## 4 Numerical Results  
In this section, we study the sensitivity of the wave frequency and phase velocity to different input parameters that could serve for design purposes of many high-tech devices made of advanced CNT-reinforced composite materials, and further computational studies on the topic. We start the analysis by accounting for the mechanical properties of the constituent materials in Table 1 for FG CNTs-reinforced composites, while validating the proposed methodology with findings by [PERSON] [38] from literature. Table 2 summarizes the results in terms of dimensionless natural frequency for a simply-supported structure and different wave numbers, with a good correspondence between results based on our model and those once from literature. One of the critical aspects for such structures, relies on the influence of the CNT distribution in the length direction. To this end, we plot the wave frequency vs. the wave number for various CNTs volume fractions, while keeping fixed \(K_{\text{FP}}=10\) and \(K_{\text{W0}}=200\), as visible in Figure 2. For low wave numbers, up to the unit value, any variation in the wave frequency is noticed, after which a monotonic increase of wave frequency occurs for increased values of wave number. This behavior is even more pronounced for an increased volume fraction of the reinforcing phase. For a constant wave number, indeed, an increased volume fraction of CNTs gets higher values of wave frequency. A further investigation accounts for the influence of the volume fraction of CNTs and wave number on the phase velocity of the structure, as plotted in Figure 3. Note that for small wave numbers up to 0.1, the volume fraction does not affect the phase velocity, with a meaningful variation in the response for higher wave numbers in a non monotonic sense, both in the ascending and descending branches of the plots. Small variations in the volume fraction of CNTs within the range 0.1 and 10, enable a severe additive effect on the phase velocity, where the structure gets stiffer. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties** & **Units** & **CNT [28]** & **Matrix [28]** \\ \hline \(E\) & GPa & 5646.6 & 2.5 \\ \hline \(\rho\) & Kg/m\({}^{3}\) & 1400 & 1190 \\ \hline \(\
u\) & - & 0.28 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of functionally graded CNTs-reinforced composite beam. \begin{table}
\begin{tabular}{c c c} \hline \hline \(n\) & **Loy [38]** & **Present** \\ \hline
1 & 0.016102 & 0.0161011 \\ \hline
2 & 0.009387 & 0.0093865 \\ \hline
3 & 0.022108 & 0.0221045 \\ \hline
4 & 0.042096 & 0.0420954 \\ \hline
5 & 0.068008 & 0.0680080 \\ \hline
6 & 0.099730 & 0.0997300 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparative evaluation of the dimensionless natural frequency. \(\Omega=\omega R.\sqrt{(1-V^{2})\rho/E}\) for a simply-supported isotropic shell and different wave numbers. Figure 2: Wave frequency vs. wave number for various volume fractions of CNTs. Figures 4 and 5 describe the variation of wave frequency and phase velocity versus the wave number for different CNT grading power terms at a fixed \(V_{\text{CNT-TOTAL}}=20\%\). Based on these plots, for high wave numbers, the influence of the power term is much more significant, where the power term plays a decreasing role in the variation of wave frequency. In both figures, the wave frequency and phase variation decrease for an increased power term, and a fixed wave number, due to a softening influence of the power term along with an overall increase in the structural deformability. Figure 4: Wave frequency vs. wave number for different values of \(k\). Figure 3: Phase velocity vs. wave number for various volume fractions of CNTs. Figure 6 also shows the effect of the Winkler coefficient on the wave frequency for a different coefficient \(\alpha\), here varied from 0.2 to 0.8, under the assumption \(V_{\rm CNT-L}=~{}5\%\), \(K_{\rm F0}=10\).