We investigate the preferential heating and acceleration of solar wind heavy ions by the resonant cyclotron interaction with parallel-propagating left-polarized hydromagnetic waves. We set up a scenario whereby the energy for this interaction is taken from saturated low-frequency Alfven waves via a cascade to the higher, resonant frequencies. In order to utilize the existing theoretical work, the particles are taken to be thermally isotropic, and the waves are taken to be dispersionless. This scenario is incorporated into a numerical solar wind code describing the flow from an inner radius (taken to be 10 solar radii) to 1 AU. Thus we present the first model of a wave-driven, three-fluid, supersonic solar wind. By varying the model parameters we test the ability of the resonant interaction in this model to produce the excess speeds and temperatures of heavy ions that are observed. We find that unrealistically steep wave spectra are required to produce differential speeds of the order of the 4.7, where &ggr; is the power law spectral index. Ions of oxygen or iron, with larger mass-per-charge ratios, are accelerated more readily than helium, but still require steeper spectra than are observed. This model is also unable to produce the mass-proportional heavy ion temperatures that are observed. We show that the production of mass-proportional temperatures is inconsistent with preferential acceleration of heavy ions by this mechanism. The model also produces a heavy ion-to-proton temperature ratio at 1 AU which is anticorrelated with solar wind speed, in contradiction to the observed behavior.