  We present simulations and a theoretical treatment of vertically vibrated
granular media. The systems considered are confined in narrow
quasi-two-dimensional and quasi-one-dimensional (column) geometries, where the
vertical extension of the container is much larger than both horizontal
lengths. The additional geometric constraint present in the column setup
frustrates the convection state that is normally observed in wider geometries.
This makes it possible to study collective oscillations of the grains with a
characteristic frequency that is much lower than the frequency of energy
injection. The frequency and amplitude of these oscillations are studied as a
function of the energy input parameters and the size of the container. We
observe that, in the quasi-two-dimensional setup, low-frequency oscillations
are present even in the convective regime. This suggests that they may play a
significant role in the transition from a density inverted state to convection.
Two models are also presented; the first one, based on Cauchy's equations, is
able to predict with high accuracy the frequency of the particles' collective
motion. This first principles model requires a single input parameter, i.e, the
centre of mass of the system. The model shows that a sufficient condition for
the existence of the low-frequency mode is an inverted density profile with
distinct low and high density regions, a condition that may apply to other
systems too. The second, simpler model just assumes an harmonic oscillator like
behaviour and, using thermodynamic arguments, is also able to reproduce the
observed frequencies with high accuracy.
