# Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

## TL;DR

This paper introduces an efficient, open-source algorithm for accurately computing steady two-dimensional surface gravity waves in any depth, suitable for various wave types and steepnesses, using conformal mapping and spectral methods.

## Contribution

The paper presents a novel, fast algorithm with O(N log N) complexity for arbitrary depth wave computation, combining conformal mapping, Babenko equation, and Petviashvili's iterations.

## Key findings

- Algorithm achieves high precision for Stokes, cnoidal, and solitary waves.
- Efficient for large steepnesses and arbitrary depths.
- Open-source code enables easy verification and adaptation.

## Abstract

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity wave in irrotational motion. The algorithm complexity is O(N log N), N being the number of Fourier modes. The algorithm allows the arbitrary precision computation of waves in arbitrary depth, i.e., it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses. The method is based on conformal mapping, Babenko equation rewritten in a suitable way, pseudo-spectral method and Petviashvili's iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.