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

Abstract: This work reports the conditions under which weak scattering assumptions can be applied in a beam loaded by multiple resonators supporting both longitudinal and flexural waves. The work derives the equations of motion of a one-dimensional elastic waveguide with several point resonators by utilizing the Green's matrix approach. The derivations include any resonator morphology, either with a discrete or continuous distribution of resonances. The method employed is based on applying multiple scattering theory. The response can be expressed as an infinite series whose convergence is closely linked to the scattering intensity provided by the resonators. The convergence conditions are reduced to studying the spectral radius of the scattering matrix. Furthermore, the the leading order of the multiple scattering expansion is associated with the Born approximation. The work also provides approximate expressions for the spectral radius, offering a physical interpretation to the concept of weak scattering in beams. Several numerical examples are presented to validate the proposed methodology.