Let $u$ be a solution of the $d$-dimensional wave equation such that $u(x, 0) =u_t(x, 0) = 0$ for $x \in B(v,R)$. Upto what time $t$ can you be sure that $u(v, t) = 0$??
My try:
I first did it for the case $d=1$. Then the ball is basically an interval $(v-R,v+R)$. I pass two characteristics through the points $v-R$ and $v+R$ namely $t=\dfrac{1}{c}x +\dfrac{R-v}{c}$ and $t=\dfrac{-1}{c}x +\dfrac{R+v}{c}$. They intersect at the point $(v, \dfrac{R}{c})$. So I can say that upto $\frac{R}{c}$, the solution is $0$.
Can I generalise this to $d-$dimension?? If so how?? I think that the time would still be the same.
Thanks for the help!!
If waves propagate at speed $c$ and there is no disturbance at time $0$ inside the ball, it is obvious that a disturbance (if any) will take a time at least $R/c$ to reach the center.