Parametric Autoresonance in Faraday waves

We develop a theory of parametric excitation of weakly nonlinear standing gravity waves in a tank, which is under vertical vibrations with a slowly time-dependent ("chirped") vibration frequency. We show that, by using a negative chirp, one can excite a steadily growing wave via parametric autoresonance. The method of averaging is employed to derive the governing equations for the primary mode. These equations are solved analytically and numerically, for typical initial conditions, for both inviscid and weakly viscous fluids. It is shown that, when passing through resonance, capture into resonance always occurs when the chirp rate is sufficiently small. The critical chirp rate, above which breakdown of autoresonance occurs, is found for different initial conditions. The autoresonance excitation is expected to terminate at large amplitudes, when the underlying constant-frequency system ceases to exhibit a non-trivial stable fixed point.