Mach cone in a shallow granular layer

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 v_R greater than a critical velocity c. We measure the half-angle, theta, of the wake as a function of v_R and layer depth, h. The angle satisfies the Mach relation, sin(theta)=c/v_R, where c=sqrt(gh), even for h as small as one particle diameter.