On Resonant Waves in Lattices

TL;DR

This paper investigates resonant wave propagation in various 2D periodic lattices, analyzing eigenfrequencies, dispersion, and wave phenomena such as beaming and localization through mathematical modeling and simulations.

Contribution

It provides a comprehensive analysis of resonant wave behavior in different lattice geometries, highlighting new wave phenomena at resonance frequencies.

Findings

Identification of multiple resonant points in frequency spectra.

Observation of wave beaming at resonance frequencies.

Detection of line-localized primitive waveforms.

Abstract

Mathematical modeling of resonant waves propagating in 2D periodic infinite lattices is conducted. Rectangular-cell, triangular-cell and hexagonal-cell lattices are considered. Eigenvalues (here eigenfrequencies) of steady-state problems are determined, and dispersion properties of free waves are described. We show that the frequency spectra of the models possess several resonant points located both at the boundary of pass/stop bands and in the interior of a pass band. Boundary value problems with a local monochromatic source are explored, and peculiarities of resonant waves are revealed. Asymptotic solutions are compared with the results of computer simulation. Special attention is given to line-localized primitive waveforms at the resonance frequencies and to the wave beaming phenomena at a resonant excitation.