## 3 Results and Discussion
### Distribution Characteristics of the Subgrade Dynamic Displacement  
The time history curves of the vertical dynamic displacements at different monitoring points on survey line 1 at the toe of the slope when the train passed over the subgrade with an underlying soil hole and that without a soil hole are shown in Figure 9. As the train passed, both conditions exhibited two peaks in vertical dynamic displacement. This  
Figure 8: Mesh division diagram of the soil hole. Figure 7: Layout of the monitoring points. indicated that after the first bogie passed, the soil experienced significant vibration, which propagated to the toe of the subgrade slope, corresponding to the first peak. Subsequently, due to soil rebound and the passage of the second bogie, the overlapping vibrations created a second peak, resulting in a double-peak phenomenon. When there was no soil hole, the peak vertical dynamic displacements at the three positions were 0.162 mm, 0.163 mm, and 0.184 mm, respectively, whereas those with the underlying soil hole were 0.142 mm, 0.112 mm, and 0.130 mm, respectively. Calculations showed that the peak vertical dynamic displacements with the underlying soil hole were reduced by 12.35%, 33.29%, and 28.99% compared to the no-soil-hole condition, indicating that the amplitude of dynamic displacement was higher when there was no soil hole. Additionally, when there was a soil hole, the peak vertical dynamic displacement at the 0 m position was reduced by 21.11% and 13.85% compared to the \(-\)16 m and 16 m positions, respectively, indicating a significant attenuation of dynamic displacement amplitude at positions farther from the soil hole compared to those closer to it. The displacement cloud diagram visually displays this characteristic. The displacement cloud diagrams of the vehicle at the same moment under both conditions are shown in Figure 10. The propagation distance of dynamic displacement was significantly farther when there was no soil hole. At the same positions, the dynamic displacement was notably greater without the soil hole, indicating that the presence of the soil hole increased the attenuation of elastic waves. That was because the integrity of the normal subgrade was better, resulting in less attenuation of elastic wave propagation, whereas the presence of a soil hole led to a looser soil structure, causing greater attenuation of elastic wave propagation. Figure 9: Time history curve of the dynamic displacement at different positions at the toe of the high-speed railway subgrade in karst areas. The distribution characteristics of dynamic displacement amplitude in the subgrade of a high-speed railway in a karst area are shown in Figure 11. The vertical dynamic displacement amplitude was significantly greater when there was no soil hole compared to a soil hole. The dynamic displacement amplitudes at measurement points on survey line 1 were greater than those on survey line 3, indicating that the dynamic displacement significantly attenuated as the distance increased. Along the train's running direction, the dynamic displacement amplitude without a soil hole showed a fluctuating decreasing trend, while when there was a soil hole, the dynamic displacement amplitude had no clear pattern. However, the vertical dynamic displacement amplitude curve on survey line 1 showed a concave shape, indicating that the attenuation coefficient of dynamic displacement amplitude at the soil hole location was greater than at other locations. In contrast, survey line 3, being farther from the soil hole, showed no clear pattern. Table 4 shows the attenuation coefficient of dynamic displacement amplitude along the train running direction under different conditions. The attenuation range at the toe of the slope was approximately 30-40%, while at the position 10 m from the toe of the slope, the attenuation range was about 30-45%, with greater attenuation near the soil hole compared to other locations.