# Scattering Reduction and Resonant Trapping of Flexural Waves: Two Rings to Rule Them
## 4 Conclusions  
In this paper, we have studied the effect of rings of point resonators on the dynamic behaviour of a thin elastic plate, modelled as a Kirchhoff plate.  
First, we have demonstrated that the scattering produced by a single point mass \(M\) can be decreased significantly if that mass is split into a large number of smaller masses, arranged along a circle. This is especially true for medium and high frequencies.  
Second, we have constructed an asymptotic model for two rings of spring-mass resonators attached to an elastic plate. This model allows determining the frequencies at which localised modes, confined between the two rings, can occur in the structure. We have also outlined the systematic procedure to vary the frequency of the trapped modes and to accordingly determine the geometrical and constitutive parameters of the two rings of resonators. In addition, we have verified the analytical results with an independent finite element code.  
It is noted that for high frequencies, the discrete nature of the structure plays a significant role, and the relative rotation of the rings of point masses may be of interest. In particular, when the distance between the rings becomes small, the system involves a structure of dipoles of different orientations. In problems of wave scattering, such systems can also be viewed as inertial interfaces.  
We envisage that the results of this work will be useful in the design of scattering systems and energy absorbers in structures supporting the propagation of flexural waves. We also note that the inertial double ring of point masses (or resonators) is a structured inertial interface system, and the homogenisation procedure can also address the disconti  
Figure 9: Frequencies \(\omega\) of localised modes (left) and ratios of total masses \(M_{1}/M_{2}\) (right) as functions of the ratio \(R_{1}/R_{2}\) between the radii of the two rings, calculated for (**a**) \(q=4\) and (**b**) \(q=10\). For any ratio \(R_{1}/R_{2}\), the first four solutions (labelled as a, b, c and d) are shown and indicated by dots. In the computations, \(R_{2}=4.5\) m and (**a**) \(M_{2}=802.686\) kg and (**b**) \(M_{2}=10{,}609.7\) kg, as in Figures 1 and 5. The crosses represent the values of the frequencies and the mass ratios obtained in Figures 1 and 5.  
nuity of tractions across such an interface for certain regimes of scattering of time-harmonic flexural waves.  
Conceptualisation, A.B.M. and R.C.M.; analytical methodology, A.B.M. and R.C.M.; numerical validation, G.C.; writing--original draft preparation, A.B.M. and G.C.; writing--review and editing, A.B.M., R.C.M. and G.C. All authors have read and agreed to the published version of the manuscript.  
The authors would like to acknowledge the support of the Liverpool Research Centre for Mathematics and Modelling (RCMM) for the academic visits of [PERSON] to the University of Liverpool. At the initial stage of analysis leading to this paper, the work was supported by the UK Engineering and Physical Sciences Research Council Programme grant EP/L024926/1; EPSRC support is gratefully acknowledged.  
Not applicable.  
Data Availability Statement: Data will be provided upon request to the authors.  
The authors declare no conflict of interest.