Wave equation with compact supported Cauchy data

Question

Consider the wave equation $u_{tt}-\Delta u =0$ in $\mathbb{R}^n \times [0,\infty)$ and let $u(x,0)=g(x) \in C^{\infty}_C(\mathbb{R}^n)$, $u_t(x,0)=0$. What is a rigurous argument that the solution will be compact supported too for every $t\geq 0$? Using the fact that for a compact set where the Cauchy data vanishes there is a cone where the solution is identically zero.