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

TL;DR

This paper establishes conditions for weak scattering in beams with multiple resonators, deriving equations using Green's matrix and multiple scattering theory, and validating the approach with numerical examples.

Contribution

It introduces a comprehensive method to analyze wave scattering in beams with multiple resonators, including conditions for the Born approximation and weak scattering regimes.

Findings

Derived equations of motion for beams with multiple resonators.

Identified spectral radius conditions for weak scattering.

Validated methodology with numerical examples.

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.