= One-dimensional wave equation
{title2=$u_{tt}-c^2u_{xx}=0$}
On the whole line, the homogeneous one-dimensional <wave equation> with constant <wave speed> $c>0$ is
$$
u_{tt}-c^2u_{xx}=0,\qquad u(x,0)=f(x),\qquad u_t(x,0)=g(x).
$$
Here $u$ is <displacement> in the string interpretation; $f$ is initial <displacement> and $g$ is initial <velocity>. Its two directions of travel give
$$
u(x,t)=F(x-ct)+G(x+ct)=\frac{f(x-ct)+f(x+ct)}2+\frac1{2c}\int_{x-ct}^{x+ct}g(s)\,ds.
$$
This is <D'Alembert's formula>, including both <Cauchy data>. The <countertravelling decomposition of the one-dimensional wave equation> identifies $F(x-ct)$ as the right-moving <travelling wave> and $G(x+ct)$ as the left-moving <travelling wave>. Their sum is permitted by the <superposition principle>. For $f\in C^2$ and $g\in C^1$, it is a <classical solution>. More generally, <travelling waves> with profiles $F$ and $G$ that are <locally integrable functions> solve the equation as <distributions>, with derivatives interpreted distributionally.<18.303 Linear Partial Differential Equations, Fall 2006 by Matthew J. Hancock/The 1-D Wave Equation>{ref}
\Image[/wave-equation-countertravelling-pulse-components.png]
{title=Synthetic pulse solutions: release from rest splits a profile, while matched initial velocity launches it rightward}
The synthetic diagram uses the smooth <bump function> $b$ defined in <D'Alembert's formula>, with $f(x)=b(x)$ and $c=1$. Releasing this <displacement> from rest, $g=0$, gives $F=G=f/2$. Each emerging <travelling wave> has half the original peak. Launching the identical initial <displacement> solely to the right instead requires $g=-cf'$ and gives $F=f$, $G=0$. At $t=1.5$, the released profiles have peaks $1/2$ at $x=\pm1.5$, while the launched profile has peak $1$ at $x=1.5$. These are calculated model values. A purely left-moving launch requires $g=cf'$. The initial <velocity> therefore selects the balance of directions; the initial shape alone does not.
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