  We present the first numerical simulations of gravitational waves (GWs)
passing through a potential well generated by a compact object in 3-D space,
with a realistic source waveform derived from numerical relativity for the
merger of two black holes. Unlike the previous work, our analyses focus on the
time-domain, in which the propagation of GWs is a well-posed "initial-value"
problem for the hyperbolic equations with rigorous rooting in mathematics and
physics. Based on these simulations, we investigate for the first time in
realistic 3-D space how the wave nature of GWs affects the speed and waveform
of GWs in a potential well. We find that GWs travel faster than the prediction
of the Shapiro time-delay in the geometric limit due to the effects of
diffraction and wavefront geometry. As the wave speed of GWs is closely related
to the locality and wavefront geometry of GWs, which are inherently difficult
to be addressed in the frequency-domain, our analyses in the time-domain,
therefore, provide the first robust analyses to date on this issue based on
solid physics. Moreover, we also investigate, for the first time, the
interference between the incident and the scattered waves (the "echoes" of the
incident waves). We find that such interference makes the total lensed
waveforms dramatically different from those of the original incident ones not
only in the amplitude but also in the phase and pattern, especially for signals
near the merger of the two back holes.
