Nonlinear oscillations in a thin ring - I. Three-wave resonant interactions

Nonlinear resonant interactions between planar waves in a thin circular ring are investigated. It is found that a high-frequency azimuthal wave is unstable against a pair of secondary low-frequency waves. The secondary waves are of two types; either two bending or azimuthal and bending. These are in phase with the primary wave. All three together compose a resonant triad. Such kind of instability causes the stress amplification in the ring. The stress growth constant and the period of energy exchange between the waves are estimated based on analytical solutions to the evolution equations driving the triad. The lowest-order nonlinear approximation analysis predicts stability for bending waves. A good qualitative agreement of the obtained results with some known experimental data is observed.