# Exact solution for progressive gravity waves on the surface of a deep fluid

## TL;DR

This paper derives an exact solution for unidirectional deep water gravity waves using Zakharov's variational formulation, revealing a new limiting wave with a cusp at its crest and a maximum steepness of 0.2034.

## Contribution

It presents a novel exact solution for gravity waves on deep water without truncating the Hamiltonian, expanding the class of known wave solutions.

## Key findings

- Exact solution expressed via Lambert W-function
- Maximum wave steepness identified as 0.2034
- Wave crest exhibits a zero-angle cusp similar to Gerstner's wave

## Abstract

Gerstner or trochoidal wave is the only known exact solution of the Euler equations for periodic surface gravity waves on deep water. In this Letter we utilize Zakharov's variational formulation of weakly nonlinear surface waves and, without truncating the Hamiltonian in its slope expansion, derive the equations of motion for unidirectional gravity waves propagating in a two-dimensional flow. We obtain an exact solution of the evolution equations in terms of the Lambert $W$-function. The associated flow field is irrotational. The maximum wave height occurs for a wave steepness of 0.2034 which compares to 0.3183 for the trochoidal wave and 0.1412 for the Stokes wave. Like in the case of Gerstner's solution, the limiting wave of a new type has a cusp of zero angle at its crest.