Proof of the equivalence of Tanizawa-Berkvens' and Cointe-van Daalen's formulations for the time derivative of the velocity potential for non-linear potential flow solvers

Dealing with freely-floating bodies in the framework of non-linear potential flow theory may require solving Laplace's equation for the time derivative of the velocity potential. At present, there are two competing formulations for the body boundary condition. The first one was derived by Cointe [1] in 2D. It was later extended to 3D by van Daalen [2]. The second formulation was derived by Tanizawa [3] in 2D. It was extended to 3D by Berkvens [4]. In this paper, a proof is given that the Cointe-van Daalen's and the Tanizawa-Berkvens' formulations are equivalent. It leads to a simplified version of Cointe-van Daalen's formulation. The formulation is validated against the analytical solution for a moving sphere in an unbounded water domain.