ABSTRACT In this paper a coupled-mode system of horizontal equations is derived with the aid of Luke's (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bathymetry. The vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential, Athanassoulis & Belibassakis (2000). This series contains the usual propagating and evanescent modes, plus two additional modes, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The system fully accounts for the effects of non-linearity and dispersion. In the present work the fully nonlinear coupled-mode system is simplified keeping only up to second-order terms, and the derived weakly non-linear model is applied to water waves propagating over a flat bottom and over an arbitrary bathymetry, in the time and in the frequency domain.
INTRODUCTION In the present work we consider the problem of surface gravity waves normally incident to a smooth, but possibly steep, two-dimensional shoaling; see Fig. 1. An essential feature of this problem is that, even in the linearised, time-harmonic case, the wave field is not spatially periodic. The numerical treatment in the framework of Navier-Stokes equations has recently become possible (see, e.g., Huang and Dong, 1999), being, however, extremely demanding computationally. Usually, the problem is treated in the framework of potential flow, Tsai and Yue (1996). Under the assumptions of incompressibility and irrotationality, the problem of evolution of water waves over a variable bathymetry region admits of at least two different variational formulations: A Hamiltonian one, constrained on the below-the-surface kinematics proposed by Petrov (1964) and further developed by Zakharov (1968) and his associates; and an unconstrained one, proposed by Luke (1967); see also Witham (1974) and Massel (1989).