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We consider the initial-value problem for a one-dimensional wave equationwith coefficients that are positive, constant outside of an interval, and havebounded variation (BV). Under the assumption of compact support of the initialdata, we prove that the local energy decays exponentially fast in time, andprovide the explicit constant to which the solution converges. The keyingredient of the proof is a high frequency resolvent estimate for anassociated Helmholtz operator with a BV potential.

Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation