Damping is usually associated with irreversibility. Here, we present a counterintuitive concept to achieve time reversal of waves propagating in a lossless medium using a transitory dissipation pulse. Applying a sudden and strong damping in a limited time generates a time-reversed wave. In the limit of a high damping shock, this amounts to "freezing" the initial wave by maintaining the wave amplitude while canceling its time derivative. The initial wave then splits in two counterpropagating waves with half of its amplitude and time evolutions in opposite directions. We implement this damping-based time reversal using phonon waves propagating in a lattice of interacting magnets placed on an air cushion. We show with computer simulations that this concept also applies to broadband time reversal in complex disordered systems.