The GIUH, h(t), can be convoluted with a specific rainfall event expression for the geomorphologic IUH is: Therefore, The Chezy formula is used to describe the frictional effects, results: The above equation is highly non-linear. Its linearization is per- 3.3, the following second order partial differential equation of motion formed according to the following definitions and assumptions: where C has been assumed constant and equal to the value corresponding ta the reference state, i.e., and {e]; is the linearized expression of the argument, given the specific For given initial and boundary conditions, analytical solutions of this equation may be obtained. In this study, the interest is on the Probabilistic interpretation of the upstream inflow IUH Since a closed form solution of the above integral does not exist, it h(t), as given by Equation 4.4, with a rainfall event, represented by Equa- Rodriguez-Iturbe and Valdes (1979) used Equation 2.10 to represent the PDF of the travel time in the streams of highest order. tion also has to be modified for infiltration losses, i.e. Therefore, the expressions for the IUH and the discharge hydrograph The final expression of the linearized equation of motion obtained where 5q represents the perturbation about the reference discharge qg- Equation A.1l may be written as: which is an homogeneous partial differential equation whose solution is desired for the case of a pulsed upstream inflow. A.2 Boundary and Initial Conditions steady state and the governing equation is: function and, for this particular function, the Laplace transform is 1. X,S). Following the same procedure used by Harley (1967), Equation A.13 can be rewritten as
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