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A nonlinear model of AC-field-induced parametric waves on a water surface

So-called parametric excitations occur as a result of a time-dependent change in a parameter (e.g., rigidity, gravitational acceleration, etc.) of a system. Such a wave can be formed on the surface of a conductive liquid by applying an AC electric field perpendicularly to the surface. It has previously been shown that, using a linearized analysis, this electrohydrodynamic phenomenon can be described mathematically by the Mathieu equation. This linearized analysis is successful at predicting the wavelength and frequency of the parametric wave, but it predicts unlimited growth and therefore cannot determine the resulting amplitude or phase. This paper presents a nonlinear extension of the linearized model, resulting in a nonlinear form of the Mathieu equation. The nonlinear model, which accounts for the change in the force due to surface tension as the wave amplitude grows, results in a prediction of the finite wave amplitude, and gives a prediction of the phase of the finite wave with respect to the exciting AC field. Viscous damping of the liquid is considered. A spatially sinusoidal wave shape is assumed, but the model can accommodate different curvatures of the upward and downward phases of the wave. The model explains why electric discharging above a water surface often occurs after the applied electric field has started to decline from its peak. Experimental measurements (obtained using a channel of water) of wave amplitude and phase are compared with predictions from the model.