# Scattering Reduction and Resonant Trapping of Flexural Waves: Two Rings to Rule Them
## 2 Analytical Methods
### Trapped Vibrations in a Double-Ring Cluster of Point Resonators
#### 2.2.1 Homogenised Problem for a Double-Ring Cluster
 \tag{12}\]where the position vectors of the points on \(\Gamma_{k}\) can be represented in the complex form  
\[\mathbf{X}^{(k)}=R_{k}\exp(\mathrm{i}\theta_{\mathbf{X}^{(k)}}),\ \mathbf{Y}^{(k)}=R_{k}\exp( \mathrm{i}\theta_{\mathbf{Y}^{(k)}}),\ R_{k}>0\quad(k=1,2).\]  
The flexural displacement \(u(\mathbf{X}^{(k)})\) is sought in the form  
\[u(\mathbf{X}^{(k)})=U^{(k)}\exp(\mathrm{i}q\theta_{\mathbf{x}^{(k)}}),\quad q\in\mathbb{ Z}.\]  
We also take into account the identity  
\[|R_{k}\exp(\mathrm{i}\theta_{\mathbf{X}})-R_{j}\exp(\mathrm{i}\theta_{\mathbf{Y}})|=R_ {k}\Big{|}1-\frac{R_{j}}{R_{k}}\exp(\mathrm{i}(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}} ))\Big{|}.\]  
Thus, Equation (12) yields the following system of linear algebraic equations with respect to \(U^{(1)}\) and \(U^{(2)}\):  
\[U^{(1)}\Big{(}\frac{2\pi\rho h}{\beta^{4}}+M_{1}\int_{0}^{2\pi} \mathrm{e}^{\mathrm{i}q(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}})}G\Big{(}\beta,R_{1} \Big{|}1-\mathrm{e}^{\mathrm{i}(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}})}\Big{|} \Big{)}\mathrm{d}\theta_{\mathbf{Y}}\Big{)} \tag{13}\] \[+U^{(2)}M_{2}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}q(\theta_{\mathbf{ Y}}-\theta_{\mathbf{X}})}G\Big{(}\beta,R_{1}\Big{|}1-\frac{R_{2}}{R_{1}}\mathrm{e}^{ \mathrm{i}(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}})}\Big{|}\Big{)}\mathrm{d}\theta_{ \mathbf{Y}}=0,\] \[U^{(1)}M_{1}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}q(\theta_{\mathbf{ Y}}-\theta_{\mathbf{X}})}G\Big{(}\beta,R_{2}\Big{|}1-\frac{R_{1}}{R_{2}}\mathrm{e}^{ \mathrm{i}(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}})}\Big{|}\Big{)}\mathrm{d}\theta_{ \mathbf{Y}}\] (14) \[+U^{(2)}\Big{(}\frac{2\pi\rho h}{\beta^{4}}+M_{2}\int_{0}^{2\pi} \mathrm{e}^{\mathrm{i}q(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}})}G\Big{(}\beta,R_{2} \Big{|}1-\mathrm{e}^{\mathrm{i}(\theta_{\mathbf{Y}}-\theta_{\mathbf{X}})}\Big{|} \Big{)}\mathrm{d}\theta_{\mathbf{Y}}\Big{)}=0.\]  
Here, the integer number \(q\) determines the order of the trapped flexural vibration mode. Using the notation \(\mathcal{A}(\beta,q)\) for the matrix of the above linear system, the equation for the spectral parameter is  
\[\det(\mathcal{A}(\beta,q))=0. \tag{15}\]  
The components of the matrix \(\mathcal{A}\) are  
\[\mathcal{A}_{11} = \frac{2\pi\rho h}{\beta^{4}}+M_{1}\int_{0}^{2\pi}\mathrm{e}^{ \mathrm{i}q\alpha}G\Big{(}\beta,R_{1}\Big{|}1-\mathrm{e}^{\mathrm{i}\alpha} \Big{|}\Big{)}\mathrm{d}\alpha, \tag{16}\] \[\mathcal{A}_{22} = \frac{2\pi\rho h}{\beta^{4}}+M_{2}\int_{0}^{2\pi}\mathrm{e}^{ \mathrm{i}q\alpha}G\Big{(}\beta,R_{2}\Big{|}1-\mathrm{e}^{\mathrm{i}\alpha} \Big{|}\Big{)}\mathrm{d}\alpha,\] (17) \[u(\mathbf{X}^{(k)})=U^{(k)}\exp(\mathrm{i}q\theta_{\mathbf{x}^{(k)}}),\quad q\in\mathbb{ Z}.\]0 (18) \[u(\mathbf{X}^{(k)})=U^{(k)}\exp(\mathrm{i}q\theta_{\mathbf{x}^{(k)}}),\quad q\in\mathbb{ Z}.\]1  
We note that the off-diagonal terms \(\mathcal{A}_{12}\) and \(\mathcal{A}_{21}\) tend to zero when \(R_{1}/R_{2}\to 0\), or \(R_{1}/R_{2}\to 1\), and hence the analysis of these limit configurations is reduced to the case of trapped modes in a single ring of resonators, as studied in [32].