The propagation of chemical waves in two regions, coupled together by a linear diffusive interchange of neutral species across a semi-permeable membrane is considered. The reaction scheme in region I is assumed to have Belousov--Zhabotinsky kinetics and the reduced Oregonator model is used to model this. In region II the reaction scheme is modified so that, in the absence of coupling, a Fisher--Kolmogorov travelling wave would be initiated. This leads, in a natural way, to a study of the effects of coupling between excitable and non-excitable systems. The bifurcation parameters are taken to be the strength of the coupling and the value of the stoichiometric factor in region I. The dynamical behaviour of this system is examined numerically, and asymptotically for strong coupling. In this limit it is shown that waves in both regions propagate with a profile identical to that of an uncoupled Belousov--Zhabotinsky wave with a stoichiometry which is half of that of the uncoupled system.