In this study, a single shock wave approaching a frictional sloping plane beach was investigated. Upon the method of the characteristics, the flow solution behind the shock wave was obtained by numerically solving the nonlinear shallow water equations with the Chezy frictional term. The bottom friction has a significant increase in two specified time–space domains, where the backflow approaches the seaward boundary and where the shock collapses. The large α2 (forward characteristic variable at the seaward side of shock wave) results in large incoming fluxes and maximum run–up height. The field of forward characteristic variable behind the shock wave has a similar time–space distribution with the friction. A comparison with the dam-break solution reveals that it shows similar behaviors as the present solution with α2 =2.86 except for the early uprush stage. Findings from this study are useful with respect to various aspects in the swash zone, e.g., overtopping flows, sediment transport, and wave forces exerting on coastal structures.


The swash zone is the area of beach which is covered by water during wave run–up process and exposed during backwash process. The flow properties in the swash zone are of great significance to the sediment transport and erosion of sandy beaches, which has an essential effect on the coastal morphodynamics (Elfrink and Baldock, 2002; Deng et al., 2015). In addition, the properties of swash flow also affect the dynamics forces (Antuono and Brocchini, 2008) and overtopping fluxes (Peregrine and Williams, 2001) in the swash zone. Accordingly, accurate and robust modeling of wave motion in the swash zone is very important for coastal scientists and engineers.

With respect to the wave motion in the swash zone, there are two models using the nonlinear shallow water equations (NSWEs) to investigate the swash hydrodynamics. One is that a dam–break induced bore climbing on a sloping beach, the other one is a shock wave on a sloping beach. For the dam–break problem, an asymptotic analysis is first proposed by Shen and Meyer (1963) which is only applicable to a small region near the shoreline. Later, Peregrine and Williams (2001) extended the Shen and Meyer solution into the global swash zone. Subsequently, a numerical and analytical solution with a tunable parameter defined at the seaward boundary has been found by Guard and Baldock (2007) and Pritchard et al. (2008), which allows different incident fluxes to enter the swash zone. All the aforementioned studies are in the absence of bed friction. As for frictional dam–break induced swash event, Deng et al. (2016) proposed a numerical solution under a simple seaward boundary conditional and revealed that the effect of friction is significant near the shoreline. Nevertheless, Antuono (2010) (hereafter referred to as AN10) pointed out that nature bores have no discontinuity at the shoreline so that they should not be modeled by the dam-break analogy.

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