## 4 Numerical Results
When the structure is supported by a foundation, the structural stiffness increases monotonically with \(K_{\rm W}\), since the wave frequency gets higher values for an increased Winkler coefficient, especially for lower values of the coefficient \(\alpha\). Another important parameter for structures on an elastic foundation is represented by the Pasternak coefficient in the shear layer. Figure 7 shows the influence of this parameter on the phase velocity, while varying the coefficient \(\beta\) from 0.2 up to 0.8. It is worth noticing that an increased Pasternak coefficient yields higher phase velocities for each selected \(\beta\), whereas the phase velocity becomes smaller by increasing the coefficient \(\beta\) from 0.2 up to 0.8 under the same Pasternak coefficient assumption. Figure 5: Phase velocity vs. wave number for different values of \(k\). Figure 6: Wave frequency vs. Winkler parameter for various \(\alpha\). In Figure 8 we plot the influence of the Winkler coefficient on the wave frequency for different power terms \(n\), with a monotonic hardening effect of the Winkler coefficient on the structural response, especially for increased values of \(n\). It is clearly evident that the wave frequency increases for an increased power term \(n\), and a fixed Winkler coefficient \(K_{\text{W}}\), while increasing monotonically with \(K_{\text{W}}\) under the same assumption of \(n\). Similar observations can be repeated for a varying Pasternak foundation coefficient and power term \(m\) (see Figure 9) whose response sensitivity is more accentuated for different values of \(m\), compared to that one stemming from a Winkler foundation. A further parametric investigation accounts for the variation of the wave frequency with the rational volume fraction at the right and left side of the structure (\(V_{\text{CNT-R}}/V_{\text{CNT-L}}\)) for various power terms \(k\). As plotted in Figure 10, such rational value \(V_{\text{CNT-R}}/V_{\text{CNT-L}}\) can affect significantly the wave frequency of the structure, especially for increased values of \(k\), with an overall hardening effect for increased values of \(V_{\text{CNT-R}}/V_{\text{CNT-L}}\) and fixed power term \(k\). Such effect is also observed for an increased value of \(k\) from one to five, while keeping fixed the rational value \(V_{\text{CNT-R}}/V_{\text{CNT-L}}\). The last two parametric investigations account for the sensitivity of the structural wave frequency (Figure 11) and phase velocity (Figure 12) for different volume fractions (in percentage) of CNTs on the left side of the beam, and different CNTs power terms. Based on both figures, an increased volume fraction of CNTs as reinforcing phase within the material makes the structure stiffer, as visible from the increased wave frequency and phase velocity. Both values get higher for an increased power term \(k\), especially for higher volume fractions of CNTs. Figure 7: Phase velocity vs. Pasternak parameter for different \(\beta\). Figure 8: Wave frequency vs. Winkler parameter for various \(n\). Figure 9: Phase velocity vs. Pasternak parameter for different \(m\). Figure 11: Wave frequency vs. \(V_{\rm CNT-L}\) for various \(k\). Figure 2:0 Wave frequency vs. the rational value \(V_{\rm CNT-R}/V_{\rm CNT-L}\) for different \(k\).