Building on the weakly nonlinear amplitude equation of the saturated Taylor vortices developing in a Taylor–Couette cell with a rotating inner cylinder and a fixed outer one, the physical mechanism underlying the destabilization of these vortices resulting in azimuthal waviness is addressed using Floquet analysis. For narrow gap configurations, analysis and direct numerical simulations together with existing experimental results support the idea that the waviness is generated by the axial shear in the azimuthal velocity due to the alternate advection by the Taylor vortices of azimuthal momentum between the cylinders. For wide gap configurations, this mechanism is no longer able to drive the azimuthal waviness and a different mechanism tends to select a subharmonic instability.