Second-order perturbation analysis of low-amplitude traveling waves in a periodic chain with quadratic and cubic nonlinearity

Traveling waves in one-dimensional nonlinear periodic structures are investigated for low-amplitude oscillations using perturbation analysis. We use second-order multiple scales analysis to capture the effects of the quadratic nonlinearity. Comparisons with the linear and cubical nonlinear cases are presented in the propagation and attenuation of the wave as well as the dispersion relationships, group and phase velocities and their dependence on wave number and amplitude of oscillation. Quadratic nonlinearity is shown to have a significant effect on the wave propagation behavior in the chain. Given the wavenumber, the quadratic nonlinearity (with nonzero linear and zero cubic terms) has shown to produce higher frequencies than linear system. At lower wave numbers, the quadratic system has higher phase and group velocities than the linear case. At higher wave numbers, however, the group velocities for the quadratic system are lower as compared to the linear case.