  Lying between traditional parabolic and hyperbolic equations, time-fractional
wave equations of order $\alpha\in(1,2)$ in time inherit both decaying and
oscillating properties. In this article, we establish a long-time asymptotic
estimate for homogeneous time-fractional wave equations, which readily implies
the strict positivity/negativity of the solution for $t\gg1$ under some sign
conditions on initial values. As a direct application, we prove the uniqueness
for a related inverse source problem on determining the temporal component.
