Viscous Effects on Evolution of Stokes Waves

The effects of laminar viscosity on the evolution of a steady Stokes wave train are investigated by employing a boundary‐layer method. It is assumed that the nonlinearity of the second order in the wave slope, $0(\mathrm{ka}{)}^2\text{,}$ is in the same order of magnitude as the viscous effect; i.e., $0(\mathrm{ka}{)}^2=0(k\delta )\text{,}$ where $\delta =\sqrt{v/2\omega }=\mathrm{the}$ Stokes boundary layer thickness. A nonlinear Schrödinger equation with a dissipation term is derived as the governing equation for the wave envelope. It is shown that the viscosity not only reduces the wave amplitude but also causes a phase shift in the same order of magnitude.