Mach cone in a shallow granular layer

TL;DR

This study investigates the formation of Mach cones in a shallow vibrated granular layer, demonstrating that the cone angle follows the Mach relation and occurs at a critical velocity related to gravity and layer depth.

Contribution

It provides experimental evidence that Mach cone angles in granular layers obey the classical Mach relation, even at minimal layer depths comparable to a single particle diameter.

Findings

Mach cones form at velocities exceeding a critical value c=sqrt(gh)

The cone angle follows the Mach relation sin(theta)=c/v_R

Mach cones are observed even in layers as thin as one particle diameter

Abstract

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.