= Countertravelling decomposition of the one-dimensional wave equation
{title2=$u(x,t)=F(x-ct)+G(x+ct)$}
Let $H$ be any <antiderivative> of the initial <velocity>, $H'=g$. One choice of profiles is
$$
F(s)=\frac{f(s)}2-\frac{H(s)}{2c},\qquad G(s)=\frac{f(s)}2+\frac{H(s)}{2c}.
$$
Opposite 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 $u_{tt}-c^2u_{xx}=-4c^2u_{\xi\eta}$ by the <chain rule>. Thus $u=F(\xi)+G(\eta)$, and the <Cauchy data> imply $F+G=f$ and $-cF'+cG'=g$, giving the profiles above.
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