## 2 Method
\begin{array}{l}\lambda_{f}=\lambda_{p}+i\omega\lambda_{f}{{}^{\prime}}= \lambda_{p}[1+i\frac{Q_{\rm p}V_{\rm s}^{p2}-2Q_{\rm p}V_{\rm s}^{p2}}{Q_{\rm p }Q_{\rm s}(V_{\rm p}^{p2}-V_{\rm s}^{p2})}]\\ \mu_{f}=\mu_{p}+i\omega\mu_{f}{{}^{\prime}}=\mu_{p}(1+i\frac{1}{Q_{\rm c}})\\ V_{p}^{f}=\sqrt{\frac{\lambda_{f}+2\mu_{f}}{\rho_{f}}}\\ V_{s}^{f}=\sqrt{\frac{\mu_{f}}{\rho_{f}}}\end{array}  \tag{18}\]  
Substituting the Equation (18) into Equations (15) and (16), we can obtain the following equations:  
\[\widetilde{\sigma}_{z}=\lambda_{f}\frac{\partial\widetilde{u}_{x}}{\partial x }+\left(\lambda_{f}+2\mu_{f}\right)\frac{\partial\widetilde{u}_{z}}{\partial z} \tag{20}\]  
\[\widetilde{\tau}_{xz}=\mu_{f}(\frac{\partial\widetilde{u}_{x}}{\partial z}+ \frac{\partial\widetilde{u}_{z}}{\partial x}) \tag{21}\]  
It can be seen that Equations (20) and (21) have the same form as those of purely elastic wave equations. Based on the Helmholtz decomposition theory [41], the 2-D dynamic equation can be expressed as two displacement potential functions. Assume that the \(\widetilde{\varphi_{i}}\)(the displacement potential of the P wave) and \(\widetilde{\psi}_{2i}\)(the second component of the displacement potential of the S wave) are the potential functions of the \(ith\) layer in the frequency domain which can be expressed by following two Equations [21; 39]. \[\widetilde{\varphi_{i}}(x,z,\omega)=U_{p_{i}}e^{i(-k_{\rm z}x+k_{\rm z}z)}+D_{ pi}e^{i(-k_{\rm z}x-k_{\rm z}yz)} \tag{22}\]  
\[\widetilde{\psi}_{2i}(x,z,\omega)=U_{si}e^{i(-k_{\rm z}x+k_{\rm z}z)}+D_{si}e^{ i(-k_{\rm z}x-k_{\rm z}z)} \tag{23}\]  
where \(k_{\rm xi}\), \(k_{\rm zpi}\) are the P-wave numbers of the \(ith\) layer in the \(x\) and \(z\) directions, and \(k_{\rm zsi}\) is the S-wave number of the \(ith\) layer in the \(z\) direction. As shown in Figure 1b, the sandwiched element consists of two types of materials in turn. When the wave propagates in the rock layer, the wave numbers \(k_{\rm xi}\), \(k_{\rm zpi}\) and \(k_{\rm zsi}\) can be expressed as \(k_{\rm x}{{}^{\rm r}}=k_{\rm p}{{}^{\rm r}}\,sin\,\theta\), \(k_{\rm zp}{{}^{\rm r}}=k_{\rm p}{{}^{\rm r}}\,cos\,\theta\), \(k_{\rm zs}^{r}=\sqrt{\left(\omega/V_{s}^{r}\right)^{2}-k_{\rm x}^{2}}\), where \(k_{\rm p}{{}^{\rm r}}=\omega/V_{\rm p}{{}^{\rm r}}\) is the wave number of the P wave. Similarly, when the wave propagates in the filled layer, the wave numbers \(k_{\rm xi}\), \(k_{\rm zpi}\) and \(k_{\rm zsi}\) can be expressed as \(k_{\rm x}{{}^{\rm f}}=k_{\rm p}{{}^{\rm f}}\,sin\,\theta\), \(k_{\rm zp}{{}^{\rm f}}=k_{\rm p}{{}^{\rm f}}\,cos\,\theta\) and \(k_{\rm zs}^{f}=\sqrt{\left(\omega/V_{s}^{f}\right)^{2}-k_{\rm x}^{f^{2}}}\), where \(k_{\rm p}{{}^{\rm f}}=\omega/V_{\rm p}{{}^{\rm f}}\) is the wave number of the P-wave propagation through the filled medium. The particle displacement can be written as:  
\[\widetilde{u}_{xi}=\frac{\partial\widetilde{\varphi_{i}}}{\partial x}-\frac{ \partial\widetilde{\psi}_{2i}}{\partial z} \tag{24}\]  
\[\widetilde{u}_{zi}=\frac{\partial\widetilde{\varphi_{i}}}{\partial z}+\frac{ \partial\widetilde{\psi}_{2i}}{\partial x} \tag{25}\]Substituting Equations (22) and (23) into Equations (24) and (25), we can form:  
\[\widetilde{u}_{xi}=[(-ik_{xi}e^{ik_{xi}x^{2}})U_{pi}+(-ik_{xi}e^{-ik_{xi}x^{2}})D_{ pi}+(-ik_{xzi}e^{ik_{xi}x^{2}})U_{si}+(ik_{xzi}e^{-ik_{xi}x^{2}})]D_{si}]e^{-ik_{xi}x ^{2}} \tag{26}\]  
\[\widetilde{u}_{zi}=[(ik_{zpi}e^{ik_{xi}x^{2}})U_{pi}+(-ik_{zpi}e^{-ik_{z}x^{2}})D _{pi}+(-ik_{xzi}e^{ik_{xi}x^{2}})U_{si}+(-ik_{xzi}e^{-ik_{xi}x^{2}})]D_{si}]e^{-ik _{xi}x} \tag{27}\]  
Based on the constitutive Equations (20) and (21) and Equations (26) and (27), we can also form:  
\[\widetilde{\sigma}_{zi}=\left[ \begin{array}{c}(-\frac{\lambda_{i}\omega^{2}} {V_{pi}^{2}}-2\mu_{i}k_{zpi}^{2})e^{ik_{xi}x^{2}}U_{pi}+(-\frac{\lambda_{i} \omega^{2}}{V_{pi}^{2}}-2\mu_{i}k_{zpi}^{2})e^{-ik_{xi}x^{2}}D_{pi}\\ +2\mu_{i}k_{xi}k_{z\pi}e^{ik_{xi}x^{2}}U_{si}+2\mu_{i}(-k_{xi}x_{zi}E_{si})e^{- ik_{xi}x^{2}}D_{si}\end{array}