Variational Principles Associated with a Free Surface Flow

Abstract

A variational principle for water wave problem was proposed by Luke. The functional of this principle is the volume integral of the pressure, and the velocity potential, which satisfies the equation of continuity and the equations of the free surface, is distinguished by the stationary condition of this functional. In the present paper, the theory is developed on the basis of Hamilton's principle. The irrotationality of the flow and the mechanical condition of the free surface are the Euler equation of the Hamilton principle. A Legendre transformation of the variables leads to a Luke-type variational principle.