A Unified Coupled-Mode Approach to Nonlinear Waves in Finite Depth: Potential Flow

Abstract

A non-linear coupled-mode system of horizontal equations is presented, as derived from Luke’s (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bottom topography. The vertical structure of the wave field is represented by means of a complete local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and has the following main features: (i) various standard models of water-wave propagation are recovered by appropriate simplifications, and (ii) it exhibits fast convergenge, and thus, a small number of modes (up to 5) are usually enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady traveling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate to shallow-water wave conditions and its results are compared vs. Stokes and cnoidal wave theories, respectively. Also, numerical results are presented for waves propagating over variable bathymetry regions and compared with second-order Stokes theory and experimental data.