Crossover of the behavior of surface waves in a vibrated granular material

We study the formation of standing waves on the surface of a vibrated granular material by molecular dynamics simulations. Two distinct regimes are observed at low and high vibration frequencies. At low frequency, the standing waves are maintained by a sizable fraction of particles traveling long distances. Such movements are significantly reduced at high frequency due to frequent interparticle collisions, and more coherent motion of particles is observed. We also describe in detail the time evolution of the standing waves, especially on the collision process of the granular layer with the bottom plate, during which horizontal flux of particles is generated.