The transformation between the third-order Eulerian and Lagrangian solutions for irrotational progressive gravity waves

Abstract

This study reports the transformation between the third-order Eulerian and Lagrangian solutions for the progressive water gravity waves propagating on water of uniform depth. Regarding to the motion of a marked fluid particle, the instantaneous velocity, mass conservation and free surface must be the same for solufions of either Eulerian or Lagrangian method. Using a successive Taylor series expansion to the path and the period of particle motion, the given conventional Eulerian solutions can be transformed into the completely unknown Lagrangian solutions and the reversible process is also identified. In the asymptotic solution, the explicit parametric equation of water particles can be obtained. In particular, the Lagrangian mean level and the Lagrangian wave frequency which differ from those in the Eulerian approach are found as part of the solutions. It shows that the present technique provides a modified method to obtain the third-order Lagrangian solution from the known Eulerian solutions.