# Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium +
## 3 Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Porosity \(n_{0}\)** & **0.1** & **0.34** & **0.36** \\ \hline \(\rho_{00}^{\rm S}\) (kg/m\({}^{3}\)) & 2650 & 2650 & 2650 \\ \(\rho_{0}^{\rm U}\) (kg/m\({}^{3}\)) & 1000 & 1000 & 1000 \\ \(\rho_{0}^{\rm U}\) (kg/m\({}^{3}\)) & 1.1 & 1.1 & 1.1 \\ \(K_{\rm S}\) (kPa) & 3.6 \(\times\) 10\({}^{7}\) & 3.6 \(\times\) 10\({}^{7}\) & 3.6 \(\times\) 10\({}^{7}\) \\ \(K_{\rm W}\) (kPa) & 2.0 \(\times\) 10\({}^{6}\) & 2.0 \(\times\) 10\({}^{6}\) & 2.0 \(\times\) 10\({}^{6}\) \\ \(K_{\rm N}\) (kPa) & 110 & 110 & 110 \\ \(v^{\rm W}\) (Pa\(\cdot\)s) & 1.0 \(\times\) 10\({}^{-3}\) & 1.0 \(\times\) 10\({}^{-3}\) & 1.0 \(\times\) 10\({}^{-3}\) \\ \(v^{\rm N}\) (Pa\(\cdot\)s) & 1.8 \(\times\) 10\({}^{-5}\) & 1.8 \(\times\) 10\({}^{-5}\) & 1.8 \(\times\) 10\({}^{-5}\) \\ \(x\) (m\(\cdot\)s\({}^{-1}\)) & 2.5 \(\times\) 10\({}^{-12}\) & 2.5 \(\times\) 10\({}^{-12}\) & 2.5 \(\times\) 10\({}^{-12}\) \\ \(G\) & 1.203 \(\times\) 10\({}^{7}\) & – & 9.23 \(\times\) 10\({}^{4}\) \\ \(a_{\rm B}\) & 1 & 1 & 1 \\ \(\hat{K}\) (kPa) & 2.6055 \(\times\) 10\({}^{7}\) & 2.189 \(\times\) 10\({}^{6}\) & 2.0 \(\times\) 10\({}^{5}\) \\ \(v\) & 0.3 & 0.1, 0.2, 0.3 & 0.3 \\ \(\alpha\) & 2.0 \(\times\) 10\({}^{-5}\) & 2.0 \(\times\) 10\({}^{-5}\) & 2.0 \(\times\) 10\({}^{-5}\) \\ \(d\) & 4.15 & 4.15 & 4.15 \\ \(S_{\rm FW}\) & 0.05 & 0.05 & 0.05 \\ \(S_{\rm FN}\) & 1.0 & 1.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters of the unsaturated porous medium. Figure 2: Effects of saturation on distributions of \(\sigma_{\theta\theta}^{\rm S}/\sigma_{0}\) around the surface of the cavity for P\({}_{1}\) wave incidence with the second set of parameters in Table 1 with \(v=0.3\) for an impermeable boundary. (a) \(k_{0}a=0.5\). (b) \(k_{0}a=1.0\). (c) \(k_{0}a=3.0\). the incident wave (\(\theta=0^{\circ}\)). At high-frequency incidence (\(k_{0}a=3.0\)), for \(S_{\rm r}<1.0\), the stress peaks at \(\theta=35^{\circ}\) and \(325^{\circ}\), and for \(S_{\rm r}=1.0\), the stress peaks at \(\theta=30^{\circ}\) and \(330^{\circ}\). The minimum stress appears in the opposite direction of the incident wave (\(\theta=180^{\circ}\)) for \(S_{\rm r}<1.0\) under all incident frequencies and \(S_{\rm r}=\)1.0 under \(k_{0}a=0.5\), 1.0. Under \(k_{0}a=3.0\), The minimum stress for \(S_{\rm r}=1.0\) occurs at \(\theta=130^{\circ}\) and \(230^{\circ}\). With the increase in the incident frequency, the effects of the saturation on the dynamic stress concentration factor increase. Under \(k_{0}a=0.5\), 1.0, the dynamic stress concentration factors for \(S_{\rm r}=1.0\) are larger than that for \(S_{\rm r}<1.0\). However, under \(k_{0}a=3.0\), the result is the opposite. When \(S_{\rm r}<1.0\), the dynamic stress concentration factor increases slightly with the increase in the saturation degree. By comparing the permeable and impermeable boundary conditions, the dynamic stress concentration factor of an impermeable cavity is slightly larger than that of a permeable cavity for \(S_{\rm r}=1.0\). For \(S_{\rm r}<1.0\), there are no obvious differences between the dynamic stress concentration factors of the permeable and impermeable cavities. (2) The influence of Poisson's ratio on the dynamic responses of the cavity  
Taking the second set of parameters in Table 1 with \(S_{\rm r}=0.6\), Figure 4 gives the distributions of \(\sigma_{\theta\theta}^{S}/\sigma_{0}\) around the cavity surface for P\({}_{1}\) wave incidence for different Poisson's ratios of \(v=0.1\), 0.2, and 0.3 with \(k_{0}a=0.5\), 1.0, and 2.0 under impermeable conditions, respectively. As can be seen from the figure, with the increase in Poisson's ratio, the stress increases gradually under different frequencies, and the angle at which the maximum value occurs gradually approaches the direction of the incident wave direction.