  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.
