The differential equation of equilibrium for the bed sediment movement for clear-water conditions is used and a new mathematical model for computing the scour development with time at the elliptical guide banks is elaborated. We suggest to solve initial equations numerically using software packages available for the solution of the Cauchy problem. In this case the time step can be chosen in order to guarantee desired accuracy of the solution. In particular, using adaptive integration formulas, one can select the step size automatically. There are at least two advantages of the proposed approach in comparison with the our method used before (Gjunsburgs et al.,2006): the next time step can be selected by analyzing the properties of the solution at the previous time steps, that is, one can choose a smaller time steps, where the solution changes quickly and larger time steps where the solution does not change too much, and there is no need to approximate scour depth s h by a constant at ea