Lagrangian-Eulerian approach to modelling of wave transformation and flow velocity in the swash zone and its seaward vicinity

One-dimensional long waves approaching a beach face are investigated herein. In particular, flow velocity distribution, wave profile transformation and extreme positions of a water uprush-backwash are analysed. For simplicity, non-dissipative waves, waves influenced by linear or quadratic bottom friction, as well as bore-like breaking waves are considered separately. Bore formation and propagation in a shoaling water is modelled by geometrical limitation of a local slope of the wave front. Analysed phenomena are described using the shallow-water wave theory, although in this paper application of the model has been restricted to the swash zone and its seaward vicinity. Governing equations are expressed in hybrid Lagrangian-Eulerian co-ordinates. The Lagrangian approach gives a precise mathematical description of both an orbital motion and the moving position of a water tongue on a beach slope. The Eulerian contribution to the model enables easy comparison of results of wave and water motion by method of conducting of measurements. Moreover, the hybrid description affords the possibility of simple prediction of mean water flow caused by propagating waves. The following description also presents results of numerical computations especially concerning wave run-up height, flow and orbital velocities and water surface transformation. For simplified bathymetric conditions some analytical solutions are also presented.