A Lagrangian asymptotic solution for finite-amplitude standing waves

A third-order asymptotic solution in a Lagrangian description is presented for the finite-amplitude standing waves in water of uniform depth. Regarding the frequency of particle motion to be a function of the wave steepness and using a successive Taylor series expansion to the path and the period of particle motion, the explicit parametric equation of water particles and the Lagrangian wave frequency up to third-order could be obtained. In particular, the Lagrangian mean level which differs from that in the Eulerian approach is also found as a part of the solutions. The variations in the wave profile and the water particle orbits for the nonlinear standing waves are also investigated. Comparison on the third-order wave profiles given by the Eulerian and Lagrangian solution with the experiments reveals that the latter is more accurate than the former in describing the shape of the wave profile.

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