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 inhomogeneous, both in the presence of a constant shaking intensity and when the granular bed is subjected 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 experiments.