We consider the question: what can be determined about the stiffness distribution inbiological tissue from indirect measurements? This leads us to consider an inverse problemfor the identification of coefficients in the second-order hyperbolic system that modelsthe propagation of elastic waves. The measured data for our inverse problemare the time-dependent interior vector displacements. In the isotropic case, weestablish sufficient conditions for the unique identifiability of wave speeds andthe simultaneous identifiability of both density and the Lamé parameters. In theanisotropic case, counterexamples are presented to exhibit the nonuniqueness and toshow the structure of the set of shear tensors corresponding to the same givendata.