## 2 Dispersion Curve Calculation
### Wave and Finite Element Scheme  
The WFE method has been developed over the last three decades and has been applied to a wide variety of structural calculations. The reader is referred to classical literature regarding this methodology [28; 29; 30] in order to obtain more information. With regard to two-dimensional periodic media as shown in Figure 1a, and a close-up example of such a structure shown in Figure 1b, the first step is to model a single periodic section using standard finite element software. There is no limitation with regard to the complexity of the cross-section to be modelled, making the method suitable for modelling layered media. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline
**K, C, M** & Stiffness, damping, and mass matrices \\
**D** & Dynamic stiffness matrix \\ \(\omega\) & Angular frequency \\
**q** & Vector of nodal degrees of freedom \\
**f** & Vector of internal forces \\
**T** & Transformation matrix \\ \(\lambda\) & Propagation constant \\ \(k\) & Wavenumber \\ \(\Xi\) & Transfer matrix \\ \(\phi\) & Modeshape vector \\ \(E,P\) & Energy and power \\ \(c_{g},c_{p}\) & Group velocity and phase velocity \\ \(\theta\) & Propagation direction \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of symbols. The finite element model of the periodic section is used to obtain the stiffness (**K**), mass (**M**), and damping (**C**) matrices to set up the dynamic equilibrium as follows:  
\[\left[\textbf{K}+\textbf{i}\omega\textbf{C}-\omega^{2}\textbf{M}\right]\textbf{ q}=\textbf{f}, \tag{1}\]  
where \(\textbf{q}=\{\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \}^{\text{T}}\ \}^{\text{T}}\) is the vector of nodal degrees of freedom. Each entry in this vector is formed by concatenating all nodes through the thickness, where \((. )^{\text{T}}\) refers to a transpose. The size of each entry equals the number of nodal degrees of freedom times the number of concatenated nodes. The vector of internal forces **f** is ordered in the same way as **q**. The coefficient matrix multiplying with **q** in Equation (1) is commonly known as the dynamic stiffness matrix **D**. The dimension of Equation (1) can be reduced by applying the periodicity condition in the \(y\)-direction. For a free wave propagation scenario, where \(k_{y}\) refers to the component of wavenumber in the \(y\)-direction, a transformation matrix **T** can be specified that reduces the nodal degrees of freedom vector as follows:  
\[\textbf{q}=\textbf{T}\textbf{q}_{\text{red}},\quad\text{where}\ \ \textbf{q}_{\text{red}}=\{\textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\ \textbf{q}_{in}^{\text{T}}\}^{\text{T}}. \tag{2}\]  
The entries of matrix **T** are determined from the wave propagation constant \(\lambda_{y}\), which, according to [PERSON]'s theorem, is given as \(\lambda_{y}=e^{-ik_{y}l_{y}}\), where \(l_{y}\) is the height of the periodic section. The complete matrix depends on the ordering of the **q** vector; an example can be found in [28]. This projects all degrees of freedom on the nodes lying on the \(x\)-axis of the waveguide by imposing the periodic boundary conditions. A similar dimensional reduction is performed to obtain the reduced force vector \(\textbf{f}_{\text{red}}\). Since there are no forces on the internal nodes, the internal nodal force vector \(\textbf{f}_{b}=0\). This helps in simplifying Equation (1) as follows [31]:  
\[\left[ \begin{array}{cc}\textbf{D}_{ll}&\textbf{D}_{lr}\\ \textbf{D}_{rl}&\textbf{D}_{rr}\end{array} \right]\left\{ \begin{array}{cc} \textbf{q}_{lb}\\ \textbf{q}_{rb}\end{array} \right\}=\left\{ \begin{array}{cc}\textbf{f}_{lb} \\ \textbf{f}_{rb}\end{array}