A modified Euler–Lagrange transformation for particle orbits in nonlinear progressive waves

Abstract

This paper presents a modified Euler–Lagrange transformation method to obtain the third-order trajectory solution in a Lagrangian form for the water particles in nonlinear water waves. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness and an arbitrary vertical position for each water particle. 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, the third-order asymptotic expressions for the Lagrangian particle trajectories, the mass transport velocity and the period of particle motion can be derived directly in Lagrangian form. The wave frequency and mean level of the particle motion in Lagrangian form differ from those of the Eulerian. Finally, the third-order asymptotic solution obtained is uniformly valid in contrast with early works containing resonant terms presented by Wiegel [1964. Oceanographical Engineering. Prentice-Hall, New Jersey, pp. 37–40] (Eqs. (B.1) and (B.2) in Appendix B) or Chen et al.[2006. Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2 nonlinear waves. Wave motion, 43, 356–369] based on a straightforward expansion for two-dimensional progressive waves.