  We consider space-times with two isometries which represent gravitational
waves with distinct wavefronts which propagate into exact
Friedmann-Robertson-Walker (FRW) universes. The geometry of possible wavefronts
is analysed in detail in all three types of FRW models. In the spatially flat
and open universes, the wavefronts can be planar or cylindrical; in the closed
case they are toroidal. Exact solutions are given which describe gravitational
waves propagating into the FRW universes with a fluid with a stiff equation of
state. It is shown that the plane-fronted waves may include impulsive or step
(shock) components, while the cylindrical waves in the spatially flat and open
universes and the toroidal waves in closed universes may contain steps. In
general, wavefronts may exist which have an arbitrary finite degree of
smoothness. In all cases, the waves are backscattered. The head-on collision of
such waves is also briefly mentioned.
