Nonlinear oscillations in a thin ring. I. Three-wave resonant interactions (Q1384824)

The authors investigate nonlinear resonant interactions between planar waves in a thin circular ring. It is found that 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 bending and azimuthal. These are in phase with the primary wave. All three waves together compose a resonant triad. Such kind of instability causes the stress amplification of a ring. The authors estimate the stress growth constant and the period of energy exchange between the waves, based on the analytical solution to the evolution equation governing the triad. This work establishes that bending waves in a ring are stable against small perturbations. The results obtained are in good qualitative agreement with known experimental data.