## 2 Analytical Model
### Displacement Fields
For \(P_{\theta}\), the radial components \(u_{r\theta}\),  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]1  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]2  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]3  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]4  
the tangential components \(u_{\theta\theta}\),  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]5  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]6  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]7  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]8and the axial component \(u_{z\theta}\),  
\[\Psi\hat{\mathbf{e}}_{\eta}=-\frac{1}{\xi}\Big{[}\frac{\partial(\xi\mathbf{P}\psi)}{ \partial\xi}-\frac{\partial(\mathbf{P}\psi)}{\partial\theta}\Big{]}\hat{\mathbf{e}}_{ \eta}, \tag{27}\]9  
\[\phi(\xi,\theta,\eta)=A_{m}J_{m}(\alpha\xi)\text{cos}(m\phi)e^{-ik_{\eta}\eta }-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{G_{01}}{\alpha^{2}-\kappa_{2}^{2}}J_{ m}(\kappa_{2}\xi), \tag{28}\]0  
\[ \begin{array}{c}F^{3}_{z\theta}=a\Big{[}\Big{(}\frac{1-m^{2}}{\xi^{3}}\Big{)} J_{m}(\beta\xi)-\Big{(}\frac{1+m^{2}}{\xi^{2}}\Big{)}\frac{\partial J_{m}(\beta\xi)} {\partial\xi}+\frac{2}{\xi}\frac{\partial^{2}J_{m}(\beta\xi)}{\partial\xi^{2}} \\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{\partial^{2}J_{m}(\beta \xi)}{\partial\xi^{2}}\Big{]}\text{cos}(m\theta)e^{-ik_{\eta}\eta},\end{array}  \tag{56}\]  
\[F^{4}_{z\theta}=0, \tag{57}\]  
the remaining task is to evaluate the coupling constants \(A_{m}\), \(B_{m}\) and \(C_{m}\). Let us apply boundary conditions to the outer surface in the same way as in Ref. [18]. There is no stress on the outer surface of the cylinder because the effect of the atmosphere pressure on the displacement field is negligible  
\[\sigma_{rr}=\sigma_{r\theta}=\sigma_{rz}=0. \tag{58}\]  
Introducing Equation (58) to the stress-strain displacement relations for the circular cylindrical shell studied in this work yields a system of three linear algebraic equations, given as follows:  
\[ \begin{bmatrix}a_{11f}&a_{12f}&a_{13f}\\ a_{21f}&a_{22f}&a_{23f}\\ a_{31f}&a_{32f}&a_{33f}\end{bmatrix}\begin{bmatrix}A_{mf}\\ B_{mf}\\ C_{mf}\end{bmatrix} = \begin{bmatrix}b_{1f}\\ b_{2f}\\ b_{3f}\end{bmatrix} . \tag{59}\]  
Elements \(a_{ijf}\) and \(b_{if}\) of the matrix in Equation (59) are given in Appendix A. For the \(P_{z}\) CF, Equation (59) becomes very simple because all elements except \(a_{12}\), \(b_{1z}\) and \(b_{3z}\) are zero. We obtain  
\[B_{mz}=0, \tag{60}\]  
\[A_{mz}=\frac{b_{1z}}{a_{11z}}=\frac{b_{3z}}{a_{31}}. \tag{61}\]  
The values of \(k_{\eta}\) at a given location of the PS can be obtained by solving the roots of the function  
\[f\big{(}k_{\eta}\big{)}=\frac{b_{1z}}{a_{11z}}-\frac{b_{3z}}{a_{31z}}=0. \tag{62}\]  
To evaluate the displacements at position \((\xi,\ \eta)\) on the outer surface, the arrival times \((\tau\text{s})\) of the AE signal generated by the PS force must be introduced into Equation (37).