## 4 Discussion
### Differences between the Results from the Spectral Method Algorithm and Baik [26]  
Figure 8 shows the phase velocity curves of the axisymmetric modes (\(n=0\)) and the non-axisymmetric modes (\(n=1,2,3,...\)) of a water-filled PMMA tube propagating along the axis, as obtained from the spectral method algorithm. Figure 9 shows the group velocity curves of the corresponding modes. In the graphs, the horizontal axis is the dimensionless \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{**Axial Wavenumber \(k_{z}\)**} \\ \hline COMSOL & 4.3629 & 21.9621 & 30.1711 & 35.6921 & 37.5116 & 84.0532 \\ Multiphysics & 107.1383 & 121.4523 & 130.5509 & 136.0542 & 138.9479 & 139.8424 \\ \hline Spectral & 4.3678 & 21.9621 & 30.1706 & 35.6932 & 37.5116 & 84.0534 \\ Method & 107.1380 & 121.4523 & 130.5509 & 136.0542 & 138.9479 & 139.8424 \\ \hline Error & \(1.12\times 10^{-3}\) & 0 & \(1.66\times 10^{-5}\) & \(3.08\times 10^{-5}\) & 0 & \(2.38\times 10^{-6}\) \\  & \(2.80\times 10^{-6}\) & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(k_{z}\) calculated using the COMSOL simulation and the spectral method algorithm at \(f=8\) kHz. Figure 7: Comparison of the displacement of the propagation modes in the tube wall between the COMSOL simulation and the spectral method algorithm. parameter \(k_{1}b\) and the vertical axis is the phase velocity or group velocity normalized by the sound velocity in water as \(c_{p}/c_{1}\) or \(c_{g}/c_{1}\). All figures use \(n\) to represent the circumferential order of the modes. The different color of lines in Figures 8 and 9 denote a different group number of the modes with the same \(n\). Figures 8 and 9 are compared with Figure 2 in Baik [26], which shows the dispersion curves of phase speed and group speed of modes for the water-filled PMMA tube obtained. Table 5 shows the modal correspondence between the dispersion curves obtained using the spectral method algorithm and results from Baik [26]. For example, the \((0,1)\) mode in Figure 8 corresponds to the \((0,0)\) mode in Figure 2 of [PERSON] [26], et cetera. This work only analyzed the modes listed in Table 5. The following analysis was based on the mode indices in Figure 8.  
Figure 8: Phase velocity curves of the water-filled PMMA tube modes: \(n=0,1,2,3\). Figure 9: Group velocity curves of the water-filled PMMA tube modes: \(n=0,1,2,3\). For the axisymmetric modes (\(n=0\)), [PERSON] [26] did not find the \((0,2)\) mode in Figure 8, obtained with the spectral method algorithm. For the non-axisymmetric modes (\(n\
eq 0\)), [PERSON] [26] lacked the modes \((1,1)\), \((1,2)\), \((2,1)\), \((2,2)\), \((3,1)\),\((3,2)\) in Figure 8. The accuracy of the spectral method algorithm was verified using the results from the COMSOL simulation in the previous section. It can be seen from Figure 8 that the phase velocity of mode \((0,2)\) did not change with the frequency, and was equal to the bulk shear wave velocity of the PMMA tube. Its group velocity was also equal to the bulk shear wave velocity, as shown in Figure 9. This is consistent with the description of the dispersion characteristics of the torsional wave provided by [PERSON] [3]. The phase velocities of modes \((0,1)\), \((1,1)\), \((2,1)\), and \((3,1)\) in Figure 8 were lower than the sound velocity in water, and the longitudinal and shear bulk wave velocities of the PMMA tube. Figures 10 and 11 show the radial distribution of the displacement components and the stress tensor components of the above four modes at \(f=8\) kHz, obtained using the spectral method algorithm. In Figure 10, the solid line represents \(u_{r}\), the dotted line represents \(u_{\theta}\), and the dashed line represents \(u_{z}\). In Figure 11, the solid line represents \(\tau_{rr}\), the dotted line represents \(\tau_{r\theta}\), and the dashed line represents \(\tau_{rz}\).