Dispersion Relation for Wind Waves with Account for the Drift Current

The presence of the drift current complicates analysis of the dispersion relation for wind waves. In the general case, this relation is derived from analysis of the Rayleigh equation which has no analytical solution for an arbitrary velocity profile. In the limiting case, when the length of the gravity–capillary wave is significantly less than the typical flow depth, the simple Doppler approximation can be used. However, this approximation is not valid in the general case and it is necessary to take into account the vertical profile of the horizontal velocity up to the depth that corresponds to the considered wavelengths. The drift velocity profile is determined using Particle Image Velocimetry. The high-resolution spatiotemporal spectra of waves are obtained using the color schlieren technique. A small addition of sodium dodecyl sulfate allows one to estimate the effect of soluble impurities on the structure of the drift current and to modify the ratio between the drift current depth and length of the gravity–capillary wave. In the present work, an algorithm for numerical calculation of the dispersion relation for a given velocity profile is used. It is shown that the Rayleigh equation adequately describes the dispersion relation in a wind channel under conditions where the influence of the profile is strong and cannot be reduced to simple Doppler corrections. Thus, deviations of dispersion relations obtained in geophysics from simple approximations can correspond to different relationships between wavelengths and drift current depth.