## 5 Geometric Effects in the Phonon Dispersion and Group Velocities for Different Numbers of Layers  
For the axially-symmetric waves (\(n=0\)), as follows from Figures 2, 4 and 6, the lowest group of the phonon dispersion curves, containing one torsional and two non-torsional modes, at the small wave vectors \(\zeta\) is only slightly changed by the number of layers \(N_{\rm L}\). In the same region of wave vectors, the second, consisting of one torsional and one non-torsional modes (the third, consisting of one torsional and two non-torsional modes) group of the phonon frequencies \(\omega\) significantly decreases from 0.57 (1.17) for \(N_{\rm L}=2\) to 0.29 (0.57) for \(N_{\rm L}=4\) and 0.19 (0.38) for \(N_{\rm L}=6\). Within the numerical accuracy, the decrease of the phonon frequencies in the long-wave limit is _inversely proportional_ to \(N_{\rm L}\). Away from the long-wave limit, a general trend of \"compression\" of the phonon energy spectrum towards lower values of phonon frequencies persists.  
As seen from Figures 3, 5 and 7, the phonon group velocity related to the fundamental (lowest) torsional mode is a weakly varying function of the wave vector \(\zeta\), while for the higher torsional modes it monotonously increases with the wave vector \(\zeta\), apparently towards a saturation. For a fixed value \(\zeta=0.05\), the phonon group velocity related to the lowest (second lowest) torsional mode depends on the number of layers \(N_{\rm L}\) as follows: 1.17 (0.13) for \(N_{\rm L}=2\); 1.17 (0.27) for \(N_{\rm L}=4\); 1.17 (0.37) for \(N_{\rm L}=6\). Within the numerical accuracy, the increase of the phonon group velocity for the second lowest torsional mode is directly propotional to \(N_{\rm L}\).  
The phonon group velocity related to the lowest two non-torsional modes is a weak function of the wave vector \(\zeta\), while for higher torsional modes it always strongly depends on \(\zeta\). For the same fixed value \(\zeta=0.05\) as above, the phonon group velocity related to the lowest (second lowest and third lowest) non-torsional mode depends on the number of layers \(N_{\rm L}\) as follows: 0.75 (2.02 and 0.43) for \(N_{\rm L}=2\); 0.75 (1.98 and 0.82) for \(N_{\rm L}=4\); and 0.78 (1.94 and 1.06) for \(N_{\rm L}=6\). Within the numerical accuracy, the phonon group velocity for the third lowest torsional mode reveals a sublinear dependence on \(N_{\rm L}\).  
Figure 7: Phonon group velocity dispersion curves for \(n=0\), \(N_{\rm L}=6\) (**a**). Phonon group velocity dispersion curves for non-torsional (**b**) and torsional (**c**) waves.  
Finally, in order to clearly vizualize the overall impact of the number of layers in a multishell on the group velocity dispersion, the average and rms phonon group velocities are calculated for the branches available from the results of the previous section. The results, shown in Figure 8, demonstrate that an increase of \(\mathrm{N_{L}}\) from two to four leads to an appreciable decrease of the average and rms phonon group velocities. A further increase of \(N_{\mathrm{L}}\) from four to six has a smaller impact on the average and RMS phonon group velocities. For the wave vector \(\zeta=0.05\), the average phonon group velocity decreases from 0.82 for \(N_{\mathrm{L}}=2\) to 0.54 for \(N_{\mathrm{L}}=4\) and further to 0.53 for \(N_{\mathrm{L}}=6\). At the same time, the RMS phonon group velocity is reduced from 0.95 for \(\mathrm{N_{L}}=2\) to 0.71 for \(N_{\mathrm{L}}=4\) and further to to 0.65 for \(N_{\mathrm{L}}=6\). At small wave vectors the trend persists: the average and RMS phonon group velocities decrease with increasing \(N_{\mathrm{L}}\).