Subharmonic capillary–gravity waves in large containers subject to horizontal vibrations

Abstract

This paper deals with nearly inviscid, capillary–gravity, modulated waves parametrically excited by monochromatic horizontal vibrations in liquid containers whose width and depth are both large compared with the wavelength of the excited waves. A general linear amplitude equation is derived with appropriate boundary conditions that provides the threshold acceleration and associated spatiotemporal patterns, which compare very well with experimental measurements and visualizations. The primary instability is associated with a pair of complex Floquet multipliers that are close to (but strictly different from) −1, meaning that the instability is not strictly (2:1) subharmonic. The resulting (quasi-periodic) waves are generally oblique, not perpendicular to the vibrating endwalls. The extension of the theory to other confined systems such as vibrating containers of arbitrary shape and vibrating drops is also considered.

Cross-waves excited by distributed forcing in the gravity-capillary regime

“…In contrast to the case of a semi-infinite domain, where steady solutions A(x) can be sought (corresponding to In la -periodic subharmonic waves in the original problem), the interaction of opposing wavetrains in the two-sided problem leads, in general, to modulated (quasiperiodic) solutions. 28 ' 33 We search for Hopf bifurcations with the ansatz, 29…”

“…Finally, it was seen how horizontal excitation in a finite-sized container, with two opposing out-of-phase wavetrains, naturally leads to a Hopf bifurcation and thus to temporally quasiperiodic eigenfunctions that cycle slowly between two-sided and one-sided patterns. 28 ' 33 If the overlap of the subharmonic waves originating at opposite endwalls is small, this can be understood as the result of a symmetry-breaking interaction that destroys the D 4 symmetry of the uncoupled (infinite length) problem. The small interaction term leads to a similarly small Hopf frequency and a small interval of forcing values where quasiperiodic solutions can be found.…”

“…It follows that the model obtained below will not apply to all subharmonic patterns excited by horizontal forcing, in particular those that are very rotated with respect to standard cross-waves. 28 It does, however, apply to a wide range of low viscosity experiments in the gravity-capillary regime including some results of Porter et al 23 where typical subharmonic patterns are rotated (i.e., modulated in the downstream direction) but often not excessively so at onset. The model obtained here has the theoretical advantage of allowing direct comparison with the many established results based on NLS models that assume a narrow forcing region.…”

Dynamics of weakly coupled parametrically forced oscillators

“…If, instead, these same two wavemakers are driven with distinct phases, then the coupled system behaves differently (it is equivalent to introducing asymmetry in the coupling between the two patterns/oscillators). We note that this second situation automatically occurs when the subharmonic surface waves are excited by rigid horizontal motion of an open fluid container [19,35,36]. In this case, while the left endwall is moving into the fluid (leading to upward displacement near the surface), the right endwall is pulling away from the fluid (causing downward displacement near the surface).…”

“…V, it is relevant to experiments on horizontally vibrated fluids in rigid containers [19,20]. In this system, the parametric forcing mechanism [34][35][36] is concentrated near opposing boundaries (endwalls or wavemakers) and, in containers of sufficiently large aspect ratio, this leads to relatively isolated patterns of subharmonic surface waves that can be considered as weakly interacting.…”