Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation

$$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$

where $f$ and $1-c$ are compactly supported in $\Omega$. 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 $\Gamma$)?