Dispersive propagation of localized waves in a mass-in-mass metamaterial lattice

Linear localized waves evolution in a discrete mass-in-mass lattice is studied. The presence of the attached mass in the model contributes to dispersion giving rise to appearance of both the acoustic and optic wave modes. Important features of the waves are described using the dispersion relation, which is obtained in a continuum limit of the original discrete equations using a harmonic wave solutions. We study numerically the localized initial perturbations evolution and compare the features of the numerical solutions with those obtained for the analytical harmonic wave ons. We have found differences in the wave dynamics depending on the parameters of the initial conditions. One scenario describes almost permanent shape and velocity counterpart localized waves propagation with oscillating standing wave around the position of the initial pulse. Another wave evolution accounts for a decrease in the moving wave amplitude with the developing oscillating tail behind the localized wave. Contrary to the periodic analytical solution, no evidence of a band gap is found in the simulations of localized waves.