The linear stability of the flow in the annular gap between two infinitely long cylinders, driven by the constant rotation and harmonic oscillation in the axial direction of the inner cylinder, is analysed using Floquet theory. Closed form solutions for the basic flow are derived, both with and without the presence of endwalls. Recent experiments and theory using the narrow gap approximation have shown that the axial oscillations significantly stabilize the flow with respect to centrifugal instabilities. However, the agreement has only been qualitative. The present analysis reproduces the experimental results to well within experimental uncertainty. We have identified the major source of discrepency with the previous theory to be due to the lack of global endwall effects enforcing a net zero axial mass flux. Analysis of the basic flow indicates that the stabilization is primarily due to waves of azimuthal vorticity propagating inwards from the cylinder boundary layers due to the axial oscillations.