# Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack
## 3 Computational Simulations Model
### Discussion of the Wave Separation in FE Simulation  
The model based on the wave separation in the P-wave and surface waves is studied in this section. As illustrated in Figure 2, the \(D_{\text{C-N1, }N2}\) represents a distance between the crack location and listening nodes in the \(x\)-direction, while \(D_{\text{C-N1}}\) represents the distance between the source point and backward listening node in the \(x\)-direction. Typically, each propagated wave (e.g., P-wave, S-wave and surface wave) has a different arrival time due to each wave's different speed. For example, in the simulation with given material property, the speed of the P-wave (\(C_{P}\)) and surface wave (\(C_{R}\)) are 4084 m/s and 2265 m/s, respectively. However, the model requires a minimum distance between the wave source and the listening nodes to obtain the pure waveforms (e.g., surface wave) so as to avoid overlapped waves.  
Figure 3 shows four different wave groups excited by an incident sine wave. The two-wave groups (P-wave and surface wave) are incident waves in the time range between 0.2 and \(0.8\times 10^{-3}\) s, while the other two-wave groups are the reflected waves at the crack in the time range between 1.2 and \(1.6\times 10^{-3}\) s. The three following groups must be considered to obtain a clear P-wave and surface wave separation: i) Reflected P-waves by the crack should not pass when the incident surface waves pass through at \(N1\), as \(\frac{D_{\text{S-N1}}+D_{\text{C-N1}}+D_{\text{C-N2}}}{C_{P}}\
eq\frac{D_{ \text{S-N1}}}{C_{R}}\). ii) the distance, \(D_{\text{C-N1}}\) should be enough to have a wave separation in P- and Surface-waves at \(N1\), (see Region A in Figure 3), \(\frac{D_{\text{S-N1}}}{C_{P}}-\frac{D_{\text{S-N1}}}{C_{R}}>T\), where T is period of the incident wave. \(D_{\text{C-N1, }N2}\) should be enough to have a wave separation between the reflected P- and Surface-waves by the crack at the \(N1\) and \(N2\), (see region B in Figure 3), \(\frac{D_{\text{C-N1},N2}}{C_{P}}-\frac{D_{\text{C-N1},N2}}{C_{R}}>T\). In addition, the simulations are designed to avoid the S-wave effect to obtain the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Damper** & **Damper** & **Damper** & **Damper** & **Damper** & **Damper** & **Damper** & **Damper** & **Damper** \\
**1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline
1000 & 6000 & 11,000 & 16,000 & 21,000 & 26,000 & 31,000 & 36,000 & 41,000 & 46,000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Damping factors for optimum damping gradient of FE model.  
Figure 2: Details of finite element (FE) model: \(N1\) is the listening node for the backward scattered waves, \(N2\) is the listening node for the forward scattered waves, \(D_{\text{C-N1}}\) and \(D_{\text{C-N1}}\) are the distances between the crack and nodes, \(N1\) and \(N2\), respectively, \(D_{\text{s-N1}}\) is the distance between the wave source and \(N1\). The figure also includes ten dampers in rainbow colors on the three sides of the solid medium and an infinite element in the thin red layer at the end of dampers. The illustration shows the required component and geometry to simulate the wave propagation in the cracked model.  
pure surface wave by receiving only the horizontal displacement, which is in a longitudinal direction, thus the S-wave was eliminated.