Analytical study of the propagation of acoustic waves in a 1D weakly disordered lattice

TL;DR

This paper develops an analytical method to study acoustic wave propagation in a weakly disordered 1D lattice of Helmholtz resonators, providing explicit expressions for transmission and localization length, and comparing results with numerical methods.

Contribution

It introduces a novel analytical approach for calculating transmission and localization length in a disordered acoustic lattice, enhancing understanding of wave behavior in such systems.

Findings

Analytical transmission coefficient matches numerical results.

Localization length depends on disorder strength and frequency.

Distinct localization behaviors observed at Bragg and Helmholtz stopbands.

Abstract

This paper presents an analytical approach of the propagation of an acoustic wave through a normally distributed disordered lattice made up of Helmholtz resonators connected to a cylindrical duct. This approach allows to determine analytically the exact transmission coefficient of a weakly disordered lattice. Analytical results are compared to a well-known numerical method based on a matrix product. Furthermore, this approach gives an analytical expression of the localization length apart from the Bragg stopband which depends only on the standard deviation of the normal distribution disorder. This expression permits to study on one hand the localization length as a function of both disorder strength and frequency, and on the other hand, the propagation characteristics on the edges of two sorts of stopbands (Bragg and Helmholtz stopbands). Lastly, the value of the localization length inside the Helmholtz stopband is compared to the localization length in the Bragg stopband.