A perturbation approach for predicting wave propagation at the spatial interface of linear and nonlinear one-dimensional lattice structures

We present analysis of dispersive wave propagation through a spatial interface between a linear and nonlinear monatomic chain using a proposed multiple scales perturbation approach. As such, we solve interface problems at each perturbation order (up to and including the second order) and assemble multi-harmonic solutions for transmitted and back-scattered waves. The perturbation approach predicts the existence of multiple nonlinear dispersion curves in the nonlinear subdomain. Using these curves, we further predict spatially-varying, higher-harmonic generation in the transmitted field. For propagating higher-harmonic waves, their amplitude is predicted to experience oscillatory spatial modulation due to the presence of multiple wavenumbers at each frequency, whereas for evanescent waves, their amplitude is predicted to undergo a saturating modulation. A transmission analysis quantifies the increase of the extra-harmonic frequency transmission, and the decrease of the fundamental frequency transmission, as the level of nonlinearity increases. Using direct numerical integration, we show that the perturbation predictions agree closely with numerical simulations for weakly nonlinear wave propagation. Lastly, informed by the perturbation results, we suggest a wave device which tailors the transmission of higher harmonics through the choice of the nonlinear subdomain's length and/or the signal amplitude.