## 4 Special Case of Low Frequency and Low Viscosity
Thus, the stresses of \(\tau_{xx}\), \(\tau_{xz}\) and \(\tau_{zz}\) could be obtained according to Equations (9)-(11), and the ratio of stresses to \(\tau_{xx}\) at depth of zero (\(\tau_{xx0}\)) is shown in Figure 4. The trend of the curves was consistent with that of the surface wave stress curve of linear elastic material, as shown in Figure 4. In addition, the three stresses all increased first and then decreased, and at the depth of twice the wavelength of the surface wave, the stress decreased to almost zero. Furthermore, the directions of stresses \(\tau_{xz}\) and \(\tau_{zz}\) remained the same, but the direction of stress \(\tau_{xx}\) changed from positive to negative at a certain depth. The tractions at the center of the cracks with different length can be calculated using Equation (35), and the results are shown in Table 1. Then, the uniform tractions shown in Table 1 were applied to the crack surface in the numerical models; the corresponding crack-opening displacement along the surface of different cracks was obtained, and the results are shown in Figure 5. It can be seen that the CODs at the same depth increased with the increase in crack length due to the increase in the normal and shear tractions. However, for a certain crack length, the COD decreased with the increase in the crack depth since the two crack surfaces connected with each other at the bottom of the crack, which made the crack surface hard to open. The farther away from the bottom of the crack, the easier it was to open and the larger the crack-opening displacement was. In addition, the maximum crack-opening displacement was several nanometers, which was sufficiently smaller than the crack width in the model, so the two surfaces of the crack did not interact with each other. It was consistent with the hypothesis of the study in the paper. It should be noted that theoretical solution of the crack-opening displacement should be determined to obtain more accurate results, since the shape of the crack in the numerical model is rectangular, which is slightly different from the real crack. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Depth (\(\upmu\)m)** & **50** & **200** & **400** & **600** & **800** & **1000** \\ \hline \(\overline{P}(\mathrm{N}/\mathrm{m}^{2})\) & 21.099 & 20.703 & 20.128 & 19.505 & 18.843 & 18.150 \\ \(\overline{T}(\mathrm{N}/\mathrm{m}^{2})\) & 0.068 & 0.286 & 0.607 & 0.958 & 1.334 & 1.729 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Magnitudes of normal and shear tractions on crack surface. Figure 4: Stress amplitudes as function of depth. For the simplicity of computation, the Newton-Coates formula was used to approximate the average COD (represented as \(\Delta\overline{U}_{x}\) and \(\Delta\overline{U}_{z}\)) of the crack, and then the theoretical amplitudes of the scattering surface waves by cracks can be represented as:  
\[A_{sc_{\alpha}}^{-}=-\frac{i}{2l}\Delta\overline{U}_{x}\int_{0}^{l}T_{xx}(z)dz= -\frac{i\mu\Delta\overline{U}_{x}}{2l}[\frac{d_{4}}{p}\Big{(}1-e^{-pl}\Big{)}+ \frac{d_{5}}{q}\Big{(}1-e^{-ql}\Big{)}\]  
\[A_{sc_{\alpha}}^{-}=-\frac{1}{2l}\Delta\overline{U}_{z}\int_{0}^{l}T_{xz}(z)dz= -\frac{\mu\Delta\overline{U}_{z}}{2l}[\frac{d_{6}}{p}\Big{(}1-e^{-pl}\Big{)}+ \frac{d_{7}}{q}\Big{(}1-e^{-ql}\Big{)} \tag{38}\]  
By substituting Equation (38) into Equations (17) and (19), the theoretical expressions of the displacements of back-scattered surface waves in \(x\) and \(z\) directions can be determined as follows:  
\[u_{x}^{-}=u_{x}^{n-}+u_{x}^{s-}=i\big{(}A_{sc_{\alpha}}^{-}-A_{sc_{\alpha}}^{ -}\big{)}W^{R}(z)e^{-ikx}\]  
\[u_{z}^{-}=u_{z}^{n-}+u_{z}^{s-}=\big{(}A_{sc_{\alpha}}^{-}-A_{sc_{\alpha}}^{-} \big{)}W^{R}(z)e^{-ikx} \tag{39}\]  
It is worth noting that the forward-scattered surface waves can be evaluated by selecting a virtual surface wave that propagates in the \(x\)-negative direction.