In this talk, I will present some recent works on the reconstruction of a potential in a wave equation from one measurement of the flux. This result is inspired in the stability results previously obtained by O. Imanuvilov and M. Yamamoto (2001, 2003) and strongly relies on suitable Carleman estimates for the waves. From these Carleman estimates, we will derive a natural iteration process allowing to reconstruct the potential for the wave equation. I will then discuss the numerical feasibility of this approach, which faces several challenging issues. I will present some ideas partially solving these issues and some numerical simulations supporting our approach. This talk is based on joint works with Lucie Baudouin (LAAS, Toulouse) and Maya de Buhan (MAP5, Univ. Paris Descartes).