## 3 Displacement Fields Generated by a Point Source
Taking into account the Laplacian operator in the \((\zeta,\vartheta,\eta)\) cylindrical coordinates, the three scalar spatial potentials have the following forms:  
\[\phi(\zeta,\vartheta,\eta)=\phi_{r}(\zeta)\phi_{\theta}(\theta)\phi_{z}(\eta), \tag{59}\]  
\[+a\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{4}(P\psi)+\rho\omega^{2} \mathbf{\
abla}^{2}(P\psi)\right] \tag{52}\]0  
\[+a\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{4}(P\psi)+\rho\omega^{2} \mathbf{\
abla}^{2}(P\psi)\right] \tag{52}\]1  
First, let us solve PDE of Equation (56). Using a similar method to that applied to the case of [PERSON]'s function, the axial and the angular parts can be obtained as  
\[+a\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{4}(P\psi)+\rho\omega^{2} \mathbf{\
abla}^{2}(P\psi)\right] \tag{52}\]2  
\[+a\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{4}(P\psi)+\rho\omega^{2} \mathbf{\
abla}^{2}(P\psi)\right] \tag{52}\]3  
respectively. Substituting the Laplacian operator and Equations (41), (59), (62), and (63), and introducing some straightforward algebra to Equation (56) leads to  
\[+a\mathbf{\
abla}\times\left[\mu\mathbf{\
abla}^{4}(P\psi)+\rho\omega^{2} \mathbf{\
abla}^{2}(P\psi)\right] \tag{52}\]4  
where \(\alpha^{2}=k_{p}^{2}-k_{\eta}^{2}\). Equation (64) is a second-order linear nonhomogeneous PDE, the solution of which is a linear combination of the homogeneous (\(\phi_{rh}\)) and particular (\(\phi_{rp}\)) solutions. The homogeneous solution of Equation (64) is  
\[\phi_{rh}=\left\{A_{mr}\left[ \begin{array}{c}J_{m}(\alpha\zeta)\\ I_{m}(\alpha\zeta)\end{array} \right]+B_{mr}\left[ \begin{array}{c}Y_{m}( \alpha\zeta)\\ K_{m}(\alpha\zeta)\end{array} \right]\right\}\qquad \begin{array}{c}(\alpha^{2} \geq 0)\\ (\alpha^{2}<0)\end{array} ,\]where \(J_{m}\) and \(I_{m}\) are the first kind and its modified Bessel functions, respectively, and \(Y_{m}\) and \(K_{m}\) are the second kind and its modified Bessel functions, respectively. The Bessel functions of the second kind are excluded due to the singularity at the origin included in the domain of the cylinder. For the case of \(\alpha^{2}\geq 0\),  
\[\phi_{rh}=A_{mr}J_{m}(\alpha\xi). \tag{65}\]  
According to [PERSON]'s solutions [28], the particular solution of the inhomogeneous equation has the form  
\[\phi_{rp}=-\frac{k_{p}^{2}}{\rho\omega^{2}}\left(\frac{1}{\phi_{\phi}\phi_{ \eta}}\right)G_{1}\frac{1}{1-\left[\frac{r_{01}}{\alpha(a-r_{0})}\right]^{2}}J _{m}\left[\frac{r_{01}}{\alpha(a-r_{0})}\xi\right],\qquad\frac{r_{01}}{\alpha (a-r_{0})}\
eq 1. \tag{66}\]  
The linear combination of Equations (65) and (66) results in  
\[\phi(\xi,\vartheta,\eta)=\phi_{r}(\xi)\phi_{\theta}(\vartheta) \phi_{z}(\eta)\] \[=[A_{mr}J_{m}(\alpha\xi)][A_{m\theta}\cos(m\vartheta)]\Big{(}A_{ z}e^{-ik_{\eta}\eta}\Big{)}-\frac{k_{p}^{2}}{\rho\omega^{2}}G_{1}\frac{1}{1- \left[\frac{r_{01}}{\alpha(a-r_{0})}\right]^{2}}J_{m}\left[\frac{r_{01}}{ \alpha(a-r_{0})}\xi\right].\]  
For further applications, the above equation is rewritten as  
\[\phi(\xi,\vartheta,\eta)=A_{m}\,J_{m}(\alpha\xi)\cos(m\vartheta)e^{-ik_{\eta} \eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}G_{1}\frac{1}{1-\left[\frac{r_{01}}{ \alpha(a-r_{0})}\right]^{2}}J_{m}\left[\frac{r_{01}}{\alpha(a-r_{0})}\xi \right], \tag{67}\]  
where the coupling constant \(A_{m}=A_{mr}A_{m\theta}A_{z}\). Following a similar procedure, the functions of \(\chi(\xi,\vartheta,\eta)\) and \(\psi(\xi,\vartheta,\eta)\) can be obtained as  
\[\chi(\xi,\vartheta,\eta)=B_{m}J_{m}(\beta\xi)\sin(m\vartheta)e^{-ik_{\eta} \eta}-\frac{k_{p}^{2}}{\rho\omega^{2}}G_{1}\frac{1}{1-\left[\frac{r_{01}}{ \beta(a-r_{0})}\right]^{2}}J_{m}\left[\frac{r_{01}}{\beta(a-r_{0})}\xi\right], \tag{68}\]  
\[\psi(\xi,\vartheta,\eta)=C_{m}\,J_{m}(\beta\xi)\cos(m\vartheta)e^{-ik_{\eta} \eta}. \tag{69}\]  
As shown in Figure 1, we introduce force vector \(\mathbf{P}\) acting in the radial and the axial directions to solve for \(\Phi\).