| Summary: | 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.
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