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New asymptotic description of nonlinear water waves in Lagrangian coordinates

Published online by Cambridge University Press:  14 August 2006

E. V. BULDAKOV
Affiliation:
Department of Civil Engineering, University College London, WC1E 6BT, UK
P. H. TAYLOR
Affiliation:
Department of Engineering Science, University of Oxford, OX1 3PJ, UK
R. EATOCK TAYLOR
Affiliation:
Department of Engineering Science, University of Oxford, OX1 3PJ, UK

Abstract

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.

Type
Papers
Copyright
© 2006 Cambridge University Press

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