Intense bending waves in a bar |
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Authors: | D A Kovriguine |
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Institution: | (1) Russian Academy of Sciences, Mechanical Engineering Institute, Nonlinear Rheology Laboratory, 85, Belinsky Str., 603024 Nizhny Novgorod, Russia, RU |
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Abstract: | 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.
Accepted for publication 20 July 1996 |
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Keywords: | nonlinear oscillations self-modulation group matching coupled soliton |
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