## 3 Implementation
### Equations of Motion  
The guided wave dynamic in the specimen is formulated using an energetic approach by inserting the kinetic and potential energies into the conservative form of [PERSON]'s principle--Equation (19):  
\[\delta U+\delta K=0. \tag{19}\] \begin{table}
\begin{tabular}{c c c c} \hline
**P** & **Legendre Polynomial** & **Lobatto Polynomials** & **Roots (Nodes) \(\eta\)** \\ \hline
1 & \(L_{0}(\eta)=1\) & 0 & \(\pm 1\) \\
2 & \(L_{1}(\eta)=\eta\) & \(L_{0}(\eta)=1\) & \(\pm 1\) 0 \\
3 & \(L_{2}(\eta)=\frac{1}{2}(3\eta^{2}-1)\) & \(L_{1}(\eta)=3\eta\) & \(\pm 1\pm 0.447\) \\
4 & \(L_{3}(\eta)=\frac{1}{2}(5\eta^{2}-3)\eta\) & \(L_{2}(\eta)=\frac{3}{2}(5\eta^{2}-1)\) & \(\pm 1\pm 0.654\) 0 \\
5 & \(L_{4}(\eta)=\frac{1}{8}(35\eta^{4}-30\eta^{2}+3)\) & \(L_{3}(\eta)=\frac{5}{2}(7\eta^{2}-3)\eta\) & \(\pm 1\pm 0.765\) \\
6 & \(L_{5}(\eta)=\frac{1}{8}(63\eta^{4}-70\eta^{2}+15)\eta\) & \(L_{4}(\eta)=\frac{15}{8}(21\eta^{4}-14\eta^{2}+1)\) & \(\pm 1\pm 0.830\) \\  & & & \(\pm 0.468\) 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Six Legendre polynomials (\(L\)), Lobatto polynomials of order \(p-1\), and roots (nodes) \(\eta\). \begin{table}
\begin{tabular}{c c} \hline \(\eta_{i}\) & **Shape Functions \(N\)** \\ \hline \(-1\) & \(N_{1}(\eta)=\left(\frac{\eta+0.447}{-1+0.447}\right)\left(\frac{\eta-0.447}{-1 -0.447}\right)\left(\frac{\eta-1}{-1}\right)\) \\ \(-0.447\) & \(N_{2}(\eta)=\left(\frac{\eta+1}{-0.447+1}\right)\left(\frac{\eta-0.447}{-0.447} \right)\left(\frac{\eta-1}{  
In Equation (20), \(U\) is the internal deformation energy and strain energy, and \(K\) is the kinetic energy given by Equation (21),  
\[\delta U=\int\int\limits_{\Omega}\delta\varepsilon^{T}\sigma d\Omega, \tag{20}\]  
\[\delta K=\int\limits_{0}^{\infty}\delta u_{n}^{T}(z)M_{0}(z)dz, \tag{21}\]  
\[M_{0}=\int\limits_{-1}^{1}N^{T}\rho N|J|\ d\eta, \tag{22}\]  
with \(\rho\) as the material density and \(M_{0}\) the mass matrix in Equation (22). By replacing Equation (6) and solving the integral, the strain energy in Equation (23) is obtained,  
\[-\delta u_{n}^{T}(z)\int\limits_{0}^{\infty}\left(E_{0}u_{,zz}(z)+(E_{1}^{T}-E_ {1})u_{n,z}(z)-E_{2}u_{n}(z)\right)dz+\delta u_{n}^{T}(0)q_{n}(0), \tag{23}\]  
Here, \(E_{0}\), \(E_{1}\) and \(E_{2}\) in Equations (24)-(26) are the coefficient matrices of the SBFEM scheme, while superscript \(T\) denotes the matrix transpose,  
\[E_{0}=\int\limits_{-1}^{1}B_{1}^{T}DB_{1}|J|\ d_{\eta}, \tag{24}\]  
\[E_{1}=\int\limits_{-1}^{1}B_{2}^{T}DB_{1}|J|\ d_{\eta}, \tag{25}\]  
\[E_{2}=\int\limits_{-1}^{1}B_{2}^{T}DB_{2}|J|\ d_{\eta}. \tag{26}\]  
Substituting Equations (21) and (23) in Equation (19) yields Equation (27) of the SBFEM, with semi-analytical equations for the displacements in the cross-section, since the nodal displacements \(u_{n}(z)\) can be arbitrary analytic functions of \(z\), while the \(y\)-direction is discretized along the element direction,  
\[-\delta u_{n}^{T}(z)\int\limits_{0}^{\infty}\left(E_{0}u_{,zz}(z)+(E_{1}^{T}-E _{1})u_{n,z}(z)-E_{2}u_{n}(z)-M_{0}(z)\right)dz=0. \tag{27}\]  
The solutions of the displacement functions in Equation (27) yield:  
\[u_{n}(z)=e^{\lambda z-i\omega}\,t\hat{u}, \tag{28}\]  
where \(\omega\) is the angular frequency, \(t\) is the time, \(i\) is the imaginary unit, and \(\hat{u}\) is the complex displacement vector. The modes that can propagate are defined by Equation (29),  
\[\delta U=\int\int\limits_{\Omega}\delta\varepsilon^{T}\sigma d\Omega, \tag{20}\]0  
where \(K\) is the wave number in the z-direction for a particular wave mode. Substituting Equation (29) into Equation (27) yields Equation (30). The generalized SBFEM governing equation for a stress-free plate is obtained as  
\[\delta U=\int\int\limits_{\Omega}\delta\varepsilon^{T}\sigma d\Omega, \tag{20}\]1For a given specific frequency \(\omega\), Equation (30) establishes a second-order polynomial eigenvalue problem for the eigenvalues \(\lambda\) and the eigenvectors \(\hat{u}_{n}\), which can be solved numerically by standard procedures.