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.
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