This paper considers Alfvén waves in a radially stratified medium where all background quantities, namely mass density, magnetic field strength and mean flow velocity, depend only on the distance from the centre, the latter two being assumed to lie in the radial direction. It is shown that the radial dependence of Alfvén waves is the same for two cases: (i) when the velocity and magnetic field perturbations are along parallels, in the one-dimensional case of only radial and time dependence; (ii) in the three-dimensional case with dependence on all three spherical coordinates and time, for velocity and magnetic field perturbations with components along parallels and meridians, represented by the radial components of the vorticity and electric current respectively. Elimination between these equations leads to the convected Alfvén-wave equation in the case of uniform flow, and an equation with an additional term in the case of non-uniform flow with mean flow velocity a linear function of distance. The latter case, namely that of non-uniform flow with flow velocity increasing linearly with distance, is analysed in detail; conservation of mass flux requires the mass density to decay as the inverse cube of the distance. The Alfvén-wave equation has a critical layer where the flow velocity equals the Alfvén speed, leading to three sets of two solutions, namely below, above and across the critical layer. The latter is used to specify the wave behaviour in the vicinity of the critical layer, where local partial transmission occurs. The problem has two dimensionless parameters: the frequency and the initial Alfvén number. It is shown, by plotting the wave fields relative to the critical layer, that these two dimensionless parameters appear in a single combination. This simplifies the plotting of the wave fields for several combinations of physical conditions. It is shown in the Appendix that the formulation of the equations of MHD in the original Elsässer (1956) form, often used in the recent literature, does not apply if the background mass density is non-uniform on the scale of a wavelength. The present theory, based on exact solutions of the Alfvén-wave equation for a inhomogeneous moving medium, is unrestricted as to the relative magnitude of the local wavelength and scale of change of properties of the background medium. The present theory shows that the rate-of-decay of wave amplitude is strongly dependent on wave frequency beyond the critical layer, i.e. the process of change with distance of the spectrum of Alfvén waves in the solar wind starts at the critical layer.