  We investigate the dynamics of a first order transition when the order
parameter field undergoes resonant oscillations, driven by a periodically
varying parameter of the free energy. This parameter could be a background
oscillating field as in models of pre-heating after inflation. In the context
of condensed matter systems, it could be temperature $T$, or pressure, external
electric/magnetic field etc. We show that with suitable driving frequency and
amplitude, the system remains in a type of mixed phase, without ever completing
transition to the stable phase, even when the oscillating parameter of the free
energy remains below the corresponding critical value (for example, with
oscillating temperature, $T$ always remains below the critical temperature
$T_c$). This phenomenon may have important implications. In cosmology, it will
imply prolonged mixed phase in a first order transition due to coupling with
background oscillating fields. In condensed matter systems, it will imply that
using oscillating temperature (or, more appropriately, pressure waves) one may
be able to sustain liquids in a mixed phase indefinitely at low temperatures,
without making transition to the frozen phase.
