Stokes Waves in Finite Depth Fluids

TL;DR

This paper investigates Stokes waves in fluids of finite depth using conformal variables, employing numerical methods to analyze bifurcations, wave profiles, and eigenvalues across different water depths.

Contribution

It introduces a numerical approach to compute and analyze Stokes waves in finite depth fluids, including bifurcation points and wave profile characteristics.

Findings

Identification of secondary bifurcation points for Stokes waves.

Wave profiles transition from broad troughs to sharp crests with decreasing depth.

Wave crest angle approaches 120 degrees at large steepness.

Abstract

We consider traveling waves on a surface of an ideal fluid of finite depth. The equation describing Stokes waves in conformal variables formulation are referred to as the Babenko equation. We use a Newton-Conjugate-Gradient method to compute Stokes waves for a range of conformal depths from deep to shallow water. In deep water, we compute eigenvalues of the linearized Babenko equation with Fourier-Floquet-Hill method. The secondary bifurcation points that correspond to double period bifurcations of the Stokes waves are identified on the family of waves. In shallow water, we find solutions that have broad troughs and sharp crests, and which resemble cnoidal or soliton-like solution profiles of the Korteweg-de Vries equation. Regardless of depth, we find that these solutions form a $2\pi/3$ angle at the crest in the limit of large steepness.