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.