The phenomenon of longitudinal waves in shallow grain flows has been studied through laboratory experiments. The transport process of spherical particles on a metallic chute has been characterized for this purpose. The wave mode of material transport could be measured within selected combinations of flow parameters such as the angular inclination of the chute, the mean size of the grains and the mass flow rate. It has been observed that the moving particles tend to redistribute systematically in the direction of mean flow. As a result, nonlinear longitudinal waves evolve on the surface of the chute. Observations of the predominantly rolling mode of particle motion revealed significant particle dispersion away from the wavefronts. The frequency of inter-particle collisions was low in the dispersed flow regions but increased rapidly near the wavefronts to dissipate the excess kinetic energy, thus resulting in a large increase in the average volumetric solid fraction. In order to explain the appearance of discontinuities in the volumetric solid fraction, a theoretical model that preserves the overall balance of energy and allows a discontinuous periodic solution is examined here. The depth-averaged dispersed flow of the grains has been approximated by equations of motion similar to those of shallow fluid flow. The resistance to the rolling motion of the particles is expressed in terms of the hydrodynamic drag force. The theoretical model predicts the flow criterion for which the longitudinal waves would be self-sustaining.