Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis

P. A. Madsen, H. B. Bingham, H. A. Schäffer

Abstract

Boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity field as well as the linear and nonlinear properties. First, an exact solution to the Laplace equation is given in terms of infinite–series expansions from an arbitrary z–level which is a constant fraction of the still–water depth. This defines the fully dispersive and fully nonlinear water–wave problem in terms of five variables: the free–surface elevation and the horizontal and vertical velocities evaluated at the free surface and at the arbitrary z–level. Next, the infinite series operators are replaced by finite–series (Boussinesq–type) approximations. Three different approximations are introduced, each involving up to fifth–derivative operators, and these formulations are analysed with respect to the linear–velocity profile, linear dispersion and linear shoaling. Nonlinear characteristics are investigated by a perturbation analysis to third order for regular waves and to second order for bichromatic waves. Finally, a numerical spectral solution is made for highly nonlinear steady waves in deep and shallow water. It can be concluded that the best of the new formulations (method III) allows an accurate description of dispersive nonlinear waves for kh (wavenumber times water depth) as high as 40, while accurate velocity profiles are restricted to kh < 10. These results represent a major improvement over existing Boussinesq formulations from the literature.