Ring-Shaped Linear Waves and Solitons in a Square Lattice of Acoustic Waveguides

TL;DR

This paper investigates ring-shaped sound waves in a 2D lattice of acoustic waveguides with Helmholtz resonators, deriving an effective equation and confirming the existence of cylindrical solitons through analytical and numerical methods.

Contribution

It introduces a novel approach to analyze radially symmetric sound waves in acoustic lattices using an effective cylindrical KdV equation.

Findings

Helmholtz resonators suppress anisotropy in the system.

Low-amplitude waves resemble Airy functions.

High-amplitude waves form cylindrical solitons.

Abstract

We study the propagation of both low- and high-amplitude ring-shaped sound waves in a 2D square lattice of acoustic waveguides with Helmholtz resonators. We show that the inclusion of the Helmholtz resonators suppresses the inherent anisotropy of the system in the low frequency regime allowing for radially symmetric solutions. By employing the electroacoustic analogue approach and asymptotic methods we derive an effective cylindrical Korteweg de Vries (cKdV) equation. Low-amplitude waveforms are self-similar structures of the Airy function profile, while high-amplitude ones are of the form of cylindrical solitons. Our analytical predictions are corroborated by results of direct numerical simulations, with a very good agreement between the two.