  Inspired by recent experimental observation of patterning at the membrane of
a living cell, we propose a generic model for the dynamics of a fluctuating
interface driven by particle-like inclusions which stimulate its growth. We
find that the coupling between interfacial and inclusions dynam- ics yields
microphase separation and the self-organisation of travelling waves. These
patterns are strikingly similar to those detected in the aforementioned
experiments on actin-protein systems. Our results further show that the active
growth kinetics does not fall into the Kardar-Parisi-Zhang universality class
for growing interfaces, displaying instead a novel superposition of
equilibrium-like scaling and sustained oscillations.
