# Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer
## 3 Solution to the Problem
### Modified Transfer Matrix Method
\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_{3}^{\langle 1\rangle}}{dx_{3}^{ \langle 1\rangle}}\Big{|}_{x_{3}^{\langle 1\rangle}=h}\\ \end{array} \right)^{T}=\left( \begin{array}{cc}C_{1}^{\langle 2 \rangle}&C_{2}^{\langle 2\rangle}&C_{3}^{\langle 2\rangle}&C_{4}^{\langle 2 \rangle}\\ \end{array} \right)^{T} \tag{27}\]Subsequently, the governing equation in the second layer is solved, and the transfer matrix of the even layer \(\mathbf{B}\), which can be used to describe the relationship between the displacement components and the dimensionless derivatives of the displacement concerning coordinates at \(x_{3}^{(2)}=h\) and \(C_{j}^{(2)}(j=1-4)\), is calculated as:  
\[\left( \begin{array}{cc}U_{1}^{(2)}\Big{|}_{x_{3}^{(2)}=h^{\prime}}&h\frac{dU_{ 1}^{(2)}}{dx_{3}^{(2)}}\Big{|}_{x_{3}^{(2)}=h^{\prime}},&ih\frac{dU_{3}^{(2)}} {dx_{3}^{(2)}}\Big{|}_{x_{3}^{(2)}=h}\end{array} \right)^{T}=\mathbf{B}\Big{(} \begin{array} []{cc}C_{1}^{(2)},&C_{2}^{(2)}&C_{3}^{(2)}&C_{4}^{(2)}\end{array} \Big{)}^{T}. \tag{28}\]  
Therefore, the relationship of \(C_{j}^{(l)}(j=1-4)\), where \(l=1-N\), is expressed as:  
\[\left( \begin{array}{cc}C_{1}^{(2m+1)},&C_{2}^{(2m+1)}&C_{3}^{(2m+1)}&C_{4}^{ (2m+1)}\end{array} \right)=\mathbf{B}\Big{(} \begin{array}{cc}C_{1}^{(2m)},&C_{2}^ {(2m)}&C_{3}^{(2m)}&C_{4}^{(2m)}\end{array} \Big{)}^{T} \tag{29}\] \[\left( \begin{array}{cc}C_{1}^{(2m)},&C_{2}^{(2m)}&C_{3}^{(2m)}&C_ {4}^{(2m)}\end{array} \right)=\mathbf{A}\Big{(} \begin{array}{cc}C_{1}^{(2m-1)},&C_ {2}^{(2m-1)}&C_{3}^{(2m-1)}&C_{4}^{(2m-1)}\end{array} \Big{)}^{T}.\]  
The displacement components and the dimensionless derivatives of the displacement concerning coordinates of the lower and upper surfaces can be expressed in terms of \(C_{j}^{(1)}(j=1-4)\). \[\left( \begin{array}{cc}U_{1}^{(1)}\Big{|}_{x_{3}=0^{\prime}}&h\frac{dU_{1}^ {(1)}}{dx_{3}}\Big{|}_{x_{3}=0},&iU_{3}^{(1)}\Big{|}_{x_{3}=0}&ih\frac{dU_{3}^ {(1)}}{dx_{3}}\Big{|}_{x_{3}=0}\end{array} \right)^{T}=\left( \begin{array}{ cc}C_{1}^{(1)},&C_{2}^{(1)}&C_{3}^{(1)}&C_{4}^{(1)}\end{array} \right)^{T} \tag{30}\] \[\left( \begin{array}{cc}U_{1}^{(N)}\Big{|}_{x_{3}=H^{\prime}},&h \frac{dU_{1}^{(N)}}{dx_{3}}\Big{|}_{x_{3}=H},&iU_{3}^{(N)}\Big{|}_{x_{3}=H}&i \frac{dU_{3}^{(N)}}{dx_{3}}\Big{|}_{x_{3}=H}\end{array} \right)^{T},\text{N is even. }\] \[=\underbrace{\mathbf{\mathrm{B}}\mathbf{\mathrm{\bullet}}\mathbf{\mathrm{A}} \mathbf{\mathrm{\bullet}}\mathbf{\mathrm{\bullet}}\mathbf{\mathrm{\bullet}}\mathbf{\mathrm{ \bullet}}}_{N}\Big{(} \begin{array}{cc}C_{1}^{(1)},&C_{2}^{(1)}&C_{3}^{(1)}&C_{4 }^{(1)}\end{array} \Big{)}^{T}\] \[\left( \begin{array}{cc}U_{1}^{(N)}\Big{|}_{x_{3}=H^{\prime}},&h \frac{dU_{1}^{(N)}}{dx_{3}}\Big{|}_{x_{3}=H},&iU_{3}^{(N)}\Big{|}_{x_{3}=H}&i \frac{dU_{3}^{(N)}}{dx_{3}}\Big{|}_{x_{3}=H}\end{array} \right)^{T},\text{N is odd. }\] \[=\underbrace{\mathbf{\mathrm{A}}\mathbf{\mathrm{\bullet}}\mathbf{\mathrm{ \bullet}}\mathbf{\mathrm{\bullet}}\mathbf{\mathrm{\bullet}}\mathbf{\mathrm{\bullet}}\mathbf{ \mathrm{\bullet}}}_{N}\Big{(} \begin{array}{cc}C_{1}^{(1)},&C_{2}^{(1)}&C_{3}^{(1) }&C_{4}^{(1)}\end{array} \Big{)}^{T}\]  
The traction-free boundary conditions in the lower and upper surfaces can be written as:  
\[\left(\lambda_{0}+2\mu_{0}\right)\frac{dU_{3}^{(1)}}{dx_{3}^{(1)}}\bigg{|}_{x_ {3}^{(1)}=0}+\mathrm{i}k\lambda_{0}\,U_{1}^{(1)}\Big{|}_{x_{3}^{(1)}=0}=0,ik\, U_{3}^{(1)}\Big{|}_{x_{3}^{(1)}=0}+\frac{dU_{3}^{(1)}}{dx_{3}^{(1)}}\bigg{|}_{x_{3}^{(1)}=0}=0 \tag{31}\] \[\left(\lambda_{H}+2\mu_{H}\right)\frac{dU_{3}^{(N)}}{dx_{3}^{(N)} }\bigg{|}_{x_{3}^{(N)}=h}+\mathrm{i}k\lambda_{H}\,U_{1}^{(N)}\Big{|}_{x_{3}^{(N )}=H}=0,ik\,U_{3}^{(N)}\Big{|}_{x_{3}^{(N)}=h}+\frac{dU_{3}^{(N)}}{dx_{3}^{(N)} }\bigg{|}_{x_{3}^{(N)}=h}=0,\]  
where subscripts \(0\) and \(H\) represent the material parameters of the lower and upper surfaces, respectively.