We have developed methods for acquiring temporally and spatially resolved spectrograms of the velocity of sliding charge-density waves (CDWs), allowing unprecedented access to CDW dynamics. Complex transients arising from the interplay between elastic and plastic processes occur when the driving field direction is reversed. A transient spectral component due to shear elasticity can be unambiguously identified, and allows the most direct determination to date of the CDW's shear elastic modulus. Near current contacts, initially elastic displacements are followed by an elastic-to-plastic transition. A simple model provides a semiquantitative account of many aspects of these transients. © 2015 American Physical Society.