# Acoustic waveguide filters made up of rigid stacked materials with elastic joints

## TL;DR

This paper analytically investigates the dispersion properties of rigid stacked acoustic waveguide filters with elastic joints, introducing models that predict pass and stop bands, including ultra-low frequency stop bands, using both discrete and homogenized continuum approaches.

## Contribution

It provides a closed-form analytical solution for dispersion relations of stacked waveguides with elastic joints and introduces homogenized micropolar continuum models for accurate approximation.

## Key findings

- Analytical dispersion relations for shear and moment waves are derived.
- Ultra-low frequency stop bands are achieved with elastic half-space coupling.
- Homogenized models accurately approximate dispersion spectra.

## Abstract

The acoustic dispersion properties of monodimensional waveguide filters can be assessed by means of the simple prototypical mechanical system made of an infinite stack of periodic massive blocks, connected to each other by elastic joints. The linear undamped dynamics of the periodic cell is governed by a two degree-of-freedom Lagrangian model. The eigenproblem governing the free propagation of shear and moment waves is solved analytically and the two dispersion relations are obtained in a suited closed form fashion. Therefore, the pass and stop bandwidths are conveniently determined in the minimal space of the independent mechanical parameters. Stop bands in the ultra-low frequency range are achieved by coupling the stacked material with an elastic half-space modelled as a Winkler support. A convenient fine approximation of the dispersion relations is pursued by formulating homogenised micropolar continuum models. An enhanced continualization approach, employing a proper Maclaurin approximation of pseudo-differential operators, is adopted to successfully approximate the acoustic and optical branches of the dispersion spectrum of the Lagrangian models, both in the absence and in the presence of the elastic support.

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Source: https://tomesphere.com/paper/1812.01352