## 4 Results and Analysis  
### Dispersion Characteristics of Particle Rotation Waves  
This section explains the effect of the degree of anisotropy on the dispersion characteristics of particle rotation waves. To facilitate the analysis without the loss of generality, the non-dimensional wave number \(\overline{k}=\frac{l}{2\pi^{2}}k\) and the non-dimensional frequency \(\overline{\omega}=\sqrt{\frac{4\pi\rho R^{3}}{3G_{\rm n}^{4}}}\omega\) are adopted in the following analysis. Figure 3 shows the dispersion curves of the three kinds of particle rotation waves under different anisotropic parameter \(a\). Unless stated otherwise, the model parameters adopted in this paper are listed in Table 1 and are the same as that used in [25]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(l/\)m & \(10^{-3}\) & \(G_{\rm n}^{\rm a}/\)N\(\cdot\)m & 5 \\ \(R/\)m & \(5\times 10^{-4}\) & \(G_{\rm n}^{\rm a}/\)N\(\cdot\)m & 5 \\ \(N_{p}/\)m\({}^{-3}\) & \(1.0\times 10^{9}\) & \(G_{\rm n}^{\rm b}/\)N\(\cdot\)m & \(1.25\times 10^{-11}\) \\ \(\rho/\)kg/m\({}^{3}\) & \(1570\) & \(\eta=G_{\rm g}^{\rm c}/G_{\rm n}^{\rm c}\) & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of model parameters. Figure 3: Dispersion curves of particle rotation waves under different \(a\). Figure 3a-c show the variation of dispersion relations of rotation longitudinal waves with \(a\). It can be seen that the rotation longitudinal waves include one acoustic branch (LA), the cutoff frequency of which equals zero at \(k=0\), and two optical branches (LO), the cutoff frequency of which is non-zero. For the LA branch, with the wave number increasing, its frequency increases rapidly at long-wavelength stage and then increases slightly to an asymptote, showing significant dispersion characteristics. Besides, the anisotropic parameter \(a\) has little effect on the LA branch for long-wavelength regime (\(\overline{k}<0.1\)), while, as the wave number increases, the effect of \(a\) becomes significant: the frequency of LA branch increases apparently with the increase in \(a\) (see Figure 3a). The LO branch associated with \(\overline{B}\) first increases slightly and then approximately linearly increases with the wave number increasing. As \(a\) increases, the LO branch associated with \(\overline{B}\) also increases (see Figure 3b). However, the frequency of the LO branch associated with \(B^{\text{D}}\) experiences a relatively small increase and eventually reaches an asymptotic value, which implies a weak dependence of wave number (see Figure 3c). Different from other longitudinal waves, there is overall decrease in the frequency of the LO branch associated with \(B^{\text{D}}\), except for the long-wavelength regime, with an increase in \(a\). Besides, as \(a\) increases, the dependence of its frequency on wave number gradually decreases. Figure 3d-f give the dispersion curves of rotation transverse waves. Similarly, the rotation transverse waves consist of one acoustic branch (TA) and two optical branches (TO). The TA branch and TO branch associated with \(B_{(12)}\) show similar dispersion relations with the LA branch and LO branch associated with \(\overline{B}\), respectively. However, the frequencies of these two transverse waves are smaller than that of longitudinal waves and the effect of anisotropic parameter \(a\) are also relatively smaller. As for the TO branch associated with \(B_{[12]}\), with the wave number increasing, the frequency slightly increases to an asymptotic value, showing a weak dependence of wave number similar with LO branch related to \(B^{\text{D}}\), and its frequency also increases with respect to \(a\). Figure 3g-i show the dispersion curves of rotation in-plane shear waves. The three rotation in-plane shear waves are all optical (TSO), and their frequencies remain almost unchanged with the wave number increasing.