Frequencies of gravity-capillary waves on highly curved interfaces with edge constraints

A recently developed technique to calculate the natural frequencies of gravity-capillary waves in a confined liquid mass with a possibly highly curved free surface is extended to the case where the contact line is pinned. The general technique is worked out in detail for the cases of rectangular and cylindrical containers of circular section, the cases for which experimental data are available. The results of the present method are in excellent agreement with all earlier experimental and theoretical data for the flat static interface case [Benjamin and Scott, 1979. Gravity-capillary waves with edge constraints. J. Fluid Mech. 92, 241-267; Graham-Eagle, 1983. A new method for calculating eigenvalues with applications to gravity-capillary waves with edge constraints. Math. Proc. Camb. Phil. Soc. 94, 553-564; Henderson and Miles, 1994. Surface-wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285-299]. However, the present method is applicable even when the contact angle is not π/2 and the static interface is curved. As a consequence we are able to work out the effects of a curved meniscus on the results of Cocciaro et al. [1993. Experimental investigation of capillary effects on surface gravity waves: non-wetting boundary conditions. J. Fluid Mech. 246, 43-66] where the measured contact angle was 62∘. We find that the meniscus does indeed account, as suggested by Cocciaro et al., for the earlier discrepancy between theory and experiment of about 20 mHz and there is now excellent agreement between the two.