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

•New analytical solutions of the linearized parabolic wave are presented.•The new solutions account for downstream boundary condition and lateral inflows.•The effects induced on discharge propagation and backwater curve are analyzed.

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, which take into account upstream and lateral inflows, are found considering two types of boundary conditions at the downstream end, namely a stage–discharge relationship and a time dependent flow depth. The solutions, for both discharge and water depth, 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. Finally, the effects induced on the flow propagation by the downstream boundary condition are analyzed using the new analytical solutions.