# Free and Forced Wave Motion in a Two-Dimensional Plate with Radial Periodicity
## 2 Stepwise WFE Approximation of a Radially Periodic Plate
### Forced Wave Amplitude  
Forced wave propagation and structural response are evaluated following the theory presented in [27; 28]. Wave properties are grouped into positive and negative going waves, which can be described by the two sets of results \(\big{(}\mathbf{k}^{+},\ \mathbf{\Phi}^{+},\ \mathbf{a}^{+}\big{)}\) and \(\big{(}\mathbf{k}^{-},\ \mathbf{\Phi}^{-},\ \mathbf{a}^{-}\big{)}\). Here \(\mathbf{k}^{+},\mathbf{k}^{-}\) and \(\mathbf{a}^{+},\ \mathbf{a}^{-}\) are wavenumbers and waves' amplitudes travelling in the positive and negative direction, while \(\mathbf{\Phi}^{+}=[\mathbf{\Phi}_{\mathbf{q}}^{+},\mathbf{\Phi}_{\mathbf{f}} ^{+}]^{T}\) and \(\mathbf{\Phi}^{-}=[\mathbf{\Phi}_{\mathbf{q}}^{-},\mathbf{\Phi}_{\mathbf{f}} ^{-}]^{T}\) are the corresponding nodal displacements and nodal forces, that is the FE discretisation of wavemodes. For evaluating the forced response, it is advantageous to obtain also the left eigenvectors of the WFE eigenvalue problem in Equation (5). These are partitioned in the same manner as the wavemodes, that is \(\mathbf{\Psi}^{+}=[\mathbf{\Psi}_{\mathbf{f}}^{+},\ \mathbf{\Psi}_{\mathbf{q}} ^{+}]\) and \(\mathbf{\Psi}^{-}=[\mathbf{\Psi}_{\mathbf{f}}^{-},\mathbf{\Psi}_{\mathbf{q}} ^{-}]\). Left and right eigenvectors are orthogonal and they can be normalised so that  
\[\mathbf{\Psi}^{+}\mathbf{\Phi}^{+}=\mathbf{I},\mathbf{\Psi}^{-}\mathbf{\Phi} ^{-}=\mathbf{I} \tag{7}\]  
In an analogous manner to the modal analysis, the total displacement and force at the junction of a cell is described by the sum of the positive and negative wavemodes so that  
\[\left[ \begin{array}{c}\mathbf{q}\\ \mathbf{f}\end{array} \right]\!=\!\!\left[ \begin{array}{cc}\mathbf{\Phi}_{ \mathbf{q}}^{+}&\mathbf{\Phi}_{\mathbf{q}}^{-}\\ \mathbf{\Phi}_{\mathbf{f}}^{+}&\mathbf{\Phi}_{\mathbf{f}}^{-}\end{array}  \right]\!\left[ \begin{array}{c}\mathbf{a}^{+}\\ \mathbf{a}^{-}\end{array} \right]\!. \tag{8}\]  
Equation (8) defines the transformation between the physical domain, where the motion is described in terms of nodal displacements and forces, and the wave domain, where the motion is described in terms of waves of amplitudes \(\mathbf{a}^{+},\ \mathbf{a}^{-}\) travelling in the positive and negative directions. A point force \(\mathbf{f}_{e}\) will generate excited positive and negative going waves, propagating away from the excitation point. Compared to the Cartesian periodicity, in the case of a radially periodic plate, the position of this point force is an independent parameter. Here, we assume a transverse harmonic point force exciting the structure at the left nodes of the first cell in Figure 1c. With reference to Figure 1, continuity and equilibrium equations can be written at the left side of the first cell, and excited wave amplitudes can be recovered from Equation (9)  
\[\left[ \begin{array}{cc}\mathbf{\Phi}_{\mathbf{d},1}^{+}&\mathbf{\Phi}_{\mathbf{ d},1}^{-}\\ \mathbf{\Phi}_{\mathbf{f},1}^{-}&\mathbf{\Phi}_{\mathbf{f},1}^{-}\end{array}  \right]\left[ \begin{array}{c}\mathbf{a}_{1}^{+}(r_{1})\\ \mathbf{a}_{1}^{-}(r_{1})\end{array} \right]=\left[ \begin{array}{c}0\\ \mathbf{f}_{e}\end{array} \right] \tag{9}\]  
In practice, as in modal analysis, only \(m\) pairs of (positive- and negative-going) waves are retained (a reduced wave basis is assumed in most cases). Moreover, the number of assumed modes can be different at each frequency. It is noteworthy to mention that all the waves propagating in the structure (real wavenumbers) should be considered. Evanescent and attenuating waves (corresponding to pure imaginary or complex wavenumbers) should be also assumed since they play an important role in scattering and forcing problems. Consequently, the wavemodes' matrix is typically rectangular and the use of standard pseudoinverse operation can lead to numerical errors. To overcome this issue, the orthogonality between right and left eigenvectors in Equation (7) can be exploited, and the excited wave amplitudes at \(r_{1}\) can be obtained from  
\[\left[