Cluster model for wave patterns of a 3D vertically vibrated granular system

Based on the cluster model for wave-like motions of a 2D vertically vibrated granular system we established previously (Chin Phys Lett, 2010, 27: 124501), a generalization of the cluster model for a 3D granular system is presented in this paper. The 3D cluster model proposes frustums of pyramids as clusters of the 3D granular system, and explains wave patterns as the result of the cluster-boundary and cluster-plate collisions. By analyzing the movement of one cluster in a collision period, we derive a basic equation, which relates the internal characteristic parameter to the external driving parameters. The theoretical results reproduce the behaviors of wave patterns as changes in the driving parameters, and the 3D dispersion relation, which agree with the experimental results.