Analytical solutions of the linearized parabolic wave accounting for downstream boundary condition and uniform lateral inflows

In this paper, new analytical solutions of the linearized parabolic approximation (LPA) of the De Saint Venant equations (DSVEs) are derived for the case of finite channel length. The new solutions for both discharge and water depth are found taking into account upstream and lateral inflows, and some of possible boundary conditions at the downstream end (a stage-discharge relationship and a time dependent flow depth). The solutions are first determined in the Laplace Transform domain, and the Laplace Transform Inversion Theorem is used in order to find the corresponding time domain expressions. The effects induced on the flow propagation by the downstream boundary condition are analyzed using the new analytical solutions. Finally, a simple flood routing based on the new analytical solutions is applied to some schematic cases and compared with the solution of the full DSVEs, showing that the new solutions can actually account for the effects of the downstream boundary condition on the flow dynamics