  We present a framework which enables the analysis of dynamic inverse problems
for wave phenomena that are modeled through second-order hyperbolic PDEs. This
includes well-posedness and regularity results for the forward operator in an
abstract setting, where the operators in an evolution equation represent the
unknowns. We also prove Fr\'echet-differentiability and local ill-posedness for
this problem. We then demonstrate how to apply this theory to actual problems
by two example equations motivated by linear elasticity and electrodynamics.
For these problems it is even possible to obtain a simple characterization of
the adjoint of the Fr\'echet-derivative of the forward operator, which is of
particular interest for the application of regularization schemes.
