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Faraday waves are gravity-capillary waves that emerge on the surface of avertically vibrated fluid when the energy injected via vibration exceeds theenergy lost due to viscous dissipation. Because this dissipation primarilyoccurs in the free surface boundary layer, their emergence is particularlysensitive to free surface properties including the surface tension, elasticity,and viscosity of surfactants present at the free surface. We study thissensitivity by considering a Newtonian fluid bath covered by an insolublesurfactant subject to vertical vibrations which produce sub-harmonic Faradaywaves. By assuming a finite-depth, infinite-breadth, low-viscosity bulk fluidand accounting for surface tension, Marangoni, and Boussinesq effects, wederive an expression for the onset acceleration up to second order in theexpansion parameter $\Upsilon = \sqrt{\tfrac{1}{\mathcal{R}e}}$. We recover theresults of previous numerical investigations, but only by modifying theMarangoni and Boussinesq numbers to account for the low-viscosity limit. Theanalytic expression allows us to consider a range of parameters not previouslystudied, including a wide variety of fluid depths and driving frequencies. Inaddition, we uncover regions of parameter space for which our model predictsthat the addition of surfactant would lower, rather than elevate, the onsetacceleration. We discuss the possible use of this model in developing a surfaceviscometer for surfactant monolayers.

The Onset Acceleration for Surfactant Covered Faraday Waves