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

Abstract

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 wavewave interaction problems.

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

“…Tuning the linear inter-chain coupling stiffness shifted the magnitude of a critical wavenumber associated with resonant energy transfer between the chains. Narisetti et al [7] and Manktelow et al [8], [9] investigated wave propagation in cubically nonlinear monoatomic and diatomic chains using Lindstedt-Poincaré and multiple scales analyses, respectively, with an emphasis on amplitude-dependent dispersion shifts. Further, they identified wave-based devices which exploit bandgap shifting to enable tunable filtering and wave-guiding.…”

Modal interaction and higher harmonic generation in a weakly nonlinear, periodic mass–spring chain

“…For the first, long-wave limit point (blue plus-sign), the higher harmonic is very close to the linear dispersion curve. Hence, it represents a limit where the analytical model might not be valid since internal resonance could be an issue, as investigated for a mono-atomic model in [9]. The results of the mentioned reference however, indicate only modest effects of internal resonance on dispersion in the long-wave limit, but the possible self-excitation from resonant higher harmonic loading is the focus here.…”

“…al. in [9], where they study the effect of interaction between propagating waves on the dispersion in a cubically nonlinear, homogeneous chain. They consider both the general case as well as the special case of internal resonance where there is a 3:1 ratio between wavenumbers and frequencies for the two waves.…”

Collective dynamics of periodic nonlinear oscillators under simultaneous parametric and external excitations

“…11, where we plotted the average response intensity, defined in Eq. (17), which is I = 4 3 (|a 2 1 | + |a 2 2 | + |a 2 3 |) for N = 3.…”

“…These nonlinearities can be due to the interaction between periodic structure and its neighbors. For instance, in the field of acoustics, Manktelow et al [16,17] focused on the interaction of wave propagation and analyzed the wave-wave interaction in a cubically nonlinear monoatomic chain, while Marathe et al [18] studied wave attenuation in nonlinear periodic structures. Lazarov et al [19] focused on the influence of nonlinearities on the filtering properties of one-dimensional chain around the linear natural frequency of the attached nonlinear local oscillators.…”