Acoustically induced micro-scale capillary wave turbulence

The theory of non-linear wave interactions leading to so-called interfacial wave turbulence, where a broadband distribution of capillary wave phenomena may be induced by a monofrequency oscillator, is well known, but experimental results are rare. In particular, it is challenging to set up a physical system where both capillary wave amplitudes are easy to measure and capillary forces dominate gravitational forces. Though capillary forces dominate at small scales, the small oscillation amplitudes and generally high oscillation frequencies preclude measurement via cameras or other traditional means. Instead, we use a laser Doppler vibrometer, capable of measuring oscillations up to 40 MHz, and providing a minimum detectable deflection of picometres. Using ultrasonic surface acoustic wave excitation at 19.5 MHz, we generate wave turbulence on the free surface of a water drop. Energy at the driving frequency does not directly enter the cascade ; rather the driving frequency excites a low-frequency resonance. This resonance appears to, in turn, excite higher harmonics, forming the cascade of length scales seen in the frequency spectrum of wave heights. The initial low-frequency resonance, contrary to expectations via Faraday wave theory, is not at one-half the excitation frequency. Instead, we find the low-frequency resonance to be on the order of 100 Hz, which probably arises due to a balance of capillary and inertial forces; the Faraday wave is not observed due to the high frequency of the excitation. By condensing each spectrum to the value of its power exponent, we find that the turbulence decays as the electrical input power increases beyond 500 mW, a SAW amplitude of about 1 nm. At these powers the probability of very large waves deviates strongly from the Gaussian distribution, indicative of strong non-linearity. Wave turbulent theory is therefore invalid in this high-power regime as the highly non-linear nature of the waves violates the theory’s fundamental assumption of weak non-linearity. THEORY OF WAVE TURBULENCE The scaling of kinetic energy E against wavenumber k in isotropic hydrodynamic turbulence,