Ring-shaped linear waves and solitons in a square lattice of acoustic waveguides

We study the propagation of both low- and high-amplitude ring-shaped sound waves in a two-dimensional 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 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.