## 3 Dynamic Response of the Slope
### Fourier Spectra  
The acceleration histories can be transformed into Fourier spectra by applying a fast Fourier transformation (FFT) algorithm to the data. The Fourier spectra allow certain features of the dynamic response of the slope to be readily visualized. We are specifically interested in the variation of the Fourier amplitude with frequency and the earthquake dominant frequency (EDF). Three points in a model with \(\alpha\) = 50\({}^{\circ}\) and \(\beta\) = 70\({}^{\circ}\) were chosen for Fourier analysis (A8, A10, and A12 along profile 2). Figure 10 shows the resulting Fourier spectra together with the corresponding ones generated for the input seismic wave.  
Figure 8: Variation of the peak acceleration amplification factors at four different points in slopes subjected to the action of sinusoidal waves. The values are plotted as functions of the angle \(\gamma\) and the points chosen are: (**a**) A5 and A9, and (**b**) A7 and A8.  
Figure 9: Variation of the three-directional peak AAF values at point A8 with slope angle \(\alpha\) and joint angle \(\beta\): (**a**) under the action of sinusoidal waves, and (**b**) under the action of natural seismic waves.  
(The data in Figure 10 are arranged to emphasize how the form of the seismic vibrations changes as it migrates upwards from the input level to points A12, A10, and then A8.)  
Figure 10: Fourier spectra components and their dominant frequencies at different elevations within a slope subjected to the action of a natural seismic wave. The data are arranged in order of increasing elevation from bottom to top (points A12, A10, and A8 along profile 2).  
Consider first the component of the Fourier spectrum in the \(X\)-direction. As can be seen, the EDF and amplitude of the Fourier spectrum in the \(X\)-direction vary little from the bottom of the slope (point A12) up to point A10: the former remains around 1-6 Hz and the latter remains around 9. However, as we move up to point A8, the Fourier amplitude is sharply amplified to 68.3 and the EDF is reduced to 0.29 Hz.  
In the \(Y\)-direction, the EDF of the Fourier spectrum gradually increases as the input wave migrates to A10 while the Fourier amplitude slightly decreases. At point A8, however, the EDF shows double- or even triple-peak characteristics and the Fourier amplitude of the seismic component around 10 Hz is amplified to about 32. Moreover, the low frequency part (\(\sim\)1.27 Hz) is also amplified. The EDF of the Fourier spectrum in the \(Z\)-direction seems to decrease from bottom to top and reaches a minimum value of 1.2 Hz at point A8 (top of the slope). The Fourier amplitude essentially just fluctuates between 1.2 and 2.  
In the planar problem, the \(Y\)- and \(Z\)-directions are equivalent to the horizontal and vertical directions, respectively. It is not surprising, therefore, that the patterns of Fourier amplitude variation found in these two directions are similar to the results obtained in previous table-shaking experiments [26]. On the whole, the seismic wave propagates from the bottom of the slope to the upper parts through reflection and refraction at the structural planes. The superposition effect of the seismic waves leads to the displacement response of the slope and hence relative motion of the rock masses. The more work that is conducted, the greater the amount of energy dissipation that will take place. As a result, high-frequency seismic waves with higher energy are generally more strongly attenuated and the EDF decreases correspondingly. Therefore, the dynamic response of the anti-dip bedding rock slope is stronger (and so more likely to become unstable) under the action of low-frequency seismic waves. This also explains why the AAFs of the slopes shown in Figure 7 are larger under the action of the sinusoidal wave (whose frequency is fixed at a low value of 5 Hz) than under the action of the natural seismic wave, (which contains frequency components that are much higher than 5 Hz).