= Wave speed
{title2=$c$}
= Wavespeed
{synonym}
For the homogeneous <one-dimensional wave equation> $u_{tt}=c^2u_{xx}$ 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.<18.303 Linear Partial Differential Equations, Fall 2006 by Matthew J. Hancock/The 1-D Wave Equation>{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>.
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