## 3 The Non-Uniform Spatio-Temporal Dynamic Response of the Seismic Interface
### Numerical Simulation for the Non-Uniform Spatio-Temporal Dynamic Response of the Slope  
Different scale models are established to simulate the spatio-temporal dynamic response. As mentioned in Section 2.2, the dynamic mechanisms of the slope model (Equation (10)) can be handled by the dynamic configuration of FDM. Besides this, quiet boundaries (viscous boundaries) are applied to both at the base and at the side boundaries of the models to simulate the boundary conditions as in Equations (11) and (12). Moreover, observation points \(x_{1}\), \(x_{2}\) and \(x_{3}\) on the interface at the base of the slope are set to record the variation of displacement in accordance with Equations (16) and (17).  
Variations of \(u_{x}\) with the time of slope observation points \((x_{1},\,x_{2},\,x_{3})\) on the interface \((z=0)\) with different scales are illustrated in Figures 7 and 8. The variations in displacement demonstrate that the inharmonic vibration of the observation points occurs in response to the seismic excitation. In the case of uniform incidence (Figure 7a-d), the displacements curves of each observation point vary in a synchronous manner. In the case of non-uniform incidence (Figure 8a-d), the displacements curves of each observation point exhibit different degrees of frequency and phase dispersion. Notably, as \(L/h\) increases, there is a more marked dispersion in the frequency and phase of the displacements at each observation point. As \(L/h\) decreases, the displacement curves at each observation point tend to be consistent.  
Figure 7: Variations of \(u_{x}\) with the time of slope observation points on the interface with different scales under uniform seismic incidence. (**a**) \(L=2000\) m, \(L/h=4:20\); (**b**) \(L=1500\) m, \(L/h=3:20\); (**c**) \(L=1000\) m, \(L/h=2:20\); (**d**) \(L=500\) m, \(L/h=1:20\); \(h\) denotes the focal depth; \(L\) denotes the length of the slope.  
Contours of \(u_{x}\) in the slope models with different scales at the moment \(t=15\) s are presented in Figures 9 and 10. In the case of uniform incidence (Figure 9), the values of the displacements are found to be in the same contour on the interface at the base of the slope. Conversely, in the case of non-uniform incidence (Figure 10a-d), different contours of the displacements are presented on the interface at the base of the slope. Notably, as \(L/h\) increases, the contours of the displacements tend to be more discrete. As \(L/h\) decreases, the contours of the displacements tend to be continuous.  
Figure 8: Variations of \(u_{x}\) with the time of slope observation points on the interface with different scales under non-uniform seismic incidence: (**a**) \(L=2000\) m, \(L/h=4:20\); (**b**) \(L=1500\) m, \(\mathrm{L/h}=320\); (**c**) \(L=1000\) m, \(L/h=2:20\); (**d**) \(L=500\) m, \(L/h=1:20\); \(h\) denotes the focal depth; \(L\) denotes the length of the slope.  
Figure 9: Contours of \(u_{x}\) of slope models with different scales at the moment \(t=15\) s under uniform seismic incidence (unit: m): \(L=2000\) m, \(L/h=4:20\); \(h\) denotes the focal depth; \(L\) denotes the length of the slope.  
In conclusion, the variations in the displacements on the interface at the base of the slope present a degree of spatio-temporal dispersibility in the dynamic response subjected to non-uniform seismic wave incidence. In the case of non-uniform incidence, the displacements exhibit different degrees of frequency and phase dispersion. The degree of dispersion increases with \(L/h\); in other words, the non-uniformity of the dynamic response keeps the same tendency as \(L/h\). In this paper, the numerical simulation results of the displacement contours indicate that the non-uniform spatio-temporal dynamic response cannot be disregarded in the case of \(L/h>4:20\). Slopes of this scale subjected to seismic wave incidence satisfy the BAP principle. Conversely, the dynamic response exhibits uniformity in the case of \(L/h<1:20\), which is in accordance with the plane wave assumption of Huygens' principle.