Weak scattering formulation for flexural waves in thin elastic plates with point-like resonators

Abstract: A theoretical study on the weak scattering formulation for flexural waves in thin elastic plates loaded by point-like resonators is reported. Our approach employs the Born approximation and far-field asymptotics of the Green function to characterize multiple scattering effects in this system. The response of the system to an incident wave can be expressed as a power series expansion, where each term introduces higher-order scattering components. The behavior of this series is governed by the spectral properties of a specific matrix, which dictates the convergence and accuracy of the weak scattering approximation. By decomposing the scattering response into geometric and physical contributions, we establish a condition for weak scattering in terms of these two factors. This formulation provides a systematic framework for assessing the validity of low-order approximations and understanding wave interactions in complex media. Numerical examples illustrate the accuracy of the weak scattering condition in periodic and random distributions of scatterers, highlighting its implications for wave control and metamaterial design.