Accuracy and convergence of velocity formulations for water waves in the framework of Boussinesq theory

The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the still-water datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh- and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary z-level, which can be varied from the sea bottom to the still-water datum. Second, we consider the classical possibility of expanding in terms of the depth-averaged velocity. Third, we analyse the use of a horizontal pseudo-velocity determined by interpolation between velocities at two arbitrary z-levels. Fourth, we investigate three different formulations based on two expansion variables, being the horizontal and vertical velocity variables at an arbitrary z-level. This is shown to have a remarkable influence on the convergence and to improve accuracy considerably. Fifth, we derive and analyse a new formulation which doubles the power of the vertical coordinate without increasing the order of the horizontal derivatives. Finally, we involve the kinematic and dynamic boundary conditions at the free surface and discuss the linear dispersion relation and a spectral solution for steady nonlinear waves.