Let $\Gamma$ be a hyperfurface in $R^n$ and assume $u$ solves the wave equation

$u_{tt}-\Delta u=0.$

Suppose $u(x,t)=0$ on $\Gamma \times (0,\infty)$. Can one guarantee that $u(x,t)=0$ in some open subset of $R^n$ for all time (under some suitable assumptions on $\Gamma$)?