The transformation between the Eulerian and Lagrangian solutions for irrotational standing gravity waves

Abstract

This study reports the transformations between the third-order Eulerian and Lagrangian solutions for the standing gravity waves on water of uniform depth. Regarding the motion of a marked fluid particle, the instantaneous velocity, the mass conservation and the free surface must be the same for either Eulerian or Lagrangian methods. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, we obtain the third-order asymptotic expressions for the Lagrangian particle trajectories and the period of particle motion directly in Lagrangian form. In particular, the Lagrangian mean level which differs from that in the Eulerian approach is also found as a part of the solutions. Therefore, the given Eulerian solutions can be transformed into the completely unknown Lagrangian solutions and the reversible process is also identified.