# Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium +
## 3 Numerical Results and Discussion
For example, when \(k_{0}a=2.0\), the maximum values for \(v=0.1\) occur at \(\theta=50^{\circ}\) and \(310^{\circ}\), whereas for \(v=0.2\), at \(\theta=40^{\circ}\) and \(320^{\circ}\), and for \(v=0.3\), at \(\theta=20^{\circ}\) and \(340^{\circ}\). For all [PERSON]'s ratios, the minimum values appear at \(\theta=180^{\circ}\). Moreover, with the increase in the incident frequency, the angle between the maximum value direction and the incident wave direction (\(\theta=0^{\circ}\)) decreases at the same Poisson's ratio. For \(v=0.3\), the maximum values appear at \(\theta=70^{\circ}\) and \(290^{\circ}\) when \(k_{0}a=0.5\), at \(\theta=40^{\circ}\) and \(320^{\circ}\) when \(k_{0}a=1.0\), and at \(\theta=20^{\circ}\) and \(340^{\circ}\) when \(k_{0}a=2.0\). Figure 3: Effects of saturation on distributions of \(\sigma_{\theta\theta}^{S}/\sigma_{0}\) around the surface of the cavity for P\({}_{1}\) wave incidence with the second set of parameters in Table 1 with \(v=0.3\) s for a permeable boundary. (a) \(k_{0}a=0.5\). (b) \(k_{0}a=1.0\). (c) \(k_{0}a=3.0\). According to Ref. [21], the volume modulus \(\hat{K}\) and shear modulus \(G\) of solid skeletons decrease with the increase in porosity \(n_{0}\), which will make the wave number change, and thus, the stress around the cavity will change accordingly. Next, the effects of porosity \(n_{0}\) on the dynamic responses around the cavity under two saturation conditions, \(S_{\rm r}=0.6\) and \(1.0\), are analyzed for the impermeable cavity. Taking the three sets of parameters in Table 1 with \(v=0.3\), Figure 5 gives the distributions of \(\sigma_{\theta\theta}^{S}/\sigma_{0}\) around the cavity surface for the P\({}_{1}\) wave incidence under \(n_{0}=0.1\), \(0.34\), and \(0.36\) for \(S_{\rm r}=0.6\) with \(k_{0}a=0.5\), \(1.0\), and \(2.0\), respectively. As can be seen from the figure, when \(S_{\rm r}=0.6\), the porosity \(n_{0}\) does not affect the distribution of \(\sigma_{\theta\theta}^{S}/\sigma_{0}\). That means, for \(S_{\rm r}=0.6\), although porosity will affect the stiffness of the medium, thus affecting the dynamic stress, it will not affect the dynamic stress concentration factor. This is because Poisson's ratio does not change with porosity in the analysis. According to Ref. [14], the velocity of the P1 wave has a sharp decline from a fully saturated condition (\(S_{\rm r}=1.0\)) to an unsaturated condition (\(S_{\rm r}<1.0\)). However, the velocity of the S wave changes little with the saturation. For \(S_{\rm r}<1.0\), the ratio of the P\({}_{1}\) wave velocity to the S wave velocity is only related to Poisson's ratio, but not to porosity. When the frequency is normalized as \(k_{0}a\), the effect of the stiffness on the wave number is eliminated. As Poisson's ratio does not change with the porosity, the change in the stiffness of the surrounding medium with the porosity certainly will not influence the normalized hoop stresses. Figure 6 gives the distributions of \(\sigma_{\theta\theta}^{S}/\sigma_{0}\) around the cavity surface for the P\({}_{1}\) wave incidence under \(n_{0}=0.1\), \(0.34\), and \(0.36\) for \(S_{\rm r}=1.0\) with \(k_{0}a=0.5\), \(1.0\), and \(2.0\), respectively. Here, \(k_{0}\) is taken as the wave number of the incident P\({}_{1}\) wave. It can be seen that when \(S_{\rm r}=1.0\), the porosity has a great influence on the dynamic stress concentration factor. The distribution pattern of \(\sigma_{\theta\theta}^{S}/\sigma_{0}\) around the cavity surface varies greatly for different porosities. This situation is different from that of unsaturated (\(S_{\rm r}<1.0\)). That is mainly because that the ratio of P\({}_{1}\) wave velocity to S wave velocity varies greatly with the porosity of the saturated poroelastic medium. According to Ref. [21], when \(n_{0}=0.1\), the ratio of the P\({}_{1}\) wave velocity to the S wave velocity is nearly \(\sqrt{\frac{2(1-v)}{1-2v}}=1.87\). With the increase in the porosity, the ratio of the P\({}_{1}\) wave velocity to the S wave velocity increases a lot.