Alfven waves are considered in a radially inhomogeneous medium under a radial background magnetic field: In order for the latter to be divergence free, the magnetic field strength must decay like the inverse square of the distance; for the radial dependence of the mass density an arbitrary inverse power of the distance is assumed. The Alfven wave equation is solved exactly in terms of Bessel functions, and the properties of the solutions are examined: (i) for small radial distances, when the reflection of waves by the inhomogeneity of the medium depends on frequency, thereby changing their spectrum; (ii) for large radial distances, when the ray approximation holds, the spectrum no longer changes and the equipartition of kinetic and magnetic energies is true; (iii) for intermediate radial distances the evolution between the limits (i) and (ii) is illustrated by plotting (Figs. 1 to 4) the wave forms of the velocity and magnetic field perturbations, for a variety of wave periods. The present problem involves some features of Alfven wave propagation in the solar wind (monopole background magnetic field and radially decaying density), but omits others (the presence of background flow, nonradial or spiral components of the background magnetic field and multiple ion species) and thus applies only near to the sun, viz. when the mean flow velocity is still small relative to the Alfven speed. The present model does account for the effects of nonuniform mass density for all wavelengths, whether short, long, or comparable to the length scales of variation of the mass density. It shows that the process of wave reflection can lead to Kolmogorov, Kraichnan, or similar types of spectra observed in the solar wind; thus wave reflection remains a linear theory alternative to the nonlinear turbulent cascade as a model for Alfvenic disturbances in the solar wind. Moreover, the growth of Alfven waves with distance could lead to nonlinear effects which drive the energy cascade of hydromagnetic turbulence by the interaction of direct and reflected Alfven waves; this would establish a relation between "wave" and "turbulent" models of Alfvenic disturbances in the solar wind. Some of the literature on Alfven waves and magnetohydrodynamic turbulence in the solar wind uses the Elsasser equations in original form for incompressible disturbances, overlooking the fact that there are additional terms in the case of an inhomogeneous medium, as shown in Appendix A. (C) 1999 American Institute of Physics. [S1070-664X(99)01007-1].
