Particle trajectories in linear periodic capillary and capillary-gravity water waves

Gravity-capillary waves in finite depth on flows of constant vorticity

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.

Regularity for steady periodic capillary water waves with vorticity

In the following, we prove new regularity results for two-dimensional steady periodic capillary water waves with vorticity, in the absence of stagnation points. Firstly, we prove that if the vorticity function has a Hölder-continuous first derivative, then the free surface is a smooth curve and the streamlines beneath the surface will be real analytic. Furthermore, once we assume that the vorticity function is real analytic, it will follow that the wave surface profile is itself also analytic. A particular case of this result includes irrotational fluid flow where the vorticity is zero. The property of the streamlines being analytic allows us to gain physical insight into small-amplitude waves by justifying a power-series approach.

Introduction: some recent developments of nonlinear water wave theory

The opening article of this issue is intended to provide a review of some relevant topics of the mathematical theory of water waves that have recently received considerable attention in the research literature. We also provide a brief discussion about the content and contribution of the articles that make up this issue.