Oblique cross-waves in horizontally vibrated containers

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 J. 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.