Journal of Fluid Mechanics

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On the Lagrangian description of steady surface gravity waves

DIDIER CLAMONDa1

a1 Institutt For Energiteknikk, P O Box 40, 2027 Kjeller, Norway didier.clamond@ife.no

Abstract

This paper concerns the mathematical formulation of two-dimensional steady surface gravity waves in a Lagrangian description of motion. It is demonstrated first that classical second-order Lagrangian Stokes-like approximations do not exactly represent a steady wave motion in the presence of net mass transport (Stokes drift). A general mathematically correct formulation is then derived. This derivation leads naturally to a Lagrangian Stokes-like perturbation scheme that is uniformly valid for all time – in other words, without secular terms. This scheme is illustrated, both for irrotational waves, with seventh-order and third-order approximations in deep water and finite depth, respectively, and for rotational waves with a third-order approximation of the Gerstner-like wave on finite depth. It is also shown that the Lagrangian approximations are more accurate than their Eulerian counterparts of the same order.

(Received September 14 2006)

(Revised June 10 2007)

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