Multiple scales analysis of wave-wave interactions in a cubically nonlinear monoatomic chain

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass-spring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave-wave interaction problems.