An Analytical Solution for the Frictionless Swash Hydrodynamics Under an Improved Nonlinear Seaward Boundary Condition

In the swash zone, it is essential to have a proper understanding of the spatial distribution and temporal variation of both the flow velocity and water depth to evaluate swash hydrodynamic characteristics, which have a significant impact on the local sediment movement and the design of coastal structures. In this study, with respect to the nonlinear shallow water equations with an improved nonlinear seaward boundary condition, swash hydrodynamics along a frictionless slope are theoretically investigated using the hodograph transformation and the Riemann’s method. Accordingly, an implicit analytical solution is obtained to evaluate the spatiotemporal distributions of the swash hydrodynamic features. A convex nonlinear seaward boundary condition governed by the seaward boundary parameter k is considered. Comparing the present results with the results under the improved linear seaward boundary condition (Zeng and Liu in Coast Eng 174, 2022), the forward characteristic variable α value is larger at the beginning, then smaller, and finally larger again at the original shoreline, which is caused by the difference in the propagation of the forward characteristic curves from seaward boundary toward the swash domain, while the difference of the backward characteristic variable β value is negligible. Accordingly, the flow velocity and water depth difference at the original shoreline also present similar variation patterns ascribing to the aforementioned varying characteristics of the α value.