Surface Waves on Water with Depths Less than the Boundary-Layer Thickness

Abstract

Equations are derived for the complex wavenumber for plane surface waves in liquids of low viscosity with depths from one-fifth to two-thirds of the boundary-layer thickness. For capillary waves, the values of the real wavenumber and of the spatial absorption coefficient are approximately proportional to (νσ)1/4, where ν is the kinematic coefficient of viscosity and σ is the real angular frequency. Numerical calculations agree with experimental values for a frequency of 20 Hz and a depth of one-half of the water boundary-layer thickness δ. For gravity waves with depths of about 14δ, the value of the real wavenumber equals that of the spatial absorption coefficient. Each is proportional to (νδ)1/2 but the coefficient is quite different from that of the main term for the absorption coefficient at intermediate depths.