## 2 Mathematical Solution
### Displacement Fields  
The displacement fields \(\mathbf{u}\) that resulted from the CF in the cylindrical geometry can be derived by using the NL equation [25],  
\[(\lambda+2\mu)\
abla(\
abla\cdot\mathbf{u})-\mu\
abla\times\
abla\times\mathbf{u}+ \mathbf{f}=\rho\frac{d^{2}\mathbf{u}}{dt^{2}}, \tag{11}\]  
and the [PERSON]'s model [26; 27],  
\[\mathbf{u}=\
abla\Phi+\
abla\times(X\mathbf{\hat{e}}_{z})+a\
abla\times\
abla\times( \
abla\mathbf{\hat{e}}_{z}), \tag{12}\]  
where \(\lambda\) and \(\mu\) are Lame constants, and \(\rho\) is the density of the media. In Equation (12), \(\Phi\) is the scalar potential for the P wave, \(X\mathbf{\hat{e}}_{z}\) is vector potential for the SH wave, \(\Psi\mathbf{\hat{e}}_{z}\) is vector potential for the SV wave, and \(a\) is the outer radius of the cylinder. Note that the velocities of the P and S waves are given as \(c_{P}=\sqrt{(\lambda+2\mu)/\rho}\) and \(c_{S}=\sqrt{\mu/\rho}\), respectively. For the PS excitation, the three potentials can be specified as the CFIPs. In \((\xi,\theta\eta)\) coordinates, the three CFIPs are expressed as follows:  
\[\Phi=(\
abla\cdot\mathbf{P}\phi)=P_{0}\frac{1}{\xi}\frac{\partial\phi}{\partial \phi}+P_{\eta}\frac{\partial\phi}{\partial\eta}, \tag{13}\]  
\[X\mathbf{\hat{e}}_{\eta}=-(\
abla\times\mathbf{P}\mathbf{\chi})=-\frac{P_{\theta}}{\xi} \frac{\partial(\xi\chi)}{\partial\xi}\mathbf{\hat{e}}_{\eta}, \tag{14}\]  
\[\Psi\mathbf{\hat{e}}_{z}=(\
abla\times\mathbf{P}\psi)=\frac{P_{\theta}}{\xi}\frac{ \partial(\xi\psi)}{\partial\xi}\mathbf{\hat{e}}_{\eta}, \tag{15}\]where \(\phi\), \(\chi\) and \(\psi\) are scalar functions for \(\Phi,X\) and \(\Psi\), respectively. (Notes. Corrects the erratum in the sign of \(\Psi\hat{\varepsilon}_{z}\) in Refs. [20; 23].) The combination of Equations (2) and (13)-(15) gives a second-order partial differential equation (PDE) for each scalar function. The radial PDEs of \(\phi\) and \(\chi\) are nonhomogeneous, and their solutions are a linear combination of the homogeneous and the particular solutions. These three PDEs were solved by introducing \(k_{\eta}^{2}\) (\(k_{\eta}\
eq\kappa_{\eta}\)) for the imaginary roots of the axial component and \(m\) (\(=0,\ \pm 1,\ \pm 2,\cdots\)) as an azimuthal constant for the angular component. The solutions of the \(\phi\), \(\chi\) and \(\psi\) functions are given by Equations (A26)-(A28) in Appendix C. By substituting each scale function into the corresponding potential, the CFIPs can be obtained as expressed in Equations (A29)-(A34). When the displacement is produced by the force component \(P_{f}\), \(\mathbf{u}\) in Equation (12) can be rewritten as the cylindrical components of the displacement:  
\[u_{f}=u_{\xi f}+u_{\theta f}+u_{\eta f}, \tag{16}\]  
where  
\[u_{\xi f}=\frac{\partial\Phi_{f}}{\partial\xi}+\frac{1}{\xi}\frac{\partial X_ {f}}{\partial\theta}+a\frac{\partial^{2}\Psi_{f}}{\partial\xi\partial\eta}, \tag{17}\]  
\[u_{\theta f}=\frac{1}{\xi}\frac{\partial\Phi_{f}}{\partial\theta}-\frac{ \partial X_{f}}{\partial\xi}+a\frac{1}{\xi}\frac{\partial^{2}\Psi_{f}}{ \partial\theta\partial\eta}, \tag{18}\]  
\[u_{\eta f}=\frac{\partial\Phi_{f}}{\partial\eta}-a\left(\frac{\partial^{2} \Psi_{f}}{\partial\xi^{2}}+\frac{1}{\xi}\frac{\partial\Psi_{f}}{\partial\xi}+ \frac{1}{\xi^{2}}\frac{\partial^{2}\Psi_{f}}{\partial\theta^{2}}\right). \tag{19}\]  
These three equations can be written as follows:  
\[u_{df}=P_{f}\Big{(}A_{mf}F_{df}^{1}+B_{mf}F_{df}^{2}+C_{mf}F_{df}^{3}+F_{df}^{4 }\Big{)}e^{-i\omega t}, \tag{20}\]  
where \(d=\xi\), \(\theta\) or \(\eta\) and \(f=z\) or \(\theta\) for the pinhole leakage. In Equation (20), the \(F_{df}^{n}\) terms are given in Equations (A35)-(A58) in Appendix C.  
Since the Bessell functions \(J_{m}(\alpha_{\varsigma}^{\alpha})\) and \(J_{m}(\beta_{\varsigma}^{\alpha})\) in the CFIPs become 1 as \(\xi\) approaches the PS, \(m=0\).