Two-Crested Stokes Waves

TL;DR

This paper investigates strongly nonlinear two-crested Stokes waves on an ideal fluid, identifying bifurcations and limiting wave forms with specific geometric features, expanding understanding of complex wave structures.

Contribution

It introduces the concept of class II Stokes waves, details their bifurcation structure, and describes their limiting forms with 120-degree crest angles, which are disconnected from small-amplitude solutions.

Findings

Distinct bifurcation points in velocity-steepness diagrams.

Existence of limiting waves with 120-degree crest angles.

Two primary families of class II waves identified.

Abstract

We study two-crested traveling Stokes waves on the surface of an ideal fluid with infinite depth. Following Chen and Saffman (1980), we refer to these waves as class $\mathrm{II}$ Stokes waves. The class $\mathrm{II}$ waves are found from bifurcations from the primary branch of Stokes waves away from the flat surface. These waves are strongly nonlinear, and are disconnected from small-amplitude solutions. Distinct class $\mathrm{II}$ bifurcations are found to occur in the first two oscillations of the velocity versus steepness diagram. The bifurcations in distinct oscillations are not connected via a continuous family of class $\mathrm{II}$ waves. We follow the first two families of class $\mathrm{II}$ waves, which we refer to as the secondary branch (that is primary class $\mathrm{II}$ branch), and the tertiary branch (that is secondary class $\mathrm{II}$ branch). Similar to Stokes waves, the class $\mathrm{II}$ waves follow through a sequence of oscillations in velocity as their steepness rises, and indicate the existence of limiting class $\mathrm{II}$ Stokes waves characterized by a $120$ degree angle at every other wave crest.