## 3 Displacement Fields Generated by a Point Source
These constants can be determined directly by applying a fundamental set of linear elastic boundary problems. The outer surface of the cylinder studied in the present paper is stress-free. Thus, the following stress components are zero under these circumstances, i.e., \(\xi=a-r_{0}\):  
\[=[A_{mr}J_{m}(\alpha\xi)][A_{m\theta}\cos(m\vartheta)]\Big{(}A_{ z}e^{-ik_{\eta}\eta}\Big{)}-\frac{k_{p}^{2}}{\rho\omega^{2}}G_{1}\frac{1}{1- \left[\frac{r_{01}}{\alpha(a-r_{0})}\right]^{2}}J_{m}\left[\frac{r_{01}}{ \alpha(a-r_{0})}\xi\right].\]3  
Substituting Equations (A17)-(A21) into Equation (104) yields the following algebraic equations  
\[\left[ \begin{array}{ccc}a_{11f}&a_{12f}&a_{13f}\\ a_{21f}&a_{22f}&a_{23f}\\ a_{31f}&a_{32f}&a_{33f}\end{array} \right]\left[ \begin{array}{c}A_{mf}\\ B_{mf}\\ C_{mf}\end{array} \right]=\left[ \begin{array}{c}b_{1f}\\ b_{2f}\\ b_{3f}\end{array} \right], \tag{105}\]  
where \(f=j\) for \(P_{j}\) and \(f=z\) for \(P_{z}\). The elements in Equation (105) are given in Appendix A. For \(P_{j}\), the elements of \(b_{1j}\), \(b_{2j}\) and \(b_{3j}\) are non-zero; therefore, the coupling constants can be determined by solving Equation (105). The CF in Equations (1) and (43) is the force impulse defined as  
\[\mathbf{P}(t)=P_{0}t\;\exp(-bt), \tag{106}\]where \(P_{0}\) and \(b\) are parameters that determine the amplitude and the duration of the wave, respectively [29]. The arrival time \(\tau\) of the impulse at the position \(\left(\xi_{i},\xi_{j},\eta\right)\) must be considered, since the force impulse is acting in the \(f\) direction at position \(\xi_{i}=\xi_{j}=0\) and \(\eta=0\) at \(t=0\). Replacing \(t\) with \((t-\tau)\) yields  
\[P_{f}(t-\tau)=P_{0,f}(t-\tau)e^{-b(t-\tau)}. \tag{107}\]  
The displacement components generated by the \(\Phi_{f}\) potential, and the \(X_{f}\) and \(\Psi_{f}\) potentials, correspond to the compressional (P), and the shear (SH and SV) waves, respectively; therefore, the arrival times of the two waves are given as  
\[\tau_{P}=\frac{\sqrt{\xi_{i}^{2}+\xi_{j}^{2}+\eta^{2}}}{c_{P}},\ \tau_{S}=\frac{\sqrt{\xi_{i}^{2}+\xi_{j}^{2}+\eta^{2}}}{c_{S}}, \tag{108}\]  
where \(c_{P}\) and \(c_{S}\) are the velocities of the P and S waves, respectively. The displacements generated by \(P_{f}\) can be rewritten as  
\[u_{rf}=\Big{[}P_{f}(t-\tau_{P})\ \Big{(}A_{mf}F_{rf}^{1}+F_{rf}^{4}\Big{)}+P _{f}(t-\tau_{s})\Big{(}B_{mf}F_{rf}^{2}+C_{mf}F_{rf}^{3}\Big{)}\Big{]}e^{-i \omega t}, \tag{109}\]  
\[u_{\theta f}=\Big{[}P_{f}(t-\tau_{P})\ \Big{(}A_{mf}F_{\theta j}^{1}\Big{)}+P _{f}(t-\tau_{s})\Big{(}B_{mf}F_{\theta f}^{2}+C_{mf}F_{\theta f}^{3}+F_{ \theta f}^{4}\Big{)}\Big{]}e^{-i\omega t}, \tag{110}\]  
\[u_{zf}=\Big{[}P_{f}(t-\tau_{P})\ \Big{(}A_{mf}F_{zf}^{1}+F_{zf}^{4}\Big{)}+P _{f}(t-\tau_{s})\Big{(}B_{mf}F_{zf}^{2}\Big{)}\Big{]}e^{-i\omega t}. \tag{111}\]  
Furthermore, the P, SH, and SV waves can be obtained from Equations (109)-(111) as  
\[u_{f}^{P}=P_{f}(t-\tau_{P})\Big{(}A_{mf}F_{df}^{1}+F_{rf}^{4}\Big{)}e^{-i \omega t}\ (d=r,\ z\ \text{and}\ \theta), \tag{112}\]  
\[u_{f}^{SV}=P_{f}(t-\tau_{S})\Big{(}C_{mf}F_{df}^{3}\Big{)}e^{-i\omega t}\ (d=r,\ z\ \text{and}\ \theta), \tag{113}\]  
\[u_{f}^{SH}=P_{f}(t-\tau_{S})\Big{(}B_{mf}F_{df}^{2}+F_{\theta f}^{4}\Big{)}e^{ -i\omega t}\ (d=r,\ z\ \text{and}\ \theta). \tag{114}\]