Self-similar solutions are presented for the equatorial propagation of axisymmetric, piston-driven magnetohydrodynamic shocks into an inhomogeneous ideal gas permeated by a current-free azimuthal magnetic field. Several regimes of magnetically dominated flow near the piston are possible, depending on the ambient density distribution of the unshocked gas. The strong hydrodynamic and magnetic gradients which permeate this flow indicate the need for a more refined treatment which will include the effects of diffusion due to dissipative phenomena. Global features of the flow are discussed and illustrated by numerical solutions. The use of reductive properties of the equations to simplify the problem in several special cases is indicated. Implications regarding the propagation of disturbances in stellar atmospheres are considered.