## 3 The Non-Uniform Spatio-Temporal Dynamic Response of the Seismic Interface
### Stoneley Equation Considering the Scale of the Slope on the Interface
\begin{array}{cccc}ik_{\rm ST}&\beta_{1}&-ik_{\rm ST}&\beta_{2}\\ -\alpha_{1}&ik_{\rm ST}&-\alpha_{2}&-ik_{\rm ST}\\ \mu_{1}\big{(}k_{\rm ST}^{2}+\beta_{1}^{2}\big{)}&-2i\mu_{1}\beta_{1}k_{\rm ST }&-\mu_{2}\big{(}k_{\rm ST}^{2}+\beta_{2}^{2}\big{)}&-2i\mu_{2}\beta_{2}k_{\rm ST }\\ -2i\mu_{1}\alpha_{1}k_{\rm ST}&-\mu_{1}\big{(}k_{\rm ST}^{2}+\beta_{1}^{2} \big{)}&-2i\mu_{2}\alpha_{2}k_{\rm ST}&\mu_{2}\big{(}k_{\rm ST}^{2}+\beta_{2} ^{2}\big{)}\end{array} \right]\left[ \begin{array}{c}A_{1}\\ B_{1}\\ A_{2}\\ B_{2}\end{array} \right]=0 \tag{13}\]  
where \(A\) and \(B\) denote the amplitude of the displacement potential function of the seismic wave, and their values cannot be simultaneously 0. Therefore, the homogeneous function (13) has a non-zero solution and the value of the determinant of its coefficient matrix is 0. Subsequently, the value of \(k_{\rm ST}\) can be obtained, and Equation (13) can be solved and substituted into Equation (7) to obtain the expression for the displacement component,  
\[ \begin{array}{l}u_{x}=u_{x}^{(1)}+u_{x}^{(2)}\\ =\left(\frac{\partial\varphi_{1}}{\partial x}+\frac{\partial\varphi_{2}}{ \partial x}\right)-\left(\frac{\partial\varphi_{1}}{\partial x}+\frac{ \partial\varphi_{2}}{\partial x}\right)\\ =ik_{\rm ST}(\varphi_{1}+\varphi_{2})+\beta_{1}\psi_{1}-\beta_{2}\psi_{2} \end{array}  \tag{14}\]  
\[ \begin{array}{l}u_{z}=u_{z}^{(1)}+u_{z}^{(2)}\\ =\left(\frac{\partial\varphi_{1}}{\partial z}+\frac{\partial\varphi_{2}}{ \partial z}\right)-\left(\frac{\partial\varphi_{1}}{\partial x}+\frac{ \partial\varphi_{2}}{\partial x}\right)\\ =-\alpha_{1}\varphi_{1}+\alpha_{2}\varphi_{2}+ik_{\rm ST}(\psi_{1}+\psi_{2}) \end{array}  \tag{15}\]  
By substituting Equation (4) into Equations (13) and (14), the expression for the displacement component considering the scale of the slope on the interface is obtained:  
\[ \begin{array}{l}\mbox{Re}(\left.u_{x}\right|_{z=0})=-k_{\rm ST}(A_{1}+A_{2} )\mbox{sin}\bigg{[}k_{\rm ST}x_{k}-\omega h\sqrt{1+\left[(\Delta/h)+(L_{k}/h )\right]^{2}}/c\bigg{]}\\ \hskip 14.226378 pt+(B_{1}\beta_{1}-B_{2}\beta_{2})\mbox{cos}\bigg{[}k_{\rm ST }x_{k}-\omega h\sqrt{1+\left[(\Delta/h)+(L_{k}/h)\right]^{2}}/c\bigg{]}\end{array}  \tag{16}\]  
\[ \begin{array}{l}\mbox{Re}(\left.u_{z}\right|_{z=0})=-k_{\rm ST}(B_{1}+B_{2} )\mbox{sin}\bigg{[}k_{\rm ST}x_{k}-\omega h\sqrt{1+\left[(\Delta/h)+(L_{k}/h )\right]^{2}}/c\bigg{]}\\ \hskip 14.226378 pt+(-A_{1}\alpha_{1}+A_{2}\alpha_{2})\mbox{cos}\bigg{[}k_{\rm ST }x_{k}-\omega h\sqrt{1+\left[(\Delta/h)+(L_{k}/h)\right]^{2}}/c\bigg{]}\end{array}  \tag{17}\]  
According to the Fourier transform, the displacement potential function of the actual seismic waves can be decomposed into integrals of plane waves of different \(\omega\), and Equation (7) can be written as  
\[\left\{ \begin{array}{l}\varphi=\int_{0}^{\infty}A(\omega)\mbox{exp}[\mp \alpha z+i(k_{\rm ST}x-\omega t)]\mbox{d}\omega\\ \psi=\int_{0}^{\infty}B(\omega)\mbox{exp}[\mp\beta z+i(k_{\rm ST}x-\omega t)] \mbox{d}\omega\end{array} \right. \tag{18}\]  
The displacement expressions similar to Equations (16) and (17) can be obtained by substituting the solution of Equation (13) into Equation (18). It can be concluded that the value of the displacement on the interface at the base of the slope varies with \(L_{k}\) at the same moment \(t\), and the variation patterns are subjected to \(L/h\). Such a non-uniform spatio-temporal dynamic response is caused by non-uniform seismic wave incidence, in which the slope scale plays an important role.