Point interactions in acoustics: one dimensional models

TL;DR

This paper models one-dimensional acoustic systems with mechanical oscillators, analyzing their spectral properties and band structures, providing insights into wave interactions with point-like acoustic interactions.

Contribution

It introduces a novel framework for modeling acoustic systems with point interactions using singular perturbations and analyzes their spectral properties.

Findings

Spectral properties of the models are fully characterized.

Periodic arrays exhibit band structures in their energy spectrum.

The models provide a detailed understanding of wave-oscillator interactions.

Abstract

A one dimensional system made up of a compressible fluid and several mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed for different settings of the oscillators array. The dynamical models are formulated in terms of singular perturbations of the decoupled dynamics of the acoustic field and the mechanical oscillators. Detailed spectral properties of the generators of the dynamics are given for each model we consider. In the case of a periodic array of mechanical oscillators it is shown that the energy spectrum presents a band structure.