Fourth Branch of Instability of a Stokes Wave and the Dependence of the Corresponding Growth Rate on Nonlinearity

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 height to wavelength ratio H/Λ) 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 is corrected. 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.