  This paper analyzes the well-known L1 scheme for fractional wave equations
with nonsmooth data. A new stability estimate is obtained, and the temporal
accuracy $ \mathcal O(\tau^{3-\alpha}) $ is derived for the nonsmooth initial
data. In addition, a modified L1 scheme is proposed, and stability and temporal
accuracy $ \mathcal O(\tau^2) $ are derived for this scheme with nonsmooth
initial data. The convergence of the two schemes in the inhomogeneous case is
also established. Finally, numerical experiments are performed to verify the
theoretical results.
