= D'Alembert formula with initial velocity
{c}
{title2=$u(x,t)=\tfrac1{2c}\int_{x-ct}^{x+ct}g(s)\,ds$}
For the <wave equation> $u_{tt}=c^2u_{xx}$ with zero initial displacement and initial <velocity> $g$, <D'Alembert's formula> becomes
$$
u(x,t)=\frac1{2c}\int_{x-ct}^{x+ct}g(s)\,ds.
$$
For $c=1$ this is the original speed-one form $u(t,x)=\tfrac12\int_{x-t}^{x+t}f(s)\,ds$ when the initial <velocity> is named $f$. The integral records the total initial kick inside the <domain of dependence>.
For a computed example, set $g(x)=V$ on $[-a,a]$ and zero elsewhere, with $V>0$. This discontinuous datum gives a <weak solution>:
$$
u(x,t)=\frac{V}{2c}\,\operatorname{length}\bigl([x-ct,x+ct]\cap[-a,a]\bigr),\qquad t\geq0.
$$
At the centre, $u(0,t)=V\min(t,a/c)$. For $a=1\,\mathrm m$, $c=2\,\mathrm{m\,s^{-1}}$ and $V=0.4\,\mathrm{m\,s^{-1}}$, this is $0.1\,\mathrm m$ at $t=0.25\,\mathrm s$ and $0.2\,\mathrm m$ at $t=1\,\mathrm s$. Once $ct>a$, a central plateau $u=Va/c$ occupies $|x|\leq ct-a$. In contrast, <central zero region for odd compactly supported initial velocity> follows when the full integral of $g$ vanishes. <Finite propagation speed> permits an interior tail; it does not imply the <Strong Huygens principle>.
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