Notes on the Stokes first order wave and wave-current problems

The review of the tidal turbine literature (Barltrop et al. 2007; Faudot and Dahlhaug 2012; Guo et al. 2018; Draycott et al. 2020) exposes various formulations of the Stokes (1847) wave or wave-current orbital velocities. To elucidate any misunderstanding in the hypothesis or in the applicability of the wave model, the first author conducted the complete derivation of the Stokes (1847) model with consistent framework and notations during his PhD project (Dufour 2025). Hence, the present is report based on the first chapter of Dufour (2025) PhD manuscript, with some additional elements and improvements. The demonstrations to solve the wave and wave-current Stokes problems and to obtain the velocity potential are thoroughly performed. The mathematical explanations are heavily detailed to facilitate the understanding for any reader. The results are stopped up to Stokes (1847) first order of approximation. Nevertheless, the boundary conditions are derived up to the second order of approximation to enable future works to easily reach higher accuracy. The present report first solves the wave only problem. It is based upon an abundant literature (Lamb 1932; Méhauté 1976; Phillips 1977; Dean and Dalrymple 1991; Molin 2002; Molin 2023). The wave only problem enables to derive meticulously the equations. In a second phase, the wave-current model is thoroughly demonstrated, emphasizing on the differences that are introduced by this mean upstream current. This section echoes the demonstration provided by Dean and Dalrymple (1991). However, the demonstration of the present document is established in a different manner which can then be complementary for the reader. In a final section, the wave action conservation is used to re-demonstrate the ratio of amplitude between waves over a steady medium and waves with a mean current. The obtained formula in this last section strictly equates to the one presented by Brevik and Aas (1979) while being easier to evaluate in some cases.