# Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity
## 2 Stepwise WFE Approximation of a Radially Periodic Plate
### Coupling of the Segments and Wave Amplitude Decay  
Compared to the corresponding Cartesian waveguides, wave propagation is not translational invariant in the radially periodic structures, and amplitude attenuation occurs as the waves travel from the centre in the radial direction. Since the energy flowing along the slice is constant, wave amplitude change due to changes in the geometry can be accommodated in the model according to an energy balance principle as in [31]. In this section, the procedure is presented for the first two cells. The same passages must be applied up to the cell \(j=n\), including the point at which the response must be evaluated. In the Cartesian cell, amplitudes are related at two points \(r_{1}\) and \(r_{2}=r_{1}+\overline{R}\), by \(\mathbf{a}_{1}^{+}(r_{2})=\mathbf{T}^{+}(\overline{R})\mathbf{a}_{1}^{+}(r_{ 1})\) and \(\mathbf{a}_{1}^{-}(r_{2})=\mathbf{T}^{-}_{1}(\overline{R})\mathbf{a}_{1}^{-}( r_{1})\), where \(\mathbf{T}^{+}_{1}(\overline{R})=\text{diag}\big{[}\text{exp}\big{(}-\mathbf{i} \mathbf{k}_{1}^{+}\overline{R}\big{)}\big{]}\) and \(\mathbf{T}^{-}_{1}(\overline{R})=\text{diag}\big{[}\text{exp}\big{(}-\mathbf{ i}\mathbf{k}_{1}^{-}\overline{R}\big{)}\big{]}\). Therefore, nodal displacements and nodal forces at the interface between cell 1 and cell 2 are  
\[\left[ \begin{array}{c}\mathbf{q}_{1}\\ \mathbf{f}_{1}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{\Phi}_{ \mathbf{d},1}^{+}&\mathbf{\Phi}_{\mathbf{d},1}^{-}\\ \mathbf{\Phi}_{\mathbf{f},1}^{+}&\mathbf{\Phi}_{\mathbf{f},1}^{-}\end{array}  \right]\left[ \begin{array}{cc}\mathbf{T}_{1}^{+}(\overline{R})&0\\ 0&\mathbf{T}_{1}^{-}(\overline{R})\end{array} \right]\left[ \begin{array}{c} \mathbf{a}_{1}^{+}(r_{1})\\ \mathbf{a}_{1}^{-}(r_{1})\end{array} \right] \tag{11}\]  
Using the left eigenvectors as in Equation (10), the first attempt to find the excited wave amplitude in cell 2 gives  
\[\left[ \begin{array}{c}\mathbf{a}_{2}^{+}\\ \mathbf{a}_{2}^{-}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{\Psi}_{ \mathbf{f},2}^{+}&\mathbf{\Psi}_{\mathbf{d},2}^{+}\\ \mathbf{\Psi}_{\mathbf{f},2}^{-}&\mathbf{\Psi}_{q,2}^{-}\end{array} \right] \left[ \begin{array}{c}\mathbf{q}_{1}\\ \mathbf{f}_{1}\end{array} \right] \tag{12}\]  
Nodal displacements and nodal forces at cell 2 can be obtained using the same expression of Equation (10)  
\[\left[ \begin{array}{c}\mathbf{q}_{2}\\ \mathbf{f}_{2}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{\Phi}_{ \mathbf{d},2}^{+}&\mathbf{\Phi}_{\mathbf{q},2}^{-}\\ \mathbf{\Phi}_{\mathbf{f},2}^{+}&\mathbf{\Phi}_{\mathbf{f},2}^{-}\end{array}  \right]\left[ \begin{array}{cc}\mathbf{T}_{2}^{+}(\overline{R})&0\\ 0&\mathbf{T}_{2}^{-}(\overline{R})\end{array} \right]\left[ \begin{array}{c} \mathbf{\tilde{a}}_{2}^{+}\\ \mathbf{a}_{2}^{-}\end{array} \right] \tag{13}\]  
and the time-averaged energy flows, Equation (6), of cell 1 and cell 2 can be evaluated using nodal displacement and nodal forces from Equations (11) and (13)  
\[\Pi_{1}=\frac{1}{2}\text{Re}\Big{\{}\mathbf{f}_{1}^{H}i\omega\mathbf{q}_{1} \Big{\}},\quad\Pi_{2}=\frac{1}{2}\text{Re}\Big{\{}\mathbf{f}_{2}^{H}i\omega \mathbf{q}_{2}\Big{\}} \tag{14}\]The ratio between \(\Pi_{1}\) and \(\Pi_{2}\) gives the amplitude decay \(\xi_{2}\) in waveguide 2, which is evaluated as in [32]  
\[{\xi_{2}}^{2}=\frac{\text{Re}\{\mathbf{f}_{1}{}^{H}i\omega\mathbf{q}_{1}\}}{ \text{Re}\{\mathbf{f}_{2}{}^{H}i\omega\mathbf{q}_{2}\}} \tag{15}\]  
The wave and amplitudes in waveguide 2 are therefore approximated to  
\[\left[ \begin{array}{c}\mathbf{a_{2}}^{+}\\ \mathbf{a_{2}}^{-}\end{array} \right]=\xi_{2}\left[ \begin{array}{c}\mathbf{ \tilde{a}_{2}}^{+}\\ \mathbf{\tilde{a}_{2}}^{-}\end{array} \right] \tag{16}\]  
Equations (13)-(16) can be repeated iteratively until \(\xi^{2}\) converges to one, and the final value of the wave amplitudes reach the required approximation. One passage was sufficient to converge to a useful approximation with a very low computational cost in the numerical cases studied.