## 2 Vibration Response Analysis of Different Roadbed-Foundation Systems
### Soft-Sandwiched System  
In most practical scenarios, constructing roads and seawalls on soft clay foundations results in a three-layer soft-sandwiched system. Figure 3c illustrates the schematic of such a system, and the relevant parameters are listed in Table 3. Two cases of vibration frequency are presented here.  
Considering a vibration load frequency \(f\) of 2.5 Hz, calculations indicate a wavelength \(\lambda\) of approximately 63.4 m. Figure 8 presents the variation in the displacement of a ground surface point with horizontal distance \(x\). Figure 9 shows the variation in the displacement amplitude of points at different cross-sections \(x\) with depth \(y\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Soil Layer**} & \multirow{2}{*}{**Thickness \(H\) (m)**} & **Shear Wave** & \multirow{2}{*}{**Mass Den-**} & **Poisson’s** & **Damping** \\  & & **(m/s)** & & **Ratio \(\mu\)** & **Ratio \(\xi\)** \\ \hline Roadbed & 10 & 400 & 2300 & 0.24 & 0.02 \\ Soft clay layer & 60 & 100 & 1700 & 0.32 & 0.2 \\ Hard foundation & \(\infty\) & 600 & 2500 & 0.25 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters for the soft-sandwiched system.  
Figure 7: Variation of displacement amplitude of a point at different cross-sections \(x\) with depth \(y\) in the upper-hard, lower-soft system: (**a**) \(x\) = 0.0 m, (**b**) \(x\) = 62.0 m.  
Figure 8 demonstrates that the displacement of ground surface points in the soft-sandwiched system rapidly decays with increasing horizontal distance \(x\). Figure 9 reveals that within the 10-m-thick roadbed, the vertical displacement of points is relatively large and remains almost constant with depth, while the horizontal displacement is smaller and decays with depth. Considering the low strength of soft clay, it can be assumed that the entire 10-m-thick roadbed undergoes an overall vertical vibration. In the soft clay layer beneath the roadbed, the vertical displacement of points exhibits decay with depth, but this decay is slower compared to that in the upper-soft, lower-hard system. The horizontal displacement in the soft clay layer is basically not attenuated. In the underlying hard soil layer, the vibration waves decay rapidly. The predominant vibration within the entire system is concentrated in the roadbed and the underlying soft clay layer near the vibration source.  
For the vibration frequency \(f\) of 7.2 Hz, calculations indicate a wavelength \(\lambda\) of approximately 30.0 m. Figure 10 presents the variation of the displacement of a ground surface point with horizontal distance \(x\). Figure 11 shows the variation of the displacement amplitude of points at different cross-sections \(x\) with depth \(y\).  
Figure 8: Variation of (**a**) vertical and (**b**) horizontal displacement of a ground surface point with horizontal distance \(x\) in the soft-sandwiched system when \(f\) = 2.5 Hz.  
Figure 9: Variation of displacement amplitude of a point at different cross-sections \(x\) with depth \(y\) in the soft-sandwiched system when \(f\) = 2.5 Hz: (**a**) \(x\) = 0.0 m, (**b**) \(x\) = 62.0 m.  
Figures 10 and 11 yield results similar to those presented in Figures 8 and 9. However, due to the shorter wavelength of the vibration waves at this frequency, the influence depth of the vibration load in the soft clay layer is shallower.  
Therefore, in the soft-sandwiched system, the kinetic energy generated by traffic vibration loads propagates primarily in the vertical direction, transferring energy to the soft clay layer beneath the roadbed, with minimal propagation outward in the horizontal direction. This behavior is similar to that observed in the upper-hard, lower-soft system.