## 3 Theoretical Formulation  
Due to the existence of the pile, scattered P and S waves will be generated. The soil motion can be expressed as the sum of the free-field displacement and the scattered soil displacement, expressed as:  
\[u_{z,j}(r,z)=u_{j}^{f}(z)+u_{z,j}^{s}(r,z) \tag{11}\]  
The free-field displacement is the soil displacement of the half-space under the P-wave incidence. It is obtained as:  
\[u_{z,j}^{f}(z)=\frac{cos\kappa_{j}z}{cos\kappa_{j}H}u_{0} \tag{12}\]  
where \(\kappa_{j}=\sqrt{\rho_{s,j}\omega^{2}/\left(\lambda_{j}{}^{s}+2G_{j}{}^{s} \right)}\).  
Using the separation of variables method, the scattered soil displacement can be obtained through Equation (1):  
\[u_{z,j}^{s}(r,z)=\sum_{n=1}^{\infty}\left[A_{j,n}K_{0}\big{(}\eta_{j,n}r\big{)} +B_{j,n}I_{0}\big{(}\eta_{j,n}r\big{)}\right]cos(\beta_{n}z),\ j=1,2,\ldots,N+1 \tag{13}\]  
Substituting Equations (11)-(13) into Equations (3) and (8), one can obtain:  
\[\mathbf{M}_{j-1,n}\left(r_{j-1}\right)\left[\begin{array}{c}A_{j-1,n}\\ B_{j-1,n}\end{array}\right]=\mathbf{M}_{j,n}\big{(}r_{j-1}\big{)}\left[\begin{array} []{c}A_{j,n}\\ B_{j,n}\end{array}\right]+\mathbf{a}_{j,n} \tag{14}\]  
\[B_{N+1,n}=0 \tag{15}\]  
where \(\mathbf{M}_{j,n}(r)\) and \(\mathbf{a}_{j,n}\) are presented in in Appendix A.  
Then, the following relation between \(\left[\begin{array}{c}A_{1,n}\\ B_{1,n}\end{array}\right]\) and \(\left[\begin{array}{c}A_{N+1,n}\\ B_{N+1,n}\end{array}\right]\) can be obtained:  
\[\left[\begin{array}{c}A_{1,n}\\ B_{1,n}\end{array}\right]=\textbf{Q}_{n}\left[\begin{array}{c}A_{N+1,n}\\ B_{N+1,n}\end{array}\right]+\textbf{R}_{n} \tag{16}\]  
where  
\[\textbf{Q}_{n}=\prod_{j=1}^{N}\textbf{M}_{j,n}^{-1}\big{(}r_{j}\big{)}\textbf {M}_{j+1,n}\big{(}r_{j}\big{)} \tag{17}\]  
\[\textbf{R}_{n}=\sum_{j=2}^{N}\prod_{k=1}^{j-1}\textbf{M}_{k,n}^{-1}(r_{k}) \textbf{M}_{k+1,n}(r_{k})\textbf{M}_{j,n}^{-1}\big{(}r_{j}\big{)}\textbf{a}_{j +1,n}+\textbf{M}_{1,n}^{-1}(r_{1})\textbf{a}_{2,n} \tag{18}\]  
The frictional force acting on the surface of the pile can be derived through Equation (9), written as:  
\[f(z)=2\pi r_{0}\tau_{zr}(r_{0},z)=2\pi r_{0}\sum_{n=1}^{\infty}\left[\begin{array} []{cc}M_{1,n}^{21}(r_{0})&M_{1,n}^{22}(r_{0})\end{array}\right]\left[ \begin{array}{c}A_{1,n}\\ B_{1,n}\end{array}\right]cos(\beta_{1,n}z) \tag{19}\]  
Substituting Equation (19) into Equation (2), the pile displacement is obtained as:  
\[\begin{array}{c}w_{p}(z)=a_{p}sin(\chi_{p}z)+b_{p}cos(\chi_{p}z)\\ -\frac{2\sum_{n=1}^{\infty}\left[\begin{array}{cc}M_{1,n}^{21}(r_{0})&M_{1,n }^{22}(r_{0})\end{array}\right]\left[\begin{array}{c}A_{1,n}\\ B_{1,n}\end{array}\right]}{(\rho_{p}\omega^{2}-E_{p}\rho_{n}^{2})r_{0}}cos( \beta_{n}z)\end{array} \tag{20}\]  
where \(\chi_{p}=\omega/\sqrt{E_{p}/\rho_{p}}\)  
Substituting Equation (20) into Equations (5) and (7), one can obtain:  
\[a_{p}=0 \tag{21}\]  
\[b_{p}=\frac{u_{0}}{cos(\chi_{p}H)} \tag{22}\]  
Substituting Equations (11), (16) and (20) into Equation (10), one can obtain:  
\[\begin{array}{c}\left[\begin{array}{cc}M_{1,n}^{11}(r_{0})+\frac{2M_{1,n }^{21}(r_{0})}{(\rho_{p}\omega^{2}-E_{p}\rho_{n}^{2})r_{0}}&M_{1,n}^{12}(r_{0 })+\frac{2M_{1,n}^{22}(r_{0})}{(\rho_{p}\omega^{2}-E_{p}\rho_{n}^{2})r_{0}} \end{array}\right]\textbf{Q}_{n}\left[\begin{array}{c}A_{N+1,n}\\ B_{N+1,n}\end{array}\right]\\ =\frac{2\pi r}{H}\int_{0}^{H}sin(\chi_{p}z)cos(\beta_{n}z)dz+\frac{2b_{p}}{H} \int_{0}^{H}cos(\chi_{p}z)cos(\beta_{n}z)dz\\ -\frac{2\mu_{0}}{H}\int_{0}^{H}\frac{cos(\chi_{1,n}z)cos(\beta_{n}z)dz}{cos \chi_{1,n}H} \tag{23}\]  
Then, the unknown coefficient \(A_{N+1,n}\) can be obtained through the above equation. With \(A_{N+1,n}\) and \(B_{N+1,n}\) obtained, the pile displacement can be determined. The numerical results in Section 4 are delivered through Equation (23) by MATLAB programming.  
The kinematic response factor is defined as:  
\[I_{u}=\frac{w_{p}(0)}{u_{N+1}^{f}(0)} \tag{24}\]  
The kinematic amplification factor is defined as:  
\[A=\frac{w_{p}(0)}{u_{0}} \tag{25}\]