Envelope waves propagation in a one-dimensional tetratomic model of acoustic metamaterial

Abstract

This paper explores the propagation of modulated waves in systems modelled as one-dimensional network whose elementary cell is constituted of four interacting subsystems. We derive the dispersion relation of its small amplitude plane waves as a quartic polynomial equation in the frequency squared; and numerically show the existence of four frequency bands. These include one acoustic band and three optical bands of which the middle one features the left-handedness. We also use the reductive perturbation method to reduce the discrete equations of motion of the system to the standard nonlinear Schrödinger, thereby predicting the possibility of the propagation modulated waves in the model. This prediction is firmly established through direct numerical simulations in which the analytical solutions of the nonlinear Schrödinger equation are used as input.