Oblique cross-waves in horizontally vibrated containers

Abstract

The excitation of subharmonic waves on the free surface of a horizontally vibrated, rectangular container of liquid is considered and the properties of threshold patterns are obtained and discussed. These waves are generally quasiperiodic and oblique (not aligned with the container walls). The parametric forcing mechanism generated by the harmonic oscillatory bulk flow is assumed to dominate over that associated with harmonic surface waves and a linear theory recently developed by the authors [Perez-Gracia et al 2014 /. Fluid Mech. 739 196-228] is used to compute both the threshold forcing amplitude and the pattern orientation. Two distinct regimes are considered: (1) large containers where the subharmonic waves generated at each endwall do not interact appreciably and (2) smaller containers where interaction occurs. The nature of the critical eigenfunction is examined in each case, and a contrast drawn between pure 2:1 resonance and the general case of quasiperiodic instability.

Dynamics of weakly coupled parametrically forced oscillators

“…The results of Sec. IV C 2 agree with recent theoretical work on cross waves [20,35,36,45] in predicting a primary Hopf bifurcation to modulated (quasiperiodic) solutions. The results of that section go further in, among other things, predicting the destruction of these modulated solutions in a saddle-node heteroclinic bifurcation (less often, in a standard heteroclinic bifurcation).…”

“…Purely dissipative coupling produces the opposite, a region of pitchfork bifurcations centered at ν = 0 with Hopf bifurcations outside of this, in accordance with Eqs. (42) and (45). Perturbations of these conservative and purely dissipative cases are similar, but with asymmetrically located A more complete bifurcation picture is shown in Fig.…”

Experimental study on travelling and standing pattern formation and capillary waves in a pinned liquid film: effects of multi-axis lateral (horizontal) vibrations and substrate geometry

“…They concluded that the cross-waves were oblique in nature, with angles between 50° and 80°. Similar findings related to the obliqueness of the cross-waves were reported in Pérez-Gracia et al (2013, 2014). It was further concluded that the subharmonic cross-waves form because of the parametric instability caused by the combination of primary harmonic surface waves and oscillatory bulk flow of the fluid.…”

A review of fluid instabilities and control strategies with applications in microgravity

“…The cross-wave instability is commonly observed in wavemaker experiments [8,110] when sufficient forcing is applied. As indicated by its name, the instability produces waves that are oriented (nearly, but not completely [119,120,123]) perpendicular to the forcing axis and to the directly forced harmonic waves. They were first described by Faraday [43], in an experiment with a partially submerged vibrating plate, as forming a series of apparently permanent ridges projecting outward "like the teeth of a coarse comb".…”