Branch cuts of Stokes wave on deep water. Part II: Structure and location of branch points in infinite set of sheets of Riemann surface

TL;DR

This paper analyzes the complex structure of Stokes waves on deep water, revealing an infinite set of branch points and sheets on the Riemann surface, and elucidating the transition to the limiting wave with a 2/3 power law singularity.

Contribution

It provides a detailed analysis of the Riemann surface structure of Stokes waves, including the infinite sheets and nested singularities, advancing understanding of wave singularity formation.

Findings

Identified a single square root branch point per wave period in the upper complex half-plane.

Showed the approach of the branch point to zero as wave height increases, leading to the limiting wave.

Conjectured the existence of infinite nested square root singularities in the non-limiting Stokes wave.

Abstract

Stokes wave is a finite amplitude periodic gravity wave propagating with constant velocity in inviscid fluid. Complex analytical structure of Stokes wave is analyzed using a conformal mapping of a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. There is one square root branch point per spatial period of Stokes located in the upper complex half-plane at the distance $v_c$ from the real axis. The increase of Stokes wave height results in approaching $v_c$ to zero with the limiting Stokes wave formation at $v_c=0.$ The limiting Stokes wave has $2/3$ power law singularity forming $2\pi/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but nonzero $v_c$ making the limit of zero $v_c$ highly nontrivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of Riemann surface to find coupled square root singularities at the distances $\pm v_c$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of Stokes wave into all these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that non-limiting Stokes wave at the leading order consists of the infinite number of nested square root singularities which also implies the existence in the third and higher sheets of the additional square root singularities away from the real and imaginary axes. These nested square roots form $2/3$ power law singularity of the limiting Stokes wave as $v_c$ vanishes.