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. A parametric excitation, manifested as a wave, can be formed on the surface of a conductive liquid, e.g., water, by application of an alternating electric (AC) field perpendicular to the surface. It has been shown previously, using a linearized analysis, that the time dependence of this electrohydrodynamic phenomenon can be described mathematically by the Mathieu equation. This result is useful for predicting the wave length 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 linear model, resulting in the Mathieu equation with a nonlinear (cubic) term added. The nonlinear equation enables a prediction of the phase of the finite wave with respect to the exciting AC field. A method to calculate the nonlinear coefficient is introduced, based on the nonlinear increase in the capillary force as the wave amplitude grows. Viscous dissipation in the liquid is considered, and a practical value of the damping coefficient is derived. A spatially sinusoidal wave shape is assumed, but an asymmetric variation of the model can accommodate different curvatures of the upward and a 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.