.