Dispersion relation and memory effects in dynamics of viscid gravity-capillary waves

Gravity waves and free-surface flows are of topical importance in a range of applications, the physics of which is typically addressed within the encompassing framework of the Boussinesq-type equations. Although extensions to this framework, such as capillarity, are well established, the effects of viscosity and no-slip surfaces remain elusive: a number of approaches have been developed, but a more general framework is still missing. We carry out a detailed analytical and numerical study of the interaction between the surface degrees of freedom and the vertical dynamics of the pressure and vorticity fields in the linear regime. Bypassing viscosity-perturbative and long-wavelength considerations, we identify dynamical features across a broad range of linear regimes, including the dispersion relation of viscid gravity-capillary waves as well as memory (non-instantaneous) effects.