Roll waves as relaxation oscillations

Abstract

Granular roll waves consist of a long rising flank, followed by an abrupt fall. Based on this observation, we draw a parallel between roll waves and relaxation oscillations. From the generalized Saint-Venant equations, we derive a dynamical system governing the shape of the waves. Casting this system in the Liénard form, custom-made for studying relaxation oscillations, we find an analytical expression for the wavelength of roll waves as a function of their amplitude.