# Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity
## 2 Stepwise WFE Approximation of a Radially Periodic Plate
### Free Wave Propagation Characteristics  
Wave characteristics of each segment (cell) in Figure 1c are evaluated using the WFE method as described in [14; 28]. The waves are numerically represented by the dispersion curves, \((\mathbf{k},\omega)\), which give the information about the wave vector \(\mathbf{k}\) available for each frequency \(\omega\), and the corresponding FE nodal displacements \(\mathbf{\Phi_{q}}\) and nodal forces \(\mathbf{\Phi_{f}}\), which occur under the passage of a wave. In the WFE, only the unit periodic cell is analysed. Figure 2 shows a finite element mesh of a unit cell using 4-noded rectangular elements and the correspondent WFE model. The degrees of freedom \(\mathbf{q}\) in the dicrestised equation of motion, Equation (1), are ordered and condensed as \(\mathbf{q}=\left[\mathbf{q}_{L}^{T}\quad\mathbf{q}_{R}^{T}\quad\mathbf{q}_{I }^{T}\right]^{T}\), where the subscripts \(L\), \(R\) and \(I\) are associated with the right \(\mathbf{q}_{R}=\left[\mathbf{q}_{1}^{T}\quad\mathbf{q}_{2}^{T}\right]^{T}\), left \(\mathbf{q}_{L}=\left[\mathbf{q}_{5}^{T}\quad\mathbf{q}_{6}^{T}\right]^{T}\) and interior \(\mathbf{q}_{I}=\left[\mathbf{q}_{3}^{T}\quad\mathbf{q}_{4}^{T}\right]^{T}\) nodal degrees of freedom of the cell, with a similar expression for the nodal forces \(\mathbf{f}=\left[\mathbf{f}_{L}^{T}\quad\mathbf{f}_{R}^{T}\quad\mathbf{f}_{I }^{T}\right]^{T}\). The discretised equation of motion, \((\mathbf{K}-\omega^{2}\mathbf{M})\mathbf{q}=\mathbf{D}\mathbf{q}=\mathbf{f}\), is obtained using conventional FEA and it is partitioned as  
\[\left[ \begin{array}{ccc}\mathbf{D}_{LL}&\mathbf{D}_{LR}&\mathbf{D}_{RI}\\ \mathbf{D}_{RL}&\mathbf{D}_{RR}&\mathbf{D}_{RI}\\ \mathbf{D}_{IL}&\mathbf{D}_{IR}&\mathbf{D}_{II}\end{array} \right]\left\{ \begin{array}{c}\mathbf{q}_{L}\\ \mathbf{q}_{R}\\ \mathbf{q}_{I}\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{f}_{L}\\ \mathbf{f}_{R}\\ \mathbf{f}_{I}\end{array} \right\} \tag{1}\]  
To apply the WFE method, the internal Dofs of the unit cell must be reduced. In this paper, a standard dynamic condensation is applied as  
\[\mathbf{q}_{I}=-\mathbf{D}_{II}^{-1}(\mathbf{D}_{IL}\mathbf{q}_{L}+\mathbf{D} _{IR}\mathbf{q}_{R}) \tag{2}\]  
However, according to the size of the model, more efficient methods for condensing the inner DOFs and speeding up the resolution of the WFE eigenvalue problem can be applied, e.g., [35; 36]. The resulting equation of motion is  
\[\left[ \begin{array}{cc}\mathbf{D}_{LL}&\mathbf{D}_{LR}\\ \mathbf{D}_{RL}&\mathbf{D}_{RR}\end{array} \right]\left\{ \begin{array}{c} \mathbf{q}_{L}\\ \mathbf{q}_{R}\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{f}_{L}\\ \mathbf{f}_{R}\end{array} \right\} \tag{3}\]  
Figure 2: (**a**) Finite element mesh of a unit cell using 4-noded rectangular elements and (**b**) WFE nodes condensation. The inclusion of stress, temperature effects and damping in the model can be accommodated, if necessary, as described in [37]. Periodicity conditions are applied so that  
\[\mathbf{q}_{R}=\lambda\mathbf{q}_{L};\quad\mathbf{f}_{R}=-\lambda\mathbf{f}_{L} \tag{4}\]  
where \(\lambda=e^{-ikL}\) and \(k\) is the wavenumber. Combining Equation (4) with Equation (3), the equation of free wave motion takes the form of a quadratic eigenvalue problem, as  
\[\Big{[}\lambda^{2}\mathbf{D}_{\text{LR}}+\lambda(\mathbf{D}_{\text{LL}}+ \mathbf{D}_{\text{RR}})+\mathbf{D}_{\text{RL}}\Big{]}\mathbf{q}_{\text{L}}=0 \tag{5}\]  
whose solutions yield the relationship between the wavenumber and frequency (dispersion curves) and the displacement \(\mathbf{q}_{L}\) of the cross-section due to wave motion (wave mode shapes). Equation (5) can be further recast as a standard linear eigenvalue problem as \((\mathbf{A}-\gamma\mathbf{I})\mathbf{Z}=0\),  
\[\text{where }\mathbf{A}=\left[ \begin{array}{cc}0&\mathbf{I}\\ -\mathbf{A}_{2}^{-1}\mathbf{A}_{0}&-\mathbf{A}_{2}^{-1}\mathbf{A}_{1}\end{array}