# Scattering Reduction and Resonant Trapping of Flexural Waves: Two Rings to Rule Them
## 3 Results and Discussion
### Trapped Modes  
Here, we show how to obtain localised modes in the plate, confined between the two concentric rings of point resonators. The procedure is the following. We choose the radian frequency \(\omega\) of the localised mode and the mode number \(q\). We fix the geometry of the rings, in particular the radii \(R_{1}\) and \(R_{2}\) and the number \(N\) of resonators. In particular, in the calculations, we take \(N=64\), which is sufficiently large to apply the homogenisation approach described in Section 2.2. Figure 4: Comparison of the reaction forces produced by a single mass scatterer and by a circular cluster of 12 point mass scatterers of the same combined mass. The incident field is produced by a point source. The data, referred to \(S1\) units, used in the computation are \(h=0.005\), \(\rho=2700\), \(D=818.277\), \(R=0.05\). The graph compares \(S_{F}^{(1)}\) for a point scatterer of mass 600 (red) with \(S_{F}^{(2)}\) for a cluster of 12 points, each of mass 50, on the ring of radius \(R\) (blue). The point scatterer appears to scatter more strongly at higher frequencies, as illustrated by the computation. Then, we calculate the total mass \(M_{1}\) from Equation (33) and, successively, the total mass \(M_{2}\) from Equation (32). An example has been illustrated in Figure 1 of the Introduction. The values of the geometrical and constitutive parameters are detailed in the caption of the figure. The localised modes corresponding to two values of the mode number \(q\) have been presented, namely \(q=4\) in part (a) and \(q=10\) in part (b). The total masses of the rings, \(M_{1}\) and \(M_{2}\), are obviously different in the two cases examined. In order to check the accuracy of the analytical formulation and the validity of the homogenisation approximation, we have also performed numerical simulations in Comsol Multiphysics 5.5 [40]. Since the numerical model needs to have a finite domain, we have considered a square plate with side length \(L=50\) m. The localised modes for \(\omega=10\) rad/s and for \(q=4\) (part (a)) and \(q=10\) (part (b)) are presented in Figure 5. Comparing Figures 1 and 5, we notice a very good agreement between analytical and numerical results. The displacement fields evaluated along the two perpendicular directions \(\bar{x}\) and \(\bar{y}\), shown in Figure 5, are given in Figure 6 for \(q=4\) and in Figure 7 for \(q=10\). The solid lines represent the analytical results derived from the homogenisation approximation described in the previous sections, while the dots indicate the data obtained numerically from Comsol Multiphysics. The comparison shows once again that the analytical and numerical models produce very similar results. In Figure 8, we show how to extrapolate the total masses of the two rings, \(M_{1}\) and \(M_{2}\), for different values of the radius \(R_{1}\) of the external ring. In the plots, the radius of the internal ring is fixed (\(R_{2}=4.5\) m) and the mode number is \(q=4\) (part (a)) and \(q=10\) (part (b)). The total masses \(M_{1}\) and \(M_{2}\) considered in Figures 1 and 5, corresponding to \(R_{1}=5.5\) m, are indicated by dots. In the left diagrams of Figure 9, we present the first four frequencies of the localised modes as functions of the ratio \(R_{1}/R_{2}\) between the radii of the two concentric circles, for \(q=4\) (part (a)) and \(q=10\) (part (b)). In the right diagrams, the corresponding values of the ratio \(M_{1}/M_{2}\) between the total masses of the rings are included. The possibility to change the frequency at which a localised mode occurs demonstrates the versatility of the  
Figure 8: Total mass \(M_{2}\) in the internal ring (left) and total mass \(M_{1}\) in the external ring (right) calculated for different values of the radius \(R_{1}\) of the external ring and for (**a**) \(q=4\) and (**b**) \(q=10\). The radius of the internal ring is \(R_{2}=4.5\) m and the frequency of the localised mode is \(\omega=10\) rad/s, as in Figures 1 and 5. The dots represent the values of the total masses determined in Figures 1 and 5.