Diffusion of Sediment in Long Channels

A model is presented, treating the transport of suspended material in a broad channel under simplifying assumptions. The motion of sediment is treated as a diffusion process and a differential equation is derived. The form of an equilibrium distribution g(y) is derived, corresponding to the classical distribution of Prandtl and Rouse. Mathematically the following two results are proven: (1)If a stationary (time-dependent) state prevails downstream a certain point P, then the distribution downstream P, tends (with increasing distance to P) exponentially to the equilibrium distribution; and (2) if the sediment distribution in the incoming water at a point P is stationary from a certain moment, then the distribution downstream P will tend to the stationary solution exponentially with time. Although these results seem to be physically evident, they have apparently not been proven before.