In this paper a mathematical model for the movement of water particles in a river is constructed.

Using the model it will be possible to find an analytic expression for the distribution function F(t) of the transit time, T, needed for a water particle to travel from one cross section of the river to another. This implies that we obtain a quantitative description of the longitudinal dispersion in the river.

Briefly the model can be described as follows. Firstly we divide the river into two layers, an upper layer and a lower layer. (The «border-line» between the two layers will depend on the mean velocity of the river and on the velocity profile across the river.) Secondly we assume that the longitudinal velocity is constant (and positive) in the upper layer and zero in the lower layer. Finally we assume that a water particle moves back and forth between the two layers in a random way. The probability law governing this movement will depend on a parameter characterizing the roughness of the bottom of the river.

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