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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 the homogeneous one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$ with constant $c>0$, wave speed means the speed at which a travelling wave profile translates. In $F(x-ct)$, a feature initially at $x=s$ follows $x=s+ct$, giving signed velocity $+c$; in $G(x+ct)$ it follows $x=s-ct$, giving signed velocity $-c$. The characteristic lines for the one-dimensional wave equation therefore have slopes $dx/dt=\pm c$. The speed of either profile is $c$, even though the displacement at a fixed point can change sign or pass through zero.^[ref]

The same $c$ bounds the causal cone for the constant-speed wave equation. For a wave equation on a string, $c=\sqrt{T/\mu }$ depends on tension and linear mass density; for small acoustic waves in stationary uniform air it is the speed of sound. This interpretation assumes the stated ideal wave equation. In media with wave dispersion, phase velocity, group velocity and the earliest signal arrival have different meanings: a measured wave crest speed need not be the speed of a wave envelope, and neither alone establishes a causal cone for the constant-speed wave equation.