  We consider an abstract wave equation with a propagation speed that depends
only on time. We investigate well-posedness results with finite derivative loss
in the case where the propagation speed is smooth for positive times, but
potentially singular at the initial time.
  We prove that solutions exhibit a finite derivative loss under a family of
conditions that involve the blow up rate of the first and second derivative of
the propagation speed, in the spirit that the weaker is the requirement on the
first derivative, the stronger is the requirement on the second derivative. Our
family of conditions interpolates between the two limit cases that were already
known in the literature.
  We also provide the counterexamples that show that, as soon as our conditions
fail, solutions can exhibit an infinite derivative loss. The existence of such
pathologies was an open problem even in the two extreme cases.
