# Scattering Reduction and Resonant Trapping of Flexural Waves: Two Rings to Rule Them
## 1 Introduction  
In recent years, mathematical and physical problems related to wave localisation and scattering by discrete clusters have attracted increasing interest in the scientific community. In fact, the study of localised waveforms in structured media and dynamic Green's kernels in lattices is essential in the analysis of defects in crystalline solids, transmission resonances for grating stacks, and novel designs of wave shields or \"cloaks\". The notion of a Faraday cage, representing a device capable of protecting a bounded domain from an external electric field [1], is well established in electromagnetism. This notion has been extended to acoustic waves, governed by the Helmholtz equation, in [2]. In elasticity, the localisation of elastic waves in solids with clusters of defects, such as small inertial inclusions or rigid pins, exhibits new features in the mathematical description and physical interpretation of the phenomena; in particular, it is known that structured flexural plates incorporate evanescent and propagating waveforms, which couple through the boundary conditions. The dynamics of imperfect crystals with point defects has been extensively analysed in [3, 4]. Propagation of waves in square lattices, generated by oscillating a lattice point in the out-of-plane direction, has been investigated in [5], while localised vibrations in band-gaps of continuous and discrete structures, obtained from the Green's function approach, have been examined in [6]. In elastic plates, localisation and transmission of flexural waves due to a periodic arrangement of rigid pins or point masses have been analysed in [7, 8, 9]. Unidirectional trapped modes, associated with Dirac points in the dispersion diagrams, have been shown to exist in thin plates with a finite cluster of pins [10]. Cloaking of a rigid inclusion in a Kirchhoff plate by means of active sources has been proposed in [11]. Localisation in semi-infinite grating stacks of rigid pins or resonators has been investigated in [12, 13]. A platonic structure with spiral resonators has been presented in [14], with special emphasison applications such as low-frequency filtering, transmission amplification and generation of edge waves. In [15], resonating beams arranged in concentric circles were connected to a plate to create a directional cloak. Time-harmonic waves produced by an external force in a thin plate with attached masses were studied in [16]. The band structures of thin and thick plates incorporating periodic arrays of spring-mass resonators were discussed in [17; 18]. The existence of band gaps in plates with different types of cavities has been numerically and experimentally demonstrated in [19; 20]. The low-frequency vibrations and radiation performance of homogeneous and sandwich plates containing periodically distributed resonators have been investigated in [21; 22]. Effective band-gaps and wave conversion in plates have been obtained in [23] by inserting a periodic array of perpendicular rod-like resonators. Gyroscopic resonators attached to plates have been employed in [24; 25] to create flexural waveforms propagating in one direction. Topologically protected edge and interfacial modes in microstructured plates have also been discovered in [26; 27; 28; 29]. An asymptotic analysis has been developed in [30; 31] to describe Bloch modes localised around a circular array of inclusions, showing similarities with whispering gallery modes. The analysis has been carried out for the planar Helmholtz equation and assuming Neumann boundary conditions at the boundaries of the inclusions. Recently, an asymptotic homogenised model for the bi-harmonic operator was presented in [32] to predict the existence of flexural vibrations confined along a ring of spring-mass resonators, which yield \"negative inertia\" in the dynamic regime. In that model, the frequency of the localised flexural modes along the ring could not be chosen ad libitum.