  We put forward a general field theory for membranes with embedded activators
and analyse their critical properties using renormalization group techniques.
Depending on the membrane-activator coupling, we find a crossover between
acoustic and diffusive scaling regimes, with mean-field dynamical critical
exponents z = 1 and 2 respectively. We argue that the acoustic scaling, which
is exact in all spatial dimensions, is a suitable candidate for the universal
description of the spatiotemporal patterns observed at the leading edge of
motile cells. Furthermore, one-loop corrections to the diffusive mean-field
exponents reveal universal behaviour distinct from the Kardar-Parisi-Zhang
scaling of passive interfaces and signs of strong-coupling behaviour.
