On Hamilton's principle for surface waves

Abstract

The boundary-value problem for irrotational surface waves is derived from a variational integral I with the Lagrangian density [Lscr ] = Ξ ηt - [Hscr ] where Ξ (X, t) is the value of the velocity potential at the free surface, y = η(x, t), and [Hscr ] is the energy density in x space. [Hscr ] then is expressed as a functional of Ξ and η, qua canonical variables, by solving a reduced boundary-value problem for the potential, after which the requirement that I be stationary with respect to independent variations of Ξ and η yields a pair of evolution equations for Ξ and η. The Fourier expansions Ξ = pn(t)ϕn*(x) and η = qn(t)ϕn(x), where {ϕn} is an orthogonal set of basis functions, reduce I to Hamilton's action integral, in which the complex amplitudes pn and qn appear as canonically conjugate co-ordinates, and yield canonical equations for pn and qn that are the spectral transforms of the evolution equations for Ξ and η. The evolution equations are reduced (asymptotically) to partial differential equations for pn and qn by expanding [Hscr ] in powers of α = a/d and β = (d/l)2, where a and l are scales of amplitude and wavelength. Explicit third approximations are developed for β = O(α).