This is a question from a past exam I'm currently studying for, very little idea on how to approach this but the marks for it are lower than most long questions so I think I'm missing some fundamental idea.
I'm assuming the general solution looks like $u(x,t) = F(x - ct) + G(x + ct)$ and the string is governed by $u_{tt} = c^2u_{xx}$, with c > 0, x is real and t > 0.
So the initial conditions are: $u(x,0) = 0$ for $x \notin[-100,100]$, $u_t(x,0)= 0$ for $x \notin[-100,100]$.
The actual problem is to show that for any point $x_0 \in \Bbb{R}$ there exists a $T_0 > 0$ such that $u(x_0,t)$ is constant for all $t > T_0$, and to find an expression for this constant value.
I'd appreciate any help on this, or insight on what I should be thinking about when approaching this kind of problem.
Use the initial conditions to show $F(x)=G(x)=0$ for $|x|>100$.
If $t$ is big enough, $x_0-ct<-100$ and $x_0+ct>100$...