Lagrangian description of gravity-capillary waves propagating on a sloping bottom

An asymptotic solution that describes a small amplitude gravity-capillary wave propagating on the surface of a gentle sloping beach is derived in the Lagrangian coordinates. The analytical solution in Lagrangian form satisfies the zero pressure at the free surface. In the Lagrangian approximation, the parametric expression of water particles can be obtained directly and explicitly as a function of the wave steepness, the bottom slope and surface tension. The analytical solution for wave asymmetry parameter up to the breaker line for an arbitrary bottom slope can also be derived. The Lagrangian solution enables the description of the features of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the process of successive deformation of a wave profile which leads to wave breaking. Furthermore, by comparing the theoretical values of wave asymmetry with experimental results, it is found that theoretical results of the present solution are in good agreement with the experimental data. It is also found that surface tension lower the breaking wave height, lengthen the wave length and increase the breaking water depth.