# Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer
## 4 Numerical Results and Discussion
### Lamb Waves in a Multilayered FG Nano Copper Wafer  
The dispersion curves of the Lamb waves in a three-layer wafer are calculated (Figure 7). Numerous modes of Lamb wave propagation exist in a multilayered FG nano copper wafer. The dispersion curves are similar to that of a simple FG nano copper wafer. For each mode, the larger the value of \(p\), the smaller the phase velocity. The dispersion curves for \(p=0\) and \(p=1\) almost coincide, and the phase velocity at \(p=3\) slightly decreases. To provide a theoretical basis for nondestructive evaluation, the variations of the change and relative change rate of the phase velocity with \(p\) at \(kh=2\pi\) are plotted in Figure 8a,b, respectively.  
Figure 5: Dispersion curves of Lamb waves in a single FG nano copper wafer.  
Figure 6: Relationship between the change in phase velocity and \(p\) in a simple FG nano cooper wafer: (**a**) change in phase velocity and (**b**) relative change rate of the phase velocity.  
When the thickness of the wafer is equal to the wavelength, the absolute values of the change in phase velocity \(|\Delta c|\) and the relative change rate \(\frac{|\Delta c|}{c_{0}}\) increase with the increase in \(p\). If the wafer thickness is 5 mm and \(p\) is 3, the absolute values of the change in phase velocity for the first four modes are 8.349, 6.503, 10.839, and 25.781 m/s, respectively. The frequencies of Lamb waves with a given wavelength vary with different phase velocities. The frequency reduction is defined as \(\triangle\omega=-\bigtriangleup ck\), and the frequency reduction values for the first four modes are 3497.23, 2723.97, 4540.24, and 10,799.15 Hz, respectively. Although the relative variable of phase velocity is limited to less than 1.2% (Figure 8b), the absolute value of the frequency change is expressed in kilohertz. The gradient properties lead to the significant changes in frequency. The most obvious phase velocity change with the gradient parameters is observed in Mode 4, followed by Modes 3, 1, and 2 (Figure 8a). The \(\frac{|\triangle c|}{c}\) values of Modes 1 and 3 almost coincide, whereas those of Modes 4 and 2 are the largest and lowest values among the four, respectively.  
To reveal the Lamb wave properties in a multilayered FG nano copper wafer, the Lamb waves in an effective homogenous wafer are selected for comparison. The material parameters of the latter, which are selected as the average material parameters of the FG nano copper wafer, satisfy the following conditions.  
\[\overline{\lambda}=\frac{\int_{0}^{h}\lambda dx_{3}}{h},\overline{\mu}=\frac{ \int_{0}^{h}\mu dx_{3}}{h},\overline{p}=\frac{\int_{0}^{h}\rho dx_{3}}{h} \tag{36}\]  
where \(\overline{\lambda}\) and \(\overline{\mu}\) are the Lame coefficients and \(\overline{p}\) is the mass density of the effective homogenous wafer. At \(p=3\),  
\[\overline{\lambda}=64.41\;\mathrm{GPa},\;\overline{\mu}=42.97\;\mathrm{GPa}, \;\mathrm{and}\;\overline{p}=8866.72\;\frac{\mathrm{kg}}{m^{3}}\]  
Figure 8: Relationship between the change in phase velocity and \(p\) in a three-layer FG nano cooper wafer: (**a**) change in phase velocity and (**b**) relative change rate of the phase velocity.  
Figure 7: Dispersion curves of the Lamb waves in a three-layer FG nano cooper wafer.  
The dispersion curves of the first four modes of the two-layer, four-layer, and effective homogenous wafers are displayed in Figure 9. The dispersion curve of the four-layer FG nano copper wafer is closer to the curve of the effective homogenous wafer than that of the two-layer FG nano copper wafer. The findings suggest the difference in the Lamb wave propagation in a multilayered FG nano copper wafer weakens with the increase in the number of layers.