The Euler-Lagrange Transformation for Irrotational Progressive Gravity Waves

This study reports the mutual transformations between the third-order Eulerian and Lagrangian solutions for the irrotational progressive gravity waves propagating on the uniform depth. Based on the fundamental principle of kinematics, the instantaneous Eulerian and Lagrangian velocities are identically equal, and using a successive Taylor series expansion to the path and the period of particle motion, the given Eulerian solutions could be transformed into the corresponding Lagrangian solutions and the reversible process is also completed. In the third-order Lagrangian solution, the angular frequency of particle motion differing from the Eulerian wave period that is a function of the initial position of each individual particle can also be obtained. The third-order trajectory solution of particles exhibits that they move as non-closed orbital motion. Comparing the wave profile with that given by the third-order Eulerian solution shows the third-order Lagrangian solution is superior to the Eulerian solution for describing the shape of the gravity wave. This paper provides a new method to obtain the Lagrangian solutions from the given Eulerian solutions for water waves.