# Singular Integral Solutions of Analytical Surface Wave Model with Internal Crack
## 4 Results and Discussions
### Results of the Analytical Model and FE Simulation  
The analytical model with five different integral methods is computed using MATLAB (2019, MathWorks, Natick, MA, USA) in accordance with the procedures of Section 2. The FE simulation is designed based on the details described in Section 3. The analytical model and FE simulation results are normalized with the highest magnitude of the variation ratio to compare the shape and tendency of each case. The wavenumbers are considered to have the displacement variation curves in the \(k_{R}a<4\) range by the given '\(a\)' values (e.g., 12 mm, 18 mm and 24 mm). In the simulation result of the forward scattered, the pure scattered waves are obtained at a higher frequency range than 10 kHz. Since the lower frequency (e.g., 10 kHz or below) has a longer wavelength, the scattered P-wave overlaps the scattered surface wave, as discussed in Section 3.2. Figure 4 includes two graphs of the backward and forward scattered cases of the C18-0.2 model, which has a 72-mm crack size. In this geometry, the higher amplitude of scattered energy variation shows in the forward scattered direction (see the amplitude of the \(u^{fs}/u^{in}\) in Figure 4) than in the backward direction (see the amplitude of the \(u^{bs}/u^{in}\) in Figure 4). The displacement variation changes depending on distance between the ground and top crack, denoted as '\(a\)' and the crack size are described in Figure 5. The three models considered are C12-0.2, C18-0.2 and C24-0.2, which have depths of 12 mm, 18 mm and 24 mm, respectively and a crack size of 48 mm, 72 mm and 96 mm. The models have the same \(a/b\) ratio, 0.2. Regardless of the crack depth and size, a similar amount of wave scattering occurs in the same \(a/b\) ratio. Figure 3: FE simulation results of the time-domain waveform at the node \(N1\) in horizontal \(x\)-displacement: Waves from the incident wave and reflected waves by the crack are calculated. Both Region A and B show different arrival times as wave separations of the P-wave and surface waves. Region A is associated with the incident waves and B is associated with the reflected waves by crack. The image explains the waveform in the time-domain and the required distance of \(D_{C-N1,\ N2}\) and \(D_{S-N1}\) to obtain a pure waveform of the surface waves. The displacement variations based on crack size change is described in Figure 6, including C12-0.1 and C12-0.2 case, which have a 108-mm and 48-mm crack size. The graph result showing 0.1 of the a/b ratio (see Figure 6a) is obtained by a longer crack than the 0.2 case (see Figure 6b) and the model shows a sharper triangular shape of curves in the lower \(k_{R}a\) value. The small crack size case has its peak value at the relatively higher \(k_{R}a\). Since the lower frequency waves can pass through the smaller length or size of the crack, it shows lesser wave scattering energy in the lower \(k_{R}a\). Thus, the peak displacement variation is formed in the higher \(k_{R}a\) as part of a smaller crack model. Figure 4: Displacement variation curves from the analytical model and FE simulation by the different direction of the C18–0.2 model: (**a**) backward scattering and (**b**) forward scattering result. The curves show the shape of the displacement variation denoted as \(u^{bs}/u^{in}\) and \(u^{fs}/u^{in}\) in the analytical model and simulation result. The results indicate that the analytical models solved by different internal methods show a shape similar to that of the FE simulation in both the forward and backward case. Figure 5: Displacement variation curves of analytical model solving five different integral methods and FE simulation shown with different crack depths, _’a’_ of the forward scattered models: (**a**) result with a 12 mm \(a\) and a 0.2 _a_/_b_ ratio model (C12–0.2), (**b**) results with an 18 mm \(a\) and 0.2 _a_/_b_ ratio model (C18–0.2) and (**c**) results with 24 mm of \(a\) and a 0.2 _a_/_b_ ratio model (C24–0.2).