Wave propagation in beams with multiple resonators: Conditions for weak scattering and the Born approximation

We investigate the conditions necessary for weak scattering in a beam loaded by multiple resonators that support both longitudinal and flexural waves. Using a Green’s matrix approach we derive the equations of motion of a one-dimensional elastic waveguide with several point resonators, for any resonator morphology modeled by the transfer matrix method, even when considering the resonators to have any number of natural frequencies. The methodology is based on multiple scattering theory, expressing the response as an infinite series whose convergence is closely linked to the scattering intensity provided by the resonators. The convergence conditions are reduced to the study of the spectral radius of the scattering matrix. Furthermore, the leading order of the multiple scattering expansion is associated with the Born approximation. The results offer approximate expressions for the spectral radius, providing a clear physical interpretation of weak scattering. Several numerical examples are presented to validate the proposed approach, demonstrating its effectiveness and applicability.