Countertravelling decomposition of the one-dimensional wave equation ($u(x,t)=F(x-ct)+G(x+ct)$)

Let $H$ be any antiderivative of the initial velocity, $H^{\prime }=g$. One choice of profiles isOpposite additive constants in $F$ and $G$ cancel, so the displacement is independent of the chosen antiderivative. Even for compact support of $g$, these separate profiles can approach nonzero constants; cancellation in their sum is essential to support widening for the wave equation.

For a short derivation, the characteristic coordinates $\xi =x-ct$, $\eta =x+ct$ give $u_{tt}-c^2{u}_{xx}=-4c^2{u}_{\xi \eta }$ by the chain rule. Thus $u=F(\xi )+G(\eta )$, and the Cauchy data imply $F+G=f$ and $-c{F}^{\prime }+c{G}^{\prime }=g$, giving the profiles above.