Solving the nonlinear shallow-water equations in physical space

The boundary value problem for the nonlinear shallow-water equations with a beach source term is solved by direct use of physical variables, so that solutions are more easily inspected than those obtained by means of hodograph transformations. Beyond an overall description of the near-shoreline flows in terms of the nonlinear shallow-water equations, significant results are provided by means of a perturbation approach which enables much of the information on the flow to be retained. For sample waves of interest (periodic and solitary), first-order solutions of the shoreline motion and of the near-shoreline flows are computed, illustrated and successfully compared with the equivalent ones obtained through a hodograph transformation method previously developed by the authors. Wave-wave interaction, both at the seaward boundary and within the domain, is also accurately described. Analytical conditions for wave breaking within the domain are provided. These, compared with the authors' hodograph model, show that the first-order condition of the present model is comparable to the second-order condition of that model.