  In this article we prove quantitative unique continuation results for wave
operators of the form $\partial$ 2 t -- div(c(x)$\nabla$$\bullet$) where the
scalar coefficient c is discontinuous across an interface of codimension one in
a bounded domain or on a compact Riemannian manifold. We do not make any
assumptions on the geometry of the interface or on the sign of the jumps of the
coefficient c. The key ingredient is a local Carleman estimate for a wave
operator with discontinuous coefficients. We then combine this estimate with
the recent techniques of Laurent-L{\'e}autaud [LL19] to propagate local unique
continuation estimates and obtain a global stability inequality. As a
consequence, we deduce the cost of the approximate controllability for waves
propagating in this geometry.
