We discuss models for coronal abundance anomalies observed in the coronae of the sun and other late-type stars following a scenario first introduced by Schwadron, Fisk, & Zurbuchen of the interaction of waves at loop footpoints with the partially neutral gas. Instead of considering wave heating of ions in this location, we explore the effects on the upper chromospheric plasma of the wave pondermotive forces. These can arise when upward-propagating waves from the chromosphere transmit or reflect upon reaching the chromosphere-corona boundary, and are in large part determined by the properties of the coronal loop above. Our scenario has the advantage that for realistic wave energy densities both positive and negative changes in the abundance of ionized species compared to neutrals can result, allowing both first ionization potential (FIP) and inverse FIP effects to come out of the model. We discuss how variations in model parameters can account for essentially all of the abundance anomalies observed in solar spectra. Expected variations with stellar spectral type are also qualitatively consistent with observations of the FIP effect in stellar coronae.