Using numerical and analytic methods, we study the behavior of granular particles contained in a vibrating box. We measure, by molecular dynamics (MD) simulation, several quantities which characterize the system. These quantities-the density and the granular temperature fields, and the vertical expansion-obey scaling in the variable x = Af . Here, A and f are the amplitude and the frequency of the vibration. The behavior of these quantities is qualitatively different for small and large values of x. We also study the system using NavierStokes type equations developed by Haff. We develop a boundary condition for moving boundaries, and solve for the density and the temperature fields of the steady state in the quasi-incompressible limit, where the average separation between the particles is much smaller than the average diameter of the particles. The fields obtained from Haff's equations show the same scaling as those from the simulations. The origin of the scaling can be easily understood. The behavior of the fields from the theory is consistent with the simulation data for small x, but they deviate significantly for large x. We argue that the deviation is due to the breakdown of the quasi-incompressibility condition for large x.

Using numerical and analytic methods, we study the time dependent behavior of granular material in a vibrating box. We find, by molecular dynamics simulation, that the temporal fluctuations of the pressure and the height expansion scale in Af , where A (f ) is the amplitude (frequency) of the vibration. On the other hand, the fluctuations of the velocity and the granular temperature do not scale in any simple combination of A and f . Using the kinetic theory of Haff, we study the temporal behaviors of the hydrodynamic quantities by perturbing about their time averaged values in the quasi-incompressible limit. The results of the kinetic theory disagree with the numerical simulations. The kinetic theory predicts that the whole material oscillates roughly as a single block. However, the numerical simulations show that the region of active particle movement is localized and moves with time, behavior very similar to the propagation of a sound wave.