Like most musicians, flute players often add vibrato to their playing for expressive effect. This is accomplished by a periodic modulation of the supplied air pressure resulting in significant amplitude and timbre variations of the resulting sound. However, a sizeable modulation in pitch also results in most cases with a width ranging from 20 to 50 cents of total variation. This work investigates the physical origins of this pitch variation based on an analysis of multiple databases of flute vibrato tones. Detailed tracking of the amplitude and pitch trajectories of each overtone is used to characterize the extent of amplitude and frequency modulation of each partial as well as their relative correlation. A high degree of correlation between amplitude and frequency modulation tracks suggests that both variations are controlled by the same physical parameter. However, the mechanisms for pressure-controlled pitch modulation are not well understood and many physically modeled synthesis algorithms do not produce this pitch variation when the input pressure is modulated. As such, this work systematically modifies existing physical models to reproduce the characteristics of recorded vibrato tones and evaluate potential mechanisms that may underlay the observed pitch modulations.