## Appendix C
For \(\Phi\),  
\[\Phi_{\eta}=P_{w}\Bigg{[}\big{(}-ik_{\eta}\big{)}A_{m}J_{m}(\alpha\xi)\text{ cos}(m\theta)e^{-ik_{\eta}\eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{J_{m} \big{(}\kappa_{\eta}\xi\big{)}}{\alpha^{2}-\kappa_{\eta}^{2}}\frac{\partial G_ {0}}{\partial\eta}\Bigg{]}, \tag{104}\]  
\[\Phi_{\theta}=-P_{w}\frac{1}{\xi}\Big{[}A_{m}J_{m}(\alpha\xi)\text{sin}(m \theta)e^{-ik_{\eta}\eta}\Big{]}. \tag{105}\]  
For X,  
\[X_{\xi}=0, \tag{106}\]  
\[X_{\theta}=-P_{w}\frac{1}{\xi}\Big{\{}B_{m}\Big{[}J_{m}\big{(}\beta\xi\big{)} +\frac{\partial J_{m}\big{(}\beta\xi\big{)}}{\partial\xi}\Big{]}\text{sin}(m \vartheta)e^{-ik_{\eta}\eta} \tag{107}\]  
\[-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\beta^{2}-\kappa_{\eta}^{2}} \bigg{[}J_{m}\big{(}\kappa_{\eta}\xi\big{)}+\xi\frac{\partial J_{m}\big{(} \kappa_{\eta}\xi\big{)}}{\partial\xi}\bigg{]}\bigg{\}}. \tag{108}\]  
For \(\Psi\),  
\[\Psi_{\xi}=0, \tag{109}\]  
\[\Psi_{\theta}=P_{w}C_{m}\Big{[}\frac{1}{\xi}J_{m}(\beta\xi)+\frac{\partial J_ {m}\big{(}\beta\xi\big{)}}{\partial\xi}\Big{]}\text{cos}(m\vartheta)e^{-ik_{ \eta}\eta}. \tag{110}\]  
For \(P=P_{z}\),  
(radial component)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]0  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]1  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]2  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]3(tangential component)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]4 (A39)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]5 (A40)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]6 (A41)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]7 (A42)  
(axial component)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]8 (A43)  
\[\chi(\xi,\theta,\eta)=B_{m}J_{m}(\beta\xi)\text{sin}(m\theta)e^{-ik_{\eta}\eta }-\frac{k_{s}^{2}}{\rho\omega^{2}}\frac{G_{0}}{\left(\beta^{2}-\kappa_{\eta}^{ 2}\right)}J_{m}\big{(}\kappa_{\eta}\xi\big{)}, \tag{102}\]9 (A44)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]0 (A45)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]1 (A46)  
For \(P=P_{\theta}\),  
(radial component)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]2 (A47)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]3 (A48)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]4 (A49)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]5 (A50)  
(tangential component)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]6 (A51)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]7 (A52)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]8 (A53)  
\[\psi(\xi,\theta,\eta)=C_{m}J_{m}(\beta\xi)\text{cos}(m\theta)e^{-ik_{\eta} \eta}, \tag{103}\]9 (A54)  
(axial component)  
\[\Phi_{\eta}=P_{w}\Bigg{[}\big{(}-ik_{\eta}\big{)}A_{m}J_{m}(\alpha\xi)\text{ cos}(m\theta)e^{-ik_{\eta}\eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{J_{m} \big{(}\kappa_{\eta}\xi\big{)}}{\alpha^{2}-\kappa_{\eta}^{2}}\frac{\partial G_ {0}}{\partial\eta}\Bigg{]}, \tag{104}\]0 (A55)  
\[\Phi_{\eta}=P_{w}\Bigg{[}\big{(}-ik_{\eta}\big{)}A_{m}J_{m}(\alpha\xi)\text{ cos}(m\theta)e^{-ik_{\eta}\eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{J_{m} \big{(}\kappa_{\eta}\xi\big{)}}{\alpha^{2}-\kappa_{\eta}^{2}}\frac{\partial G_ {0}}{\partial\eta}\Bigg{]}, \tag{104}\]1 (A56)  
\[\Phi_{\eta}=P_{w}\Bigg{[}\big{(}-ik_{\eta}\big{)}A_{m}J_{m}(\alpha\xi)\text{ cos}(m\theta)e^{-ik_{\eta}\eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{J_{m} \big{(}\kappa_{\eta}\xi\big{)}}{\alpha^{2}-\kappa_{\eta}^{2}}\frac{\partial G_ {0}}{\partial\eta}\Bigg{]}, \tag{104}\]2 (A57)  
\[\Phi_{\eta}=P_{w}\Bigg{[}\big{(}-ik_{\eta}\big{)}A_{m}J_{m}(\alpha\xi)\text{ cos}(m\theta)e^{-ik_{\eta}\eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{J_{m} \big{(}\kappa_{\eta}\xi\big{)}}{\alpha^{2}-\kappa_{\eta}^{2}}\frac{\partial G_ {0}}{\partial\eta}\Bigg{]}, \tag{104}\]3  
\[\Phi_{\eta}=P_{w}\Bigg{[}\big{(}-ik_{\eta}\big{)}A_{m}J_{m}(\alpha\xi)\text{ cos}(m\theta)e^{-ik_{\eta}\eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}\frac{J_{m} \big{(}\kappa_{\eta}\xi\big{)}}{\alpha^{2}-\kappa_{\eta}^{2}}\frac{\partial G_ {0}}{\partial\eta}\Bigg{]}, \tag{104}\]4 (A58)