Let $\displaystyle u(x,y,t)$ be the solution of the Cauchy problem

$\displaystyle u_{tt}-c^2 u_{xx}-c^2 u_{yy}=0$,

$\displaystyle u(x,y,0)=f(x,y)$,

$\displaystyle u_{t}(x,y,0)=g(x,y)$,

where $\displaystyle c>0$ is a constant and $\displaystyle f(x,y)$ and $\displaystyle g(x,y)$ vanish for $\displaystyle x^2 +y^2 >r^2$ for some $\displaystyle r>0$.

Show that the solution $\displaystyle u(x,y,t)$ vanishes if $\displaystyle x^2+y^2-r^2>c^2 t^2$