Boussinesq-type equations for wave–current interaction

A new formulation of Boussinesq-type equations for wave–current interaction is presented by means of a perturbation method and in terms of a computation velocity. The research focuses on inclusion of the mean water depth change caused by the presence of a current and the wave–current interaction. Strong and moderate current cases are considered for the derivation, and this allows a perturbation method to be used for the derivation and can reveal the effect of the current on the vertical distributions of the wave velocity and the pressure explicitly. The derived model is enhanced to include the higher accuracy dispersion of Padé [4, 4] expansion and a fully nonlinear property. Numerical simulations are performed to show the effect of the mean water depth change caused by the presence of a current on wave motions.

► The wave–current equations of Boussinesq-type are formulated in a new way.
► A perturbation method for strong and moderate currents is applied.
► A computation velocity is introduced to consider the mean water depth change due to the current.
► The model’s dispersion relation is extended to Padé [4, 4] expansion.
► The effect of the mean depth change due to the current is examined by numerical examples.