epl draft Capillary-Gravity Waves on Depth-Dependent Currents: Consequences for the Wave Resistance

Abstract.- We study theoretically the capillary-gravity waves created at the water-air inter-face by a small two-dimensional perturbation in the frequently encountered case where a depth-dependent current is present in the fluid. Assuming linear wave theory, we derive a general expression of the wave resistance experienced by the perturbation as a function of the current profile in the case of an inviscid fluid. We then illustrate the use of this expression in the case of constant vorticity. Water waves are both fascinating and of great practi-cal importance [1–3]. For these reasons, they have at-tracted the attention of scientists for many centuries [4]. Water waves can for instance be generated by the wind at sea, by a moving boat on a calm lake, or simply by throwing a pebble into a pond. Their propagation at the surface of water is driven by a balance between the liq-uid inertia and its tendency, under the action of grav-ity or of surface tension (or a combination of both in the case of capillary-gravity waves), to return to a state of stable equilibrium [5]. Neglecting the viscosity of water, the dispersion relation of linear capillary-gravity waves relating the angular frequency ω to the wavenumber k is given by ω2 = gk + γk3/ρ tanh(kh), where γ is the liquid-air surface tension, ρ the liquid density, g the ac-celeration due to gravity and h the depth of water [1]. The above equation may also be written as a depen-dence of the phase velocity c = ω/k on the wavenumber: c(k) = (g/k + γk/ρ