# Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer
## 3 Solution to the Problem
### Modified Transfer Matrix Method  
The transfer matrix method is commonly used in the study of laminated structures. For example, when scientists investigate the wave propagation problem in FG structures, these structures can be simplified as multilayer structures with layers that are assumed to be homogeneous. Interface continuity conditions include stress and displacement continuity conditions. When multilayer structures are used to simulate FG structures, the strain in the interface is discontinuous because of the discontinuous properties of the elastic parameters in the interface. The elastic parameters in the interface of the layered structure depicted in Figure 1b are continuous; thus, the strain components are continuous. The present study assumes that the displacement and strain components are continuous, the displacement components can be expressed as power series forms, and the strain components are related to the derivatives of the displacements with respect to the coordinates. Mathematically, the displacement components and their derivatives are continuous in the surface. Let \(x_{3}^{\langle l\rangle}=x_{3}-(l-1)h\), where \(l\) denotes the \(l\)th layer. The derivative of an arbitrary function with respect to \(x_{3}\) is equal to the derivative of the function with respect to \(x_{3}^{\langle l\rangle}\). The governing equations for the Lamb waves in \(N\)-layered structures include \(l\) groups. These equations are similar to Equation (16), where the displacement amplitude components \(U_{1}\) and \(U_{3}\); material parameters \(\lambda\), \(\mu\), and \(\rho\); and \(x_{3}\) are replaced by \(U_{1}^{\langle l\rangle}\), \(U_{3}^{\langle l\rangle}\), \(\lambda^{\langle l\rangle}\), \(\mu^{\langle l\rangle}\), \(\rho^{\langle l\rangle}\), and \(x_{3}^{\langle l\rangle}\), respectively. On the basis of the power series solution mentioned above, the solutions for the governing equations of each layer are obtained in the same form as Equation (22). \[U_{1}^{\langle l\rangle}=\sum_{j=1}^{4}C_{j}^{\langle l\rangle}\sum_{n=0}^{ \infty}s_{nj}^{\langle l\rangle}\left(\frac{x_{3}^{\langle l\rangle}}{h} \right)^{n},U_{3}^{\langle l\rangle}=-i\sum_{j=1}^{4}C_{j}^{\langle l\rangle} \sum_{n=0}^{\infty}t_{nj}^{\langle l\rangle}\left(\frac{x_{3}^{\langle l \rangle}}{h}\right)^{n} \tag{24}\]  
where:  
\[\left( \begin{array}{cc}s_{0j}^{\langle l\rangle}&s_{1j}^{\langle l\rangle} &t_{0j}^{\langle l\rangle}&t_{1j}^{\langle l\rangle}\\ \end{array} \right)=\mathbf{I} \tag{25}\]  
The values of \(n\), \(s_{nj}^{\langle l\rangle}\), and \(t_{nj}^{\langle l\rangle}\) can be determined using Equation (20). The material parameters for odd and even layers are different. Similarly, the coefficients of the equation and the obtained solutions are different. The governing equation in the first layer is solved, and the displacement amplitude components \(U_{1}^{\langle 1\rangle}\) and \(U_{3}^{\langle 1\rangle}\) are calculated based on the solution. Then, the relationship between the displacement components and the dimensionless derivatives of the displacement concerning coordinates at \(x_{3}^{\langle 1\rangle}=h\) and \(C_{j}^{\langle 1\rangle}(j=1-4)\) is obtained as:  
\[\left( \begin{array}{cc}U_{1}^{\langle 1\rangle}\Big{|}_{x_{3}^{\langle 1 \rangle}=h}&h\frac{dU_{1}^{\langle 1\rangle}}{dx_{3}^{\langle 1\rangle}} \Big{|}_{x_{3}^{\langle 1\rangle}=h}&iU_{3}^{\langle 1\rangle} \Big{|}_{x_{3}^{\langle 1\rangle}=h}&ih\frac{dU_{1}^{\langle 1\rangle}}{dx_{3}^{ \langle 1\rangle}}\Big{|}_{x_{3}^{\langle 1\rangle}=h}\\ \end{array} \right)^{T}=\mathbf{A}\left( \begin{array}{cc}C_{1}^{\langle 1 \rangle}&C_{2}^{\langle 1\rangle}&C_{3}^{\langle 1\rangle}&C_{4}^{\langle 1 \rangle}\\ \end{array} \right)^{T} \tag{26}\]  
where superscript T represents the matrix transposition and \(\mathbf{A}\) is a \(4\times 4\) matrix named the transfer matrix of odd layers. By considering the physical meaning of \(C_{j}^{\langle 1\rangle}(j=1-4)\) and the continuous condition in the interface, we obtain:  
\[\left(