(459j) Parametric Excitation of Capillary Waves: Did Faraday Miss Something?

In 1831, Michael Faraday observed that a substrate vibrating at a frequency f induced capillary waves on the free surface of a liquid above it that oscillate at a single specific frequency that is one-half of the excitation frequency, i.e., f/2.¹ The Faraday instability theory has since been a cornerstone of wave parametric excitation and many after Faraday, including Benjamin and Ursell,² Eisenmenger,³ and Miles and Henderson,⁴ have developed elegant linear and weakly nonlinear theories for ideal (nearly inviscid) fluids that reduce the system of equations governing the evolution of the free surface of the liquid to the Mathieu equation in order to describe the specific Faraday observation of the f/2 subharmonic response as well as the Hopf bifurcations that appear upon sufficient excitation. More recently, however, using advanced measurement techniques to measure the capillary wave vibration, and especially at excitation frequencies in the MHz range much higher than that in the Faraday experiment,¹ we have noticed, possibly for the first time, that the Faraday waves were simply absent, i.e., the capillary waves did not simply respond with a single-cascade Faraday wave excitation at the f/2 subharmonic.^5,6 Instead, and, quite incredibly, the capillary waves appeared to undulate via a broadband response at 1-100 kHz, which is several orders of magnitude smaller than the MHz excitation frequency. In addition, a weak harmonic response at the excitation frequency f was also observed, although the relative dominance between the sharp harmonic and the broadband subharmonic responses appear to be sensitive to the geometry of the system, in particular, the fluid depth compared to the characteristic length scale of a viscous boundary layer over which the acoustic energy dissipates.^5,6 Whilst harmonic responses, in particular for simple piston-like vertical sinusoidal excitation, have been predicted by Kumar,^7,8 who extended the classical linear theories to allow for a finite liquid viscosity, these do not appear in the absence of other subharmonic responses in contrast to our observations. In any case, all of these theories, including the classical theories,^2-4 fail to predict the low frequency broadband subharmonic response observed in our experiments.^5,6 More concerningly, however, is that these classical theories^2-4 appear to have been widely accepted to date without much apparent controversy despite the underlying assumption of linearity, which whilst perhaps acceptable when examining low amplitude and low frequency excitations, grossly neglects important nonlinear effects that could appear as the excitation amplitude and frequency increases. Further, recent theories that predict the harmonic response, such as that by Kumar,^7,8 have, quite loosely, been collectively associated with the original description of the Faraday instability despite the fact that these were never observed in Faraday’s original experiment.¹

¹ M. Faraday. Philos. Trans. R. Soc. London 121, 299-340 (1831).

² T.B. Benjamin & F. Ursell. Proc. R. Soc. London Ser. A 225, 505-515 (1954).

³ W. Eisenmenger. Acustica 9, 327-340 (1959).

⁴ J. Miles & D. Henderson. Annu. Rev. Fluid Mech. 2, 143-165 (1990).

⁵ A. Qi, L.Y. Yeo & J.R. Friend. Phys. Fluids 20, 074103 (2008).

⁶ M.K. Tan, J.R. Friend, O.K. Matar & L.Y. Yeo. Preprint. 

⁷ K. Kumar. Proc. R. Soc. Lond. Ser. A 452, 1113-1126 (1996).

⁸ K. Kumar & L.S. Tuckerman. J. Fluid Mech. 279, 49-68 (1994).