ON RAYLEIGH INVESTIGATION OF CRISPATIONS OF FLUID RESTING ON A VIBRATING SUPPORT

Rayleigh (1883) observed that the frequency of parametrically excited capillary-gravity waves in a container of lateral dimensions large compared with the wavelength was only 75% of the frequency omega0 calculated from Kelvin's dispersion relation for waves of the observed length, and attributed this discrepancy to friction. A boundary-layer calculation on the assumption that the free surface acts as an inextensible film (as is typical for water in the laboratory) yields a 10% reduction from omega0. The remaining discrepancy may be plausibly attributed to a contamination-induced reduction of surface tension from the value assumed by Rayleigh, but the possibility remains that nonlinearity could account for a significant shift of the frequency from omega0. The solution of the weakly nonlinear problem for parametrically excited capillary-gravity waves reveals that this shift is positive for Rayleigh's data, whence the surface tension must have been even smaller than that inferred from Kelvin's dispersion relation. This solution also suggests quantitative errors in the solutions of Ezerskii et al. (1986) and Milner (1991) for the limiting case of deep-water capillary waves.