# Dynamic Behavior of an Embedded Foundation under Horizontal Vibration in a Poroelastic Half-Space
## 2 Problem Definition and Governing Equations
### General Solutions of Horizontal Vibration of Embedded Foundation  
#### 2.3.1 Reaction Force at the Side of the Embedded Foundation  
Thin layer theory is adapted to conduct the reaction force along the height of the embedded foundation (\(L\)). The integration of force along the foundation side can be conducted to calculate the reaction force along the foundation side. The horizontal displacement of foundation under horizontal excitation is \(u_{\text{h}}\) (at \(\theta=0\) direction), the foundation is under time-harmonic vibration. The normal stress on the ground surface is zero. There's no displacement at the bottom of the foundation. The model of motion is purely horizontal vibration, and the vertical and torsional displacement is neglected. Displacement at tangential and radial direction is not zero, that is, \(u_{r}\), \(u_{\theta}\
eq 0\), \(w_{r}\), \(w_{\theta}\
eq 0\), \(u_{x}\), \(w_{z}=0\). With the radiation condition, the boundary condition of the model of motion can be written as,  
\[\begin{array}{l}\sigma_{z}(r,\theta,0)=0\;r>0,\;0\leq\theta\leq 2\pi,\;0<z<L\\ u_{r}(r,\theta,L)=0,\;u_  
\(E=e_{8}=\bigtriangledown 2\varphi s\) and \(e_{\rm f}=\bigtriangledown^{2}\varphi_{\rm f}\), the following equations can be obtained by substituting Equations (5) and (6) to Equations (2) and (3),  
\[ \begin{array}{l}(\lambda+2G_{\rm s})\frac{\partial\
abla^{2}\varphi_{\rm s}}{ \partial\sigma}+\frac{G_{\rm s}}{r}\frac{\partial\
abla^{2}\psi_{\rm s}}{ \partial\theta}+\Big{(}\alpha b\omega i+\rho_{f}\omega^{2}-a\omega^{2}\alpha \Big{)}\Big{(}\frac{\partial\varphi_{f}}{\partial\sigma}+\frac{1}{r}\frac{ \partial\varphi_{f}}{\partial\theta}\Big{)}\\ =-\Big{(}\rho-a\varphi_{f}\Big{)}\omega^{2}\Big{(}\frac{\partial\varphi_{\rm s }}{\partial\sigma}+\frac{1}{r}\frac{\partial\psi_{\rm s}}{\partial\theta} \Big{)}\\ (\lambda+2G_{\rm s})\frac{1}{r}\frac{\partial\
abla^{2}\varphi_{\rm s}}{ \partial\theta}-G_{\rm s}\frac{\partial\
abla^{2}\psi_{\rm s}}{\partial \sigma}+\Big{(}\alpha b\omega i+\rho_{f}\omega^{2}-a\omega^{2}\alpha\Big{)} \Big{(}\frac{1}{r}\frac{\partial\varphi_{f}}{\partial\theta}-\frac{\partial \varphi_{f}}{\partial r}\Big{)}\\ =-\Big{(}\rho-\alpha\varphi_{f}\Big{)}\omega^{2}\Big{(}\frac{1}{r}\frac{ \partial\varphi_{\rm s}}{\partial\theta}-\frac{\partial\psi_{\rm s}}{ \partial\sigma}\Big{)}\end{array}  \tag{11}\]  
\[ \begin{array}{l}(\lambda+2G_{\rm s})\frac{1}{r}\frac{\partial\
abla^{2} \varphi_{\rm s}}{\partial\theta}-G_{\rm s}\frac{\partial\
abla^{2}\psi_{\rm s }}{\partial r}+\Big{(}\alpha b\omega i+\rho_{f}\omega^{2}-a\omega^{2}\alpha \Big{)}\Big{(}\frac{1}{r}\frac{\partial\varphi_{f}}{\partial\theta}-\frac{ \partial\varphi_{f}}{\partial r}\Big{)}\\ =-\Big{(}\rho-\alpha\varphi_{f}\Big{)}\omega^{2}\Big{(}\frac{1}{r}\frac{ \partial\varphi_{\rm s}}{\partial\theta}-\frac{\partial\psi_{\rm s}}{ \partial\sigma}\Big{)}\end{array}  \tag{12}\]  
The following equations can be derived by substituting the potential functions to the fluid constitutive relations after some manipulations. \[M\
abla^{2}\varphi_{f}+\alpha M\
abla^{2}\varphi_{\rm s}=-\rho_{f}\omega^{2} \varphi_{s}-\Big{(}a\omega^{2}-bi\omega\Big{)}\varphi_{f} \tag{13}\]  
\[\rho_{f}\omega^{2}\psi_{s}+\Big{(}a\omega^{2}-bi\omega\Big{)}\psi_{f}=0 \tag{14}\]  
After divergence and curl operation of Equations (13) and (14), the following equation set can be obtained:  
\[\left[ \begin{array}{cc}\big{(}\lambda+2G_{\rm s}+\alpha^{2}M\big{)}&\
abla^ {2}+\rho\omega^{2}\alpha M\
abla^{2}+\rho_{f}\omega^{2}\\ \alpha M+\rho_{f}\omega^{2}&M\
abla^{2}+\big{(}a\omega^{2}-bi\omega\big{)} \end{array} \right]\left[ \begin{array}{c}\varphi_{\rm s}\\ \varphi_{f}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array} \right] \tag{15}\]  
\[\left[ \begin{array}{cc}G_{\rm s}\
abla^{2}+\rho\omega^{2}&\rho_{f}\omega^{ 2}\\ \rho_{f}\omega^{2}&\big{(}a\omega^{2}-bi\omega\big{)}\end{array} \right]\left[ \begin{array}{c}\psi_{\rm s}\\ \psi_{f}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\end{array}