  We consider stochastic partial differential equations (SPDEs) on the
one-dimensional torus, driven by space-time white noise, and with a
time-periodic drift term, which vanishes on two stable and one unstable
equilibrium branches. Each of the stable branches approaches the unstable one
once per period. We prove that there exists a critical noise intensity,
depending on the forcing period and on the minimal distance between equilibrium
branches, such that the probability that solutions of the SPDE make transitions
between stable equilibria is exponentially small for subcritical noise
intensity, while they happen with probability exponentially close to $1$ for
supercritical noise intensity. Concentration estimates of solutions are given
in the $H^s$ Sobolev norm for any $s<\frac12$. The results generalise to an
infinite-dimensional setting those obtained for $1$-dimensional SDEs in [Nils
Berglund and Barbara Gentz. A sample-paths approach to noise-induced
synchronization: stochastic resonance in a double-well potential. Ann. Appl.
Probab., 12(4):1419-1470, 2002].
