Journal of Fluid Mechanics


New asymptotic description of nonlinear water waves in Lagrangian coordinates

a1 Department of Civil Engineering, University College London, WC1E 6BT, UK
a2 Department of Engineering Science, University of Oxford, OX1 3PJ, UK

Article author query
buldakov ev   [Google Scholar] 
taylor ph   [Google Scholar] 
taylor re   [Google Scholar] 


A new description of two-dimensional continuous free-surface flows in Lagrangian coordinates is proposed. It is shown that the position of a fluid particle in such flows can be represented as a fixed point of a transformation in $\mathbb{R}^2$. Components of the transformation function satisfy the linear Euler-type continuity equation and can be expressed via a single function analogous to an Eulerian stream function. Fixed-point iterations lead to a simple recursive representation of a solution satisfying the Lagrangian continuity equation. Expanding the unknown function in a small-perturbation asymptotic expansion we obtain the complete asymptotic formulation of the problem in a fixed domain of Lagrangian labels. The method is then applied to the classical problem of a regular wave travelling in deep water, and the fifth-order Lagrangian asymptotic solution is constructed, which provides a much better approximation of steep waves than the corresponding Eulerian Stokes expansion. In contrast with early attempts at Lagrangian regular-wave expansions, the asymptotic solution presented is uniformly valid at large times.

(Published Online August 14 2006)
(Received May 2 2004)
(Revised March 6 2006)