On surface waves with zero contact angle

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Abstract
The linear, inviscid reflection of a straight-crested surface wave from a vertical wall is determined on the hypothesis that the contact angle of the meniscus vanishes. The reflection coefficient is a function of the parameter γ ≡ k0l, where k0 is the wavenumber of the incident wave and l is the capillary length, and is approximated by R = exp (−4iγ2) for a gravity–capillary wave for which γ [Lt ] 1. The solution of this reflection problem is used to obtain matched-asymptotic approximations for standing waves in channels and circular cylinders. The meniscus-induced, fractional reduction of the frequency of the dominant mode in a deep circular cylinder is 0.77 γ2 (which exceeds the increase of ½γ2 associated with the capillary energy of the free surface). This decrement is within 2 mHz of the value inferred from the measurements of Cocciaro et al. (1991) after allowing for the reduction in frequency induced by the viscous boundary layers at the walls, but there are residual uncertainties (in this comparison) associated with the wetting process at the moving contact line and possible surface contamination.

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