## 2 Analytical Model
### Displacement Fields
, \tag{21}\]  
\[A_{01}=\frac{1}{\pi\big{[}e^{\kappa_{z}z_{0}}+e^{-\kappa_{z}(l-z_{0})}-2 \big{]}}\times\frac{2}{\Big{\{}a^{2}[J_{v+1}(\kappa_{z}a)]^{2}-b^{2}[J_{v+1}( \kappa_{z}b)]^{2}\Big{\}}}, \tag{22}\]  
where \(l\) is the length of the cylinder, and \(a\) and \(b\) are the outer and inner diameters, respectively. (Note: this corrects the erratum in Equation (16) for the \(\eta\) range in Ref. [18].) For gas leakage, the Green's function is non-zero on the outer surface of the cylindrical shell because the PS is located on the outer surface of the cylindrical shell. In this study, the value of \(\kappa_{z}\) was determined empirically. In Equation (20), the value of \(\xi\) is the shortest distance between the PS and the point where the detecting sensor is projected onto the equatorial layer containing the PS. However, as shown in Figure 1b, there is no linear distance between the two points across the hollow interior. Since the thickness is much shorter than the diameter of the outer circle, we simply use the value of \(\xi\), as the length of the arc around the outer circle,  
\[\xi\approx\xi_{p}=\pi a\theta/180^{\circ}, \tag{23}\]  
where \(\theta\) is angle between the PS and the projected point. From Equations (16) and (17), the force vector can be rewritten as  
\[ \begin{array}{c}\mathbf{f}=P\
abla^{2}g(\xi,\theta,\eta)e^{-i\omega t}\\ =\mathbf{\
abla}[\mathbf{\
abla}\cdot Pg(\xi,\theta,\eta)]-\mathbf{\
abla}\times[\mathbf{ \
abla}\times Pg(\xi,\theta,\eta)]e^{-i\omega t},\end{array}  \tag{24}\]  
where \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{1}{\xi}\frac{\partial}{ \partial\theta}+\frac{1}{\xi^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+ \frac{\partial^{2}}{\partial\eta^{2}}\). Previously [18], the three potential functions were specified as CFIPs generated by the PS. \[\Phi=\mathbf{\
abla}\cdot(\mathbf{P}\phi)=\frac{\partial(\mathbf{P}\phi)}{\partial\xi}+\frac{1 }{\xi}\frac{\partial(\mathbf{P}\phi)}{\partial\theta}+\frac{\partial(\mathbf{P}\phi)}{ \partial\eta}, \tag{25}\]  
\[X\hat{\mathbf{e}}_{\eta}=-\mathbf{\
abla}\times(\mathbf{P}\chi)=-\frac{1}{\xi}\Big{[}\frac {\partial(\xi\mathbf{P}\chi)}{\partial\xi}-\frac{\partial(\mathbf{P}\chi)}{\partial \theta}\Big{]}\hat{\mathbf{e}}_{\eta}, \tag{26}\]  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]  
where \(\phi,\chi\) and \(\psi\) are scalar functions. These scalar functions were determined by solving Equation (10) combined with Equations (11) and (25)-(27) as follows  
\[\phi(\xi,\theta,\eta)=A_{m}J_{m}(\alpha\xi)\text{cos}(m\phi)e^{-ik_{\eta}\eta }-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{G_{01}}{\alpha^{2}-\kappa_{2}^{2}}J_{ m}(\kappa_{2}\xi), \tag{28}\]  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\phi)e^{-ik_{\eta}z}- \frac{k_{p}^{2}}{\rho\omega^{2}}\frac{G_{01}}{\beta^{2}-k^{2}}J_{m}(\kappa_{2 }\xi), \tag{29}\]  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\phi)e^{-ik_{\eta}\eta}, \tag{30}\]  
where \(\alpha^{2}=k_{p}^{2}-k_{\eta}^{2}>0\), and \(\beta^{2}=k_{s}^{2}-k_{\eta}^{2}>0\). In Equations (28)-(30), \(A_{m}\), \(B_{m}\) and \(C_{m}\) are the coupling constants. As shown in Figure 1a, the CF vectors generated by the FRS act in the axial (\(P_{z}\)) and the tangential (\(P_{\theta}\)) directions. There is no CF in the radial (\(P_{r}\)) direction because the surface boundary is perpendicular to the radial direction.