Fourth branch of instability of Stokes' wave and dependence of corresponding growth rate on nonlinearity

TL;DR

This study computationally identified the fourth superharmonic instability branch of Stokes' wave and validated existing phenomenological formulas for growth rates across all branches, reducing the need for further complex calculations.

Contribution

The paper extends the understanding of Stokes' wave instability by computing the fourth instability branch and confirming the applicability of existing formulas to this new branch.

Findings

Identified the fourth superharmonic instability branch.

Validated that existing formulas fit the new branch.

Reported growth rates for all four instability branches.

Abstract

Through a massive computation we reached the fourth superharmonic instability branch of the Stokes' wave. Using the obtained results we checked phenomenological formulae for the dependence of the instability growth rates corresponding to different branches of instability on the nonlinearity parameter (steepness, defined as the wave \red{hight} to wavelength ratio $H/\Lambda$) in the vicinity of the new instability branch appearance and far from it. It is demonstrated, that the formulae, obtained as a least squares fit (using the information from the first three branches of instability) and a phenomenological asymptotics, work for the fourth branch as well. Range of applicability of the relations \red{is} corrected. \red{This result removes the necessity to compute further branches of instability if accuracy better than 10\% for the growth rate is acceptable.} Growth rates for all four instability branches are reported.