Let $\Gamma$ be a hypersurface in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ for all time (under some suitable assumptions on $\Gamma$)?