Some theoretical aspects of the generation of surface ripples by parametric subharmonic resonance with sound waves

Abstract

An investigation is made of the dynamics of a liquid free surface in the presence of a reflecting sound field. A finite-amplitude surface standing wave is also specified and the resulting unbalanced vertical stress at the surface is equated to the acoustic pressure. Second- and third-order terms are retained in the surface-wave description (and fourth-order in the nonlinear interaction model). By further equating the viscous dissipation in the surface wave with the net upward acoustic energy flux, the surface-wave parameters are determined in terms of the acoustic-field parameters. Three major results are obtained: no surface waves can exist below an acoustic-intensity threshold (same value as predicted by using Mathieu’s differential equation), above the threshold the square of the surface-wave amplitude is linearly dependent on the acoustic-driving amplitude (for small surface-wave slopes), the sensitivity of the surface waves to driving amplitude is essentially independent of frequency except near the region of resonant gravity–capillary coupling. Comparisons are made with measurements (primarily at ultrasonic frequencies) and reasonable agreement is shown to exist.