Hysteresis, limit cycles and mode interactions of standing waves with Faraday excitation.

Complex interaction phenomena are known to arise among standing free-surface waves in fluids within containers subjected to small periodic vertical (Faraday) oscillations. Here, we review theoretical and experimental work concerning (i) hysteresis and limit-cycle behaviour of’puret standing waves; and (ii) instability of a ’pure\dagger standing wave to a pair of neighbouring modes, and the subsequent modulations. We also discuss the special case of (iii) second-harmonic resonance. 1 lntroduction Study of wave motion excited by small periodic vertical vibrations of a cylindrical container began with the pioneering studies of Faraday1 and Rayleigh.2,3 Such vertical oscillation is now known as tFaraday excitationt. The waves most prone to generation are those with frequencies close to one-half of the forcing frequency; a situation now commonly referred to as $\mathfrak{l}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{C}’, $ $ ’ \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{C}’ $ , or $|\mathrm{F}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{y}^{1}$ resonance. The linear theory developed by Beniamin&Urse11 established that, in the inviscid limit, each surface-wave mode is governed by an equation of Mathie