Scattering of flexural waves from an $N$-beam resonator in a thin plate

The impedance matrix method is applied to study the scattering of flexural waves propagating in an infinite thin plate containing an $N$-beam resonator. The resonator consists of a circular hole containing a smaller plate connected to the background plate by a number $N$ of rectangular beams. After representing the boundary conditions in a modal multipole expansion form, a compact expression is obtained for the T-matrix, which relates the incident and the scattered transverse (out-of-plane) waves. The analysis of the scattering cross-section reveals interesting scattering features, like resonances and anisotropy, associated with this type of resonators. Numerical experiments performed within the framework of the finite element method support the accuracy of the model here developed.