## 4 Vibration Attenuation Pattern of the Track Embankment and Pile-Raft Foundation  
Figure 13 shows values of the peak velocity for different locations (V1, V2, V3, V4) under different train speeds. The vibration response level and attenuation are different for different train speeds. The data fitting results are shown in Table 6 and Figure 13. The data curves of the vibration velocity peaks under the different train speeds were fitted using the following equations:  
\[V_{90 km/h}=522e^{\frac{-x}{0.008}}+1.04 \tag{4}\]  
\[V_{180 km/h}=3890e^{\frac{-x}{0.0148}}+2.56 \tag{5}\]  
\[V_{225m/h}=8781e^  
The fitting curves shows that the vibration velocity attenuation follows the exponential curve distribution from the low to high train speed. The peak velocity depends significantly on the distance, decreasing quickly to a distance of up to 0.45 m and slowing down from 0.45 to 1.66 m. The peak velocity increases with the train speed. Further prediction and analysis of the vibration velocity response of the pile-raft composite foundation can be performed on high-speed train loads in soft soil areas. The vibration velocity attenuation rates for the track embankment and pile-raft foundation in the horizontal direction are shown in Table 7. For \(v\) = 90 km/h, \(v\) = 180 km/h, \(v\) = 225 km/h, and \(v\) = 270 km/h, the amplitude of the vibration velocity from the track slab (V1) to the roadbed (V2) are attenuated by 87.36%, 88.66%, 89.02%, and 88.75%, respectively. In turn, the attenuation rates from the roadbed to the shoulder are 5.92%, 2.07%, \(-\)0.20%, and 2.05%, respectively. This indicates that the attenuation rates for different train speeds are relatively close, and vibration velocity attenuation mainly occurs from the track slab to the roadbed. Most vibration attenuation is completed in the process of vibration propagation from the track slab to the road bed. The velocity attenuation tendencies in the ballast, embankment, and ground are different due to their dynamic characteristics. Figure 14 shows the distribution of the vibration velocity with the depth from the roadbed surface in the middle cross-section of the track embankment and subsoil foundation. The amplitudes of the vibration velocity at the roadbed are 1.58 and 4.22 mm/s at train speeds of 90 and 270 km/h, respectively, which is reduced by 95% and 93.2% from the track slab V1. The existence of stiffness of the raft has a certain effect on the transmission, reflection, and superposition of the vibration wave inside the embankment, and results in slower attenuation of the vibration velocities. The amplitudes of the vibration velocity at the bottom of the subsoil are 0.08 and 0.526 mm/s, which are much smaller than those at the roadbed V1. This is because the pile-raft structure supports most of the vibration loads, and blocks the transmission of the vibration vibrating waves, resulting in comparably lower vibration vibrating velocities. The effect of the moving train load on the vibration velocities \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Train Speed** & \(v\) = 90 km/h & \(v\) = 180 km/h & \(v\) = 225 km/h & \(v\) = 270 km/h \\ \hline
**Location** & **Rate** & **Rate** & **Rate** & **Rate** \\ \hline V1 & 0 & 0 & 0 & 0 \\ V2 & 87.36\% & 88.66\% & 89.02\% & 88.75\% \\ V3 & 90.00\% & 89.62\% & 91.01\% & 91.68\% \\ V4 & 93.28\% & 90.73\% & 88.83\% & 90.80\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Vibration velocity attenuation rate in the horizontal direction. Figure 13: Distribution of the vibration velocity in the horizontal direction from the track center. of the soft soil foundation is very small. The pile-raft structure composite foundation over soft soil can effectively reduce the level of the vibration response induced by high train loads. Notice from Figure 14 that the velocities decrease along the depth from the roadbed to raft, and have a vibration enhancement region (from the raft to a 1 m depth of the foundation soil).