Recently we have observed a dependence of the strength of the antiferromagnetic (AF) coupling across Cu(001) on the Co layer thickness in a MBE-grown triple wedge sample. The results appear to be consistent with the recently predicted oscillatory behavior with Co thickness. The current paper analyzes the experimental results in more detail using model calculations within the free-electron approximation. The oscillation in the phase of the coupling predicted by these calculations has not been observed. In contrast, it was found that the phase of the coupling depends on the composition of the ferromagnetic layer. The composition-dependent behavior of the phase appears strongly orientation dependent and different for the long- and short-period oscillations. It is believed that the occurrence of energy band gaps in the k||-resolved electronic structure of the ferromagnet plays a decisive role in determining the phase of the oscillatory interlayer exchange coupling.