Using optical coherence tomography (OCT) vibrometry we studied wave propagation by recording at multiple longitudinal positions in the gerbil and guinea pig cochlea. Our aims are to (1) fully portray the unusual dispersive character of the wave; (2) derive from these properties the underlying physics of the wave and the way it is regulated by physiological processes; (3) challenge cochlear modelers to report and analyze wave propagation in their models. In the current study we report two unexpected features of the spatial profiles of sound-evoked vibrations of the basilar membrane. The profiles of low-frequency components show very little systematic phase accumulation, whereas the slow waves near the best frequency often exhibit abrupt, place-fixed transitions in propagation velocity.