# Non-interacting gravity waves on the surface of a deep fluid

## TL;DR

This paper develops a recursive Hamiltonian expansion for deep water gravity waves, revealing conditions under which nonlinear wave interactions vanish, especially for unidirectional waves, and clarifies higher-order interaction amplitudes.

## Contribution

It introduces a recursive method for Hamiltonian expansion of water waves that simplifies analysis of higher-order wave interactions and explains the absence of certain nonlinear interactions.

## Key findings

- Unidirectional waves in 2D do not interact nonlinearly under energy-momentum conservation.
- Vanishing of fourth- and fifth-order interaction amplitudes on resonant hypersurfaces.
- Procedure to identify wave configurations with zero higher-order interaction amplitudes.

## Abstract

We study the interaction of gravity waves on the surface of an infinitely deep ideal fluid. Starting from Zakharov's variational formulation for water waves we derive an expansion of the Hamiltonian to an arbitrary order, in a manner that avoids a laborious series reversion associated with expressing the velocity potential in terms of its value at the free surface. The expansion kernels are shown to satisfy a recursion relation enabling us to draw some conclusions about higher-order wave-wave interaction amplitudes, without referring to the explicit forms of the individual lower-order kernels. In particular, we show that unidirectional waves propagating in a two-dimensional flow do not interact nonlinearly provided they fulfill the energy-momentum conservation law. Switching from the physical variables to the so-called normal variables we explain the vanishing of the amplitudes of fourth- and certain fifth-order non-generic resonant interactions reported earlier and outline a procedure for finding the one-dimensional wave vector configurations for which the higher order interaction amplitudes become zero on the resonant hypersurfaces.