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

Quantitative unique continuation for wave operators with a jump discontinuity across an interface and applications to approximate control