## 4 Simulation  
In this study, we consider only the azimuthal dependence of the wave propagation arising entirely from the direction of the force vector, \(m=0\). A stainless-steel (SS) cylinder (\(a=0.50\) m, \(l=2.0\) m, \(\rho=7.80\times 10^{3}\) kg/m\({}^{3}\), \(c_{P}=5.98\) km/s, and \(c_{S}=3.30\) km/s) was used as the test specimen. Stiffness parameters (\(c_{11}=c_{22}=c_{33}=2.086\times 10^{11}\), \(c_{12}=c_{13}=c_{23}=1.465\times 10^{11}\), \(c_{44}=c_{55}=c_{66}=1.269\times 10^{11}\) N m\({}^{-2}\)) can be found for austenitic stainless steel [30]. Previously, the natural frequencies of SS active on AE in the range of 0-300 kHz were determined experimentally by a tensile test (Appendix B). The frequency of \(\
u=155.4\) kHz (the angular frequency \(\omega=2\pi\
u\)) was found to be predominant.  
First, we determined \(k_{z}\) by applying the boundary conditions to Equations (A20) and (A22) which were associated with the displacements due to \(P_{z}\). From Equation (105), two solutions of \(A_{0}\) were obtained, as follows:  
\[A_{0z}^{1}=\frac{b_{1z}}{a_{11z}}\ \ \ \text{or}\ \ \ A_{0z}^{2}=\frac{b_{3z}}{a_{31z}} \tag{115}\]  
since \(a_{13z}=a_{33z}=0\).  
When \(m=0\),  
\[\begin{array}{c}a_{11z}=ik_{z}\Big{[}c_{13}k_{z}^{2}J_{0}(\alpha\xi)-\frac{c _{12}}{\xi}\frac{\partial b_{(}\alpha\xi)}{\partial\xi}-c_{11}\frac{\partial^ {2}J_{0}(\alpha\xi)}{\partial\xi^{2}}\Big{]}e^{-ik_{z}\eta}\\ =ik_{z}\Big{\{}c_{13}k_{z}^{2}J_{0}(\alpha\xi)+\frac{c_{12}}{\xi}\alpha J_{1}( \alpha\xi)-c_{11}2\big{(}\frac{\alpha}{2}\big{)}^{2}[-J_{0}(\alpha\xi)+J_{2}( \alpha\xi)]\Big{\}}e^{-ik_{z}\eta},\end{array} \tag{116}\]  
axial displacement that is stronger than the radial and tangential displacements. These displacements result in the P wave being the main wave, with a minor SH wave and very weak SV wave (Figure 4b). As shown in Figure 4c, the displacement features generated by the \(P_{z}\) excitation differ significantly from those generated by the \(P_{j}\) excitation. For the \(P_{j}\) excitation, the maximum values of \(u_{rj}\) and \(u_{zj}\) at the (0.5 m, 45\({}^{\circ}\), 1 m) position were 0.29 and 0.71 nm, respectively, whereas for the \(P_{z}\) excitation, they were 28.1 and 5.0 nm, respectively. The amplitudes of the displacements due to the \(P_{z}\) excitation were much stronger than those due to the \(P_{j}\) excitation. It should be noted that the \(P_{z}\) excitation produces only the P wave (Figure 4d). The angular dependence of the displacement was also calculated (Figure 5). When the point source is located at the center of the circular plane, the angular dependences of \(u_{rj}\), \(u_{zj}\) and \(u_{ij}\) arise from only \(\Xi\) and \(\Sigma\) as defined in Equations (72) and (78), respectively. For \(P_{z}\) excitation, the displacements of \(u_{rz}\) and \(u_{zz}\) are free from these factors. When the distances from the point source to the circumference are not equivalent, the angular dependences of the radial and axial displacements are highly significant. These effects are due not only to \(\Xi\) and \(\Sigma\) but also the superposition of the Bessel functions involved in the displacement equations. At a certain angle, some \(a_{ij}\) values in Equation (105) become too small to cause the sudden increase in displacement.  
Figure 4: Displacements at the (0.5 m, 45\({}^{\circ}\), 1 m) position generated by the point source of 1.0 N located at the center of the cylinder: (**a**,**c**) radial (black line), axial (red line), and tangential (green line) components; and (**b**,**d**) compression P (black line) wave and vertically polarized (SV, red line) and horizontally polarized (SH, green line) shear waves.  
Figure 3: Effect of the \(b\) parameter of the point source on the envelopes of the simulated \(u_{rj}\) waveforms: (1) 6 \(\times\) 10\({}^{4}\), (2) 8 \(\times\) 10\({}^{4}\), (3) 1 \(\times\) 10\({}^{5}\), (4) 2 \(\times\) 10\({}^{5}\), and (5) 4 \(\times\) 10\({}^{5}\).