Journal of Fluid Mechanics

On Hamilton's principle for surface waves

John W.  Miles a1
a1 Institute of Geophysics and Planetary Physics, University of California, La Jolla

Article author query
miles jw   [Google Scholar] 


The boundary-value problem for irrotational surface waves is derived from a variational integral I with the Lagrangian density [script L] = Ξ ηt - [script H] where Ξ (X, t) is the value of the velocity potential at the free surface, y = η(x, t), and [script H] is the energy density in x space. [script H] 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 [script H] 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).

(Published Online April 12 2006)
(Received July 9 1976)