Fourth-order Stokes theory for capillary–gravity waves in arbitrary water depth on linear shear currents

Abstract

In this paper the two-dimensional steady surface capillary–gravity waves, incorporating the effects of linear shear currents, is studied in water of constant depth. Herein, linear shear currents are considered to be a linear combination of depth-uniform current and uniform vorticity. Employing an excellent Stokes expansion method, where the expansion parameter represents the wave steepness itself, a fourth-order perturbation series solution for plane progressive waves is developed. The key results of this work are (a) to find the influence of both co-flowing and counter-flowing currents on the wave profiles using the fourth-order approximation (b) the strong dependence of the wave velocity on both the magnitudes of the shear and depth-uniform current, (c) the Wilton singularities in the Stokes expansion in powers of wave amplitude due to a inverse Bond number of $1/2$, $1/3$ and $1/4$, which are the results of the non-uniformity in the ordering of the Fourier coefficients are observed to be influenced by vorticity and depth-uniform current, (d) distinct surface profiles of capillary–gravity waves are obtained and the effects of depth-uniform currents on these profiles are described. This analysis also shows that for any given value of the water depth, there exist a threshold value of the vorticity above which no resonances occur. For the steepest waves considered in this analysis, it is observed that when the wavenumber is not in the vicinity of certain critical values, determined by the depth and the vorticity, the present fourth-order analysis shows significant deviations on the surface profiles from the third-order analysis and provides better results consistent with the exact numerical results.