A new asymptotic solution describing nonlinear water wave propagation on the surface of a uniform sloping bottom is derived in the Lagrangian coordinates. We use the two-parameter perturbation method to develop a new mathematical derivation. The particle trajectories, wave pressure and Lagrangian velocity potential are obtained as a function of the nonlinear ordering parameter ε and the bottom slope α perturbed to second order. The analytical solution in Lagrangian form satisfies the zero pressure at the free surface. The condition of the conservation of mass flux is examined in detail for the first time. Then, the solution is used to estimate the mean return current for waves progressing over the sloping bottom. 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 and water particle trajectories leading to breaking. The nonlinear analytical solution is verified by reducing to the Lagrangian second-order solution of progressive waves in both the limit of deep water and of constant water.
The motion of a fluid particle within a propagating surface wave may be described by either observing the fluid velocity at a fixed position or the trajectory of a particle that is carried along with the flow. These alternative descriptions are called Eulerian and Lagrangian approach, respectively. For an incompressible fluid, the Eulerian approach is clearly prefered, because of the corresponding continuity equation is linear. It is also well known that the Eulerian description for a free surface can always be expressed in Taylor series at a fixed water level, which implicitly assumes that the surface profile is a differentiable single-valued function.
