D'Alembert formula with initial velocity

For $u_{tt}-u_{xx}=0$ with $u(0,x)=0$ and $u_t(0,x)=f(x)$,

If $u(x,0)=0$, $u_t(x,0)=g(x)$, and the odd function $g$ is supported on $[-a,a]$, thenThe interval $[x-t,x+t]$ then contains the entire support, whose integral vanishes by oddness.