# Scattering of Plane Waves by Cylindrical Cavity in Unsaturated Poroelastic Medium +
## 2 Model and Solution
### Solution of Dynamic Equation of Unsaturated Porous Media
\tag{10}\]  
where, \(M_{SS}^{s}=M_{SS}+2n_{0}^{S}\mu_{S}\); \(\gamma_{SS}=n_{0}^{S}\rho_{0}^{S}+\frac{i\left(\hat{\mu}^{W}+\hat{\mu}^{N} \right)}{\omega}\); \(\gamma_{SW}=\frac{i\hat{\mu}^{W}}{\omega}\); \(\gamma_{SW}=\frac{i\hat{\mu}^{W}}{\omega}\); \(\gamma_{WW}=n_{0}^{W}\rho_{0}^{W}+\frac{i\hat{\mu}^{W}}{\omega}\); \(\gamma_{NN}=n_{0}^{N}\rho_{0}^{N}+\frac{i\hat{\mu}^{N}}{\omega}\). Eliminating \(\varphi^{W}\) and \(\varphi^{N}\) from Equation (9), the scalar potential of the solid component \(\varphi^{S}\) can be expressed as:  
\[\left(A\
abla^{6}+B\omega^{2}\
abla^{4}+C\omega^{4}\
abla^{2}+D\omega^{6} \right)\varphi^{S}=0 \tag{11}\]  
where,  
\[\left. \begin{array}{l}A=M_{SS}^{s}A_{3}+M_{SW}A_{1}+M_{SN}A_{2}\\ B=M_{SS}^{s}B_{3}+M_{SW}B_{1}+M_{SN}B_{2}+\gamma_{SS}A_{3}-\gamma_{SW}A_{1}- \gamma_{SN}A_{2}\\ C=M_{SS}^{s}C_{3}+M_{SW}C_{1}+M_{SN}C_{2}+\gamma_{SS}B_{3}-\gamma_{SW}B_{1}- \gamma_{SN}B_{2}\\ D=\gamma_{SS}C_{3}-\gamma_{SW}C_{1}-\gamma_{SN}C_{2}\end{array} \right\} \tag{12}\]  
with  
\[ \begin{array}{l}A_{1}=M_{WN}M_{SN}-M_{SW}M_{NN}B_{1}=M_{NN}\gamma_{SW}-M_{SW} \gamma_{NN}-M_{WN}\gamma_{SN}C_{1}=\gamma_{NN}\gamma_{SW}\\ A_{2}=M_{SW}M_{WN}-M_{SN}M_{WW}B_{2}=M_{WW}\gamma_{SN}-M_{SN}\gamma_{WW}-M_{WN} \gamma_{SW}C_{2}=\gamma_{WW}\gamma_{SN}\\ A_{3}=M_{WW}M_{NN}-M_{WN}M_{WN},\;\;B_{3}=M_{WW}\gamma_{NN}+M_{NN}\gamma_{WW},C_{3}= \gamma_{WW}\gamma_{NN}.\end{array} \right.\end{array} \tag{13}\]  
To solve the general solution of \(\varphi^{S}\), Equation (11) can be decomposed into:  
\[\left(\
abla^{2}-k_{j}^{2}\right)\varphi^{S}_{j}=0(j=1,2,3) \tag{14}\]with  
\[\left\{ \begin{array}{l}k_{1}^{2}=-\frac{\varphi^{2}}{3A}\bigg{(}B-\frac{\left(1+ \sqrt{3}i\right)A^{*}}{2C^{*}}+\frac{\left(1-\sqrt{3}i\right)}{2}C^{*}\bigg{)} \\ k_{2}^{2}=-\frac{\varphi^{2}}{3A}\bigg{(}B-\frac{\left(1-\sqrt{3}i\right)A^{*}}{ 2C^{*}}+\frac{\left(1+\sqrt{3}i\right)}{2}C^{*}\bigg{)}\\ k_{3}^{2}=-\frac{\varphi^{2}}{3A}\bigg{(}B+\frac{A^{*}}{C^{*}}-C^{*}\bigg{)} \end{array} \right. \tag{14}\]  
where, \(A^{*}=3 AC-B^{2};B^{*}=9 ABC-2B^{3}-27A^{2}D;C^{*}=\left[\frac{B^{*}}{2}+\sqrt[ \left(A^{*}\right)^{3}+\left(\frac{B^{*}}{2}\right)^{2}\right]^{1/3}\). It can be seen from Equation (13) that there are three kinds of compression waves named P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\) waves in unsaturated porous media. \(k_{i}(i=1,2,3)\) are the wave numbers associated with the three types of waves. In cylindrical coordinates, Equation (13) can be expressed as:  
\[\frac{\partial^{2}\varphi_{j}^{\mathrm{S}}}{\partial r^{2}}+\frac{1}{r}\frac {\partial\varphi_{j}^{\mathrm{S}}}{\partial r}+\frac{1}{r^{2}}\frac{\partial ^{2}\varphi_{j}^{\mathrm{S}}}{\partial\theta^{2}}-k_{j}^{2}\varphi_{j}^{ \mathrm{S}}=0 \tag{15}\]  
By using the method of separating variables and according to Sommerfeld radiation condition [20], the solution \(\varphi_{j}^{\mathrm{S}}(r,\theta)\) is obtained as:  
\[\varphi_{j}^{\mathrm{S}}(r,\theta)=\sum_{n=0}^{\infty}A_{n,j}K_{n}\left(k_{j }r\right)\cos n\theta(j=1,2,3) \tag{16}\]  
where \(K_{n}(k_{j}r)\) denotes the \(n\)-order modified Bessel functions of the first kind, \(A_{n,j}(j=1,2,3)\) is the undetermined constant coefficient.