## 2 Numerical Simulations of Vibratory Sheet Pile Driving
### Numerical Simulation Results  
Based on the model described above, spatial wave propagations due to vibratory sheet pile driving at different depths were simulated. Figure 2 displays a representative example of wave propagations when the pile toe is located at 10 m below the ground surface. The snapshots at time steps \(t\) = 50 ms (i.e., one period after the start of the driving force), \(t\) = 300 ms (i.e., six periods after the start of the driving force), \(t\) = 550 ms (i.e., one period after the end of driving source), and \(t\) = 700 ms (i.e., four periods after the end of driving source) are presented. These snapshots show that three types of wavefronts emerged as the wave propagated away from the vibratory driving location. Because of the cyclic dynamic frictions between pile shaft and adjacent soil, cylindrical wavefronts of shear wave (S wave) with the largest amplitude are generated from the pile shaft ranging from the ground surface to the pile toe. Similarly, due to the cyclic dynamic contact force between the pile toe and the underlying soil, spherical wavefronts of body waves (P and S waves) are generated from the pile toe. There are also cylindrical wavefronts of surface wave (R wave) near the ground surface, which is always slower than the cylindrical wavefronts generated by the pile shaft. The surface waves are formed through the interaction between the reflected body waves and advancing shear waves near the ground surface [29; 30]. Similar observations of wave propagations were made when pre-embedding the pile toe at other depths. Vibration responses of pile and soil nodes were subsequently investigated. Figure 3 shows the adopted spatial Cartesian coordinate system and definitions of plan distance \(d\) and radial distance \(r\), which help position the investigated soil nodes on and below the ground surface. Figure 2: Wave propagation snapshots at different time steps during vibratory sheet pile driving when pre-embedding the pile toe 10 m below the ground surface. (**a**) 50 ms; (**b**) 300 ms; (**c**) 550 ms; (**d**) 700 ms. Figure 4 displays the vertical velocity time histories of the pile and three soil nodes at different depths (\(Z=0,-7,-25\) m) but the same plan distance (\(d=20\) m). The ToA between the pile vibration and soil vibrations can be observed. The cross-correlation function is widely used to determine the time lag between two discrete vibration signals in the time domain. The mathematic form is represented as:  
\[(f\bigstar g)[n]\triangleq\sum_{m=-\infty}^{\infty}\overline{f[m]}g[m+n] \tag{5}\]  
in which \(f\) and \(g\) represent two discrete vibration signals and \(\overline{f[m]}\) is the complex conjugate of \(f\) [\(m\)], and \(n\) is the time lag indicating that a feature in \(f\) at \(m\) occurs in \(g\) at \(m+n\). Therefore, the time lags \(n\) (i.e., ToAs) in Equation (5) for vibration signals at any soil nodes were computed by performing a cross-correlation function between the vibration signal of the pile and corresponding vibration signals of soil nodes. The ToAs at different ground surface points on the \(X\)-\(Y\) plane (\(0<X<50\) m, \(-50\) m \(<Y<50\) m and \(Z=0\)) and underground points on the \(X\)-\(Z\) plane (\(0<X<50\) m, \(Y=0\) and \(-30\) m \(<Z<-3\) m) were collected. To differentiate the wave types based  
Figure 4: Computed vertical velocity time histories of the pile and three soil nodes at different depths (\(Z=0,-7\) m, \(-25\) m) and one plan distance (\(d=20\) m) when pile toe is pre-embedded at \(10\) m below the ground surface. Figure 3: Definitions of plan distance \(d\) (m) and radial distance \(r\) (m).