On the leading nonlinear correction to gravity-wave dynamics

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Abstract
The principal nonlinear correction to the dynamics of gravity waves on an irrotational fluid is traditionally derived as a non-resonant perturbation solution to the Stokes expansion. When the problem is reformulated in the Hamiltonian description and limited to moderately collimated random waves over infinite depth, the perturbation term assumes a very simple and descriptive form. The sum-frequency component for the surface height is just a bilinear product of the height with the associated scalar strain, and the accompanying term in the potential is half the time derivative of the squared linear height. This solution is exact in one surface dimension and remains quite accurate for long-crested waves in two dimensions, with an error small to second order in the angular spread of constituent wave vectors. It is a natural generalization for random, disordered wave ensembles of the second-order Stokes solution, and its effect is to sharpen the random crests and to flatten the troughs. For wave sets of narrow relative bandwidth the difference-frequency component consists of a negligible elevation term and a non-negligible potential term whose gradient is the surface value of the volume return flow balancing the quadratic wave transport of fluid.
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