Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2: Nonlinear waves

Abstract

This is the second of two papers on the theoretical analysis of surface wave propagating on sloping bottoms. In the first paper, the linear solution for velocity potential was derived as a function of bottom slope α to the third order. In this paper, Part 2, we use the two-parameter perturbation method to develop a new mathematical derivation to describe nonlinear periodic gravity surface waves propagating over sloping bottoms. In the Eulerian coordinate system, the velocity potential is obtained as a function of the nonlinear ordering parameter ε and the bottom slope α perturbed to the fourth order ( ε m α n , m + n = 4). A time-decoupled solution is derived and 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 analytical solution for wave asymmetry and set-down valid up to the breaker line for an arbitrary bottom slope can also be obtained. Besides, the direct influence of nonlinearity and bottom slope upon the wave number and a new second-order dispersion relation is also derived. This nonlinear analytical solution is verified by reducing to the classical Stokes fourth-order solution of progressive waves in both the limit of deep water and of constant depth. Furthermore, by comparing the theoretical values of wave asymmetry and set-down with the experimental data, it is found that theoretical results of the present solution are in good agreement with the experimental data.