Suppose $u$ solves
$$ \begin{cases} \partial^2_{tt}u - \nabla\cdot\left(c^2(x)\nabla u\right) = F(t,x) & (t,x) \in \mathbb R \times \mathbb R^n \\ u(0,x) = u_t(0,x) = 0 & x \in \mathbb R^n \end{cases} $$
where $c(x) \in C^\infty$, $F(t,x) = \chi_{[a,b]}(t)g(x - \gamma(t))$ with $g = \chi_{B_1(0)}$ (i.e. the characteristic function of the unit ball) and $\gamma: [a,b] \to \mathbb R^n$ is a single line segment.
Is it possible that, for some time $t_0 > b$, $u(t,\cdot) = u_t(t,\cdot) = 0$ for all $t \geq t_0$? I.e., can a force moving along a line segment excite a solution, and then drive it back to zero?