Statistics of shock waves in a two-dimensional granular flow

We have investigated the dynamics of shock waves in a single layer of uniform balls in a small-angle two-dimensional funnel. When the funnel half-angle $0°\lesssim \beta \lesssim 2°$, the flow is intermittent and kinematic shock waves are observed to propagate against the flow. We have used fast video equipment and image analysis methods to study the statistics of the shock waves. It is found that their speed and frequency increase with the distance from the outlet. In particular, the shock speed scales as the ratio of the local funnel width to the width of the funnel outlet. Various kinds of interactions between shock waves are observed, including repulsion. New shock waves are only created at those sites where a close-packed triangular packing of the monodisperse balls fits across the funnel.