Intense bending waves in a bar

Summary

In this paper, the work presented in [1] is extended to study higher-order approximations of nonlinear effects in a bar. It has been found that long bending waves, being the low-frequency modes involved in resonant triads, are stable against small perturbations. Consequently, a bending wave with group velocity which is less than that of longitudinal waves should behave as a linear quasi-harmonic wavetrain. On the other hand, one may expect self-modulation instability of intense bending wavetrains during the long-time evolution. This paper overcomes such a contradiction. To describe the nonlinear dynamics in detail, one should allow for higher-order approximation effects in the model. Such effects are associated with the diffusion of linear wave packets due to different group velocities, and amplitude dispersion caused by nonlinearity. Within the second-order approximation analysis, an amplitude modulation is indeed experienced for intense bending waves. As a result, envelope solitons can be formed from unstable bending wavetrains. The group matching of long longitudinal and short bending waves, being a particular case of the self-modulation, is of special interest as a limit case of the triple-wave resonant interactions. It demonstrates the relation between the first- and the second-order approximation effects.