Subharmonic surface waves in vibrated granular media

Using a simplified model of a dissipative discrete-particle fluid we study the formation of surface waves under vertical vibration. The only dissipative forces included in this two-dimensional molecular dynamics simulation act normally to the line of contact during collisions; transverse frictional forces commonly used in granular models are not required. Depending on the parameters, surface waves are observed to form with a half or a quarter of the driving frequency.