  This work is motivated by recent progress in information theory and signal
processing where the so-called `spatially coupled' design of systems leads to
considerably better performance. We address relevant open questions about
spatially coupled systems through the study of a simple Ising model. In
particular, we consider a chain of Curie-Weiss models that are coupled by
interactions up to a certain range. Indeed, it is well known that the pure
(uncoupled) Curie-Weiss model undergoes a first order phase transition driven
by the magnetic field, and furthermore, in the spinodal region such systems are
unable to reach equilibrium in sub-exponential time if initialized in the
metastable state. By contrast, the spatially coupled system is, instead, able
to reach the equilibrium even when initialized to the metastable state. The
equilibrium phase propagates along the chain in the form of a travelling wave.
Here we study the speed of the wave-front and the so-called `termination
cost'--- \textit{i.e.}, the conditions necessary for the propagation to occur.
We reach several interesting conclusions about optimization of the speed and
the cost.
