Second-harmonic generation in membrane-type nonlinear acoustic metamaterials

Abstract

We study analytically and numerically the second-harmonic generation in a one-dimensional nonlinear acoustic metamaterial, composed of an air-filled waveguide periodically loaded by clamped elastic plates. Based on the transmission line approach, we derive a nonlinear dynamical lattice model which, in the continuum approximation, leads to a nonlinear dispersive wave equation. By applying the perturbation method to the latter, we derive the analytical expressions for the first- and second-harmonics, which are in excellent agreement with the numerical simulations of the nonlinear dynamical lattice model. Apart from the case of dispersionless nonlinear propagation and the Fubini solution, special attention is payed to the role of dispersion. In that regard, it is found that, once dispersion comes into play, second-harmonic beatings in space due to phase-mismatch can be identified. Our results provide many opportunities for the development of new periodic acoustic structures featuring both nonlinearity and dispersion.