Mach cone in a shallow granular fluid

We study the V -shaped wake (Mach cone) formed by a cylindrical rod moving through a thin, vertically vibrated granular layer. The wake, analogous to a shock (hydraulic jump) in shallow water, appears for rod velocities vR greater than a critical velocity c . We measure the half angle theta; of the wake as a function of vR and layer depth h . The angle satisfies the Mach relation, sin theta=c/vR , where c= square root of gh , even for h as small as one-particle diameter.