Parallel propagating finite-amplitude Alfv¿n waves in a nonhomogeneous medium are considered. The analysis is based on an ideal magnetohydrodynamic (MHD) description with an arbitrary geometry for the background magnetic field. Examination of the characteristic equations of the system reveals, as expected, six waves: forward and backward propagating Alfv¿n waves and fast/slow magnetosonic waves. However, although these waves are well classified in a homogeneous medium (in particular, the pure Alfv¿n wave is strictly incompressible and circularly polarized), such a simple classification with regard to compressibility and polarization is found to be impossible in a nonhomogeneous medium. We derive exact forms for the wave action invariant equation and the amplitude conservation law for Alfv¿n waves and find these equations to be inconsistent, reaffirming that the pure incompressible Alfv¿n wave does not exist in a nonhomogeneous medium. Construction of a consistent Alfv¿n wave solution is outlined that incorporates predominantly large-amplitude incompressible Alfv¿n waves with small-amplitude compressible fluctuations. Such a solution corresponds to strong fluctuations in the magnetic field vector and weak fluctuations in the magnetic field strength, as typically observed in the solar wind. In order to counteract the tendency of compressible fluctuations to grow without bound in the ideal MHD model, we include a Landau damping term obtained from kinetic theory.¿ 1997 American Geophysical Union