Inhomogeneous relaxation in vibrated granular media: consolidation waves

We investigate the spatial dependence of the density of vibrated granular beds, using simulations based on a hybrid Monte Carlo algorithm. We find that the initial consolidation is typically inhmogeneous, both in the presence of a constant shaking intensity and when the granular bed is submitted to `annealed shaking'. We also present a theoretical model which explains such inhomogeneous relaxation in terms of a `consolidation wave', in good qualitative agreement with our simulations. Our results are also in qualitative agreement with recent simulations.