  We investigate how GWs pass through the spacetime of a Schwarzschild black
hole using time-domain numerical simulations. Our work is based on the
perturbed 3+1 Einstein's equations up to the linear order. We show explicitly
that our perturbation equations are covariant under infinitesimal coordinate
transformations. Then we solve a symmetric second-order hyperbolic wave
equation with a spatially varying wave speed. As the wave speed in our wave
equation vanishes at the horizon, our formalism can naturally avoid boundary
conditions at the horizon. Our formalism also does not contain coordinate
singularities and, therefore, does not need regularity conditions. Then, based
on our code, we simulate both finite and continuous initially plane-fronted
wave trains passing through the Schwarzschild black hole. We find that for the
finite wave train, the wave zone of GWs is wildly twisted by the black hole.
While for the continuous wave train, unlike geometric optics, GWs can not be
sheltered by the back hole. A strong beam and an interference pattern appear
behind the black hole along the optical axis. Moreover, we find that the
back-scattering due to the interaction between GWs and the background curvature
is strongly dependent on the direction of the propagation of the trailing
wavefront relative to the black hole. Finally, for a realistic input waveform
generated by binary black holes, we find that the lensed waveform in the merger
and ringdown phases is much longer than that of the input waveform due to the
effect of back-scattering.
