title: Gravity-capillary waves in finite depth on flows of constant vorticity.
abstract: This paper considers two-dimensional periodic gravity-capillary waves propagating steadily in finite depth on a linear shear current (constant vorticity). A perturbation series solution for steady periodic waves, accurate up to the third order, is derived using a classical Stokes expansion procedure, which allows us to include surface tension effects in the analysis of wave-current interactions in the presence of constant vorticity. The analytical results are then compared with numerical computations with the full equations. The main results are (i) the phase velocity is strongly dependent on the value of the vorticity; (ii) the singularities (Wilton singularities) in the Stokes expansion in powers of wave amplitude that correspond to a Bond number of 1/2 and 1/3, which are the consequences of the non-uniformity in the ordering of the Fourier coefficients, are found to be influenced by vorticity; (iii) different surface profiles of capillary-gravity waves are computed and the effect of vorticity on those profiles is shown to be important, in particular that the solutions exhibit type-2-like wave features, characterized by a secondary maximum on the surface profile with a trough between the two maxima.
authors.affiliation: Tainan Hydraulics Laboratory , National Cheng Kung University , Tainan, 701 Taiwan, Republic of China.
authors.forename: Hung-Chu
authors.initials: HC
authors.lastname: Hsu
authors.affiliation: Université du Sud Toulon-Var, CNRS/INSU, IRD, Mediterranean Institute of Oceanography (MIO) , UM 110, 83957 La Garde, France.
authors.forename: Marc
authors.initials: M
authors.lastname: Francius
authors.affiliation: International Wave Dynamics Research Center , National Cheng-Kung University , Tainan, 701 Taiwan, Republic of China.
authors.forename: Pablo
authors.identifier: 0000-0003-4050-2024
authors.initials: P
authors.lastname: Montalvo
authors.affiliation: Aix Marseille Université, CNRS, Centrale Marseille , IRPHE UMR 7342, 13384, Marseille, France.
authors.forename: Christian
authors.initials: C
authors.lastname: Kharif
keywords: nonlinear waves; stokes expansion; surface tension; vorticity