## 4 Solutions for the Equations
### Solution for the Pile
\[U_{\rm P1}=a_{1}\Bigg{[}\cos\Big{(}\overline{\lambda_{1}z^{\prime}}\Big{)}+\sum \limits_{k=1}^{\infty}\zeta_{1k}^{\prime}\cos\Big{(}\overline{h_{11k}z^{ \prime}}-\varphi_{11k}\Big{)}\Bigg{]}+b_{1}\Bigg{[}\sin\Big{(}\overline{ \lambda_{1}z^{\prime}}\Big{)}-\sum\limits_{k=1}^{\infty}\zeta_{1k}^{\prime \prime}\cos\Big{(}\overline{h_{11k}z^{\prime}}-\varphi_{11k}\Big{)}\Bigg{]} \tag{13}\]  
where \(\zeta_{1k}^{\prime}=\zeta_{1k}\Bigg{[}\frac{\sin\big{[}(\overline{\lambda_{1} -h_{11k}})+\varphi_{11k}\big{]}-\sin(\varphi_{11k})}{\overline{\lambda_{1}-h_ {11k}}}+\frac{\sin\big{[}(\overline{\lambda_{1}+h_{11k}})-\varphi_{11k}\big{]} +\sin(\varphi_{11k})}{\overline{\lambda_{1}+h_{11k}}}\Bigg{]}\);  
\(\zeta_{1k}=-\frac{\zeta_{11k}\Big{[}\frac{\cos\big{[}(\overline{\lambda_{1}+h_ {11k}})-\varphi_{11k}\big{]}-\cos(\varphi_{11k})}{\overline{\lambda_{1}+h_{11 k}}}+\frac{\cos\big{[}(\overline{\lambda_{1}-h_{11k}})+\varphi_{11k} \big{]}-\cos(\varphi_{11k})}{\overline{\lambda_{1}-h_{11k}}}\Big{]}}\);  
\(\zeta_{1k}=-\frac{\zeta_{11k}^{\prime}\overline{\lambda_{11k}}}{\overline{ \varphi_{1}}^{\rm P1}\big{(}\mathfrak{o}_{1}^{2}-\overline{h_{11k}}\mathfrak{o }_{1}^{2}/\overline{\lambda_{1}^{\prime}}\big{)}}\varphi_{11k}\overline{ \omega_{1}\omega_{1}^{\prime}\cos^{2}\big{(}\overline{h_{11k}z^{\prime}}- \varphi_{11k}\big{)}}\overline{\big{[}M_{11k}I_{1}\big{(}\overline{q_{11k}r_{ \rm P1}}\big{)}\big{]}}\);  
\(\phi_{1k}=M_{11k}\Bigg{[}\mathfrak{l}_{0}\big{(}\overline{q_{11k}r_{\rm P1}} \big{)}+\frac{2\overline{\zeta_{11}^{\prime}}\overline{q_{11k}}}{\overline{r_ {\rm P1}}\big{(}\mathfrak{o}_{1}^{2}-\overline{h_{11k}}\mathfrak{o}_{1}^{2}/ \overline{\lambda_{1}^{\prime}}\big{)}}\mathrm{I}_{1}\big{(}\overline{q_{11k}r_ {\rm P1}}\big{)}\Bigg{]}+\mathrm{K}_{0}\big{(}\overline{q_{11k}r_{\rm P1}} \big{)}\)  
\(-\frac{2\overline{\zeta_{11}^{\prime}}\overline{q_{11k}}}{\overline{r_{\rm P1} }\big{(}\mathfrak{o}_{1}^{2}-\overline{h_{11k}}\mathfrak{o}_{1}^{2}/\overline {\lambda_{1}^{\prime}}\big{)}}\mathrm{K}_{1}\big{(}\overline{q_{11k}r_{\rm P1}} \big{)}\); \(\overline{\lambda_{1}}=\sqrt{\frac{\mathfrak{o}_{1}^{2}}{1-\overline{\mu_{1}^{ \prime}}\overline{p_{1}^{\prime}}\overline{p_{1}^{\prime}}\overline{p_{1}^{ \prime}}\overline{p_{1}^{\prime}}\overline{p_{1}^{\prime}}\overline{p_{1}^{ \prime}}}}\); \(\theta_{1}=\omega t_{1}\), \(\overline{h_{11k}}=h_{11k}l_{1}\), \(\overline{q_{11k}}=q_{11k}l_{1}\), \(\overline{q_{11k}}=\overline{\lambda_{1}}=\overline{\lambda_{1}}^{\prime}/E_{ \rm P1}\), \(\overline{z^{\prime}}=z^{\prime}/l_{1}\), and \(\overline{r_{\rm P1}}=r_{\rm P1}/l_{1}\) are all dimensionless coefficients; \(M_{11k}=A_{11k}/B_{11k}\) and can be derived according to Assumption (4). Applying the Laplace transform to Equation (10) and substituting Equation (13) into it, the dynamic impedance at the top of the fictitious soil pile (i.e., the bottom of the pile) can be obtained as  
\[Z_{\rm P1}=-\frac{E_{\rm P1}A_{\rm P1}-\rho_{\rm P1}A_{\rm P1}\big{(}\mu_{\rm P 1}r_{\rm P1}\big{)}^{2}\omega^{2}}{l_{1}}\frac{a_{1}}{k_{1}}\frac{\sum\limits_ {k=1}^{\infty}\zeta_{1k}^{\prime}\overline{h_{11k}}\sin\varphi_{11k}}{ \overline{a_{1}}\Big{(}1+\sum\limits_{k=1}^{\infty}\zeta_{1k}^{\prime}\cos \varphi_{11k}\Big{)}-\sum\limits_{k=1}^{\infty}\zeta_{1k}^{\prime\prime}\cos \varphi_{11k}} \tag{14}\]  
where \(\frac{a_{1}}{b_{1}}=-\frac{\sin\overline{\lambda_{1}}-\sum\limits_{k=1}^{ \infty}\zeta_{1k}^{\prime\prime}\cos(\overline{h_{11k}}-\varphi_{11k})}{\cos \overline{\lambda_{1}}+\sum\limits_{k=1}^{\infty}\zeta_{1k}^{\prime}\cos( \overline{h_{11k}}-\varphi_{11k})}\) is derived according to Equation (11). Using a similar approach and together with the recursive method, the displacement impedance at the head of the large-diameter-bored pile, \(Z_{\rm p1n}\), can be derived. It can be further written as  
\[Z_{\rm p1n}=K_{\rm P}+{\rm i}\cdot C_{\rm P} \tag{15}\]  
where \(K_{\rm P}\) and \(C_{\rm P}\) denote the dynamic stiffness and damping, respectively.