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We consider the initial-value problem for the one-dimensional, time-dependentwave equation with positive, Lipschitz continuous coefficients, which areconstant outside a bounded region. Under the assumption of compact support ofthe initial data, we prove that the local energy decays exponentially fast intime, and provide the explicit constant to which the solution converges forlarge times. We give explicit estimates of the rate of this exponential decayby two different techniques. The first one is based on the definition of amodified, weighted local energy, with suitably constructed weights. The secondone is based on the integral formulation of the problem and, under a morerestrictive assumption on the variation of the coefficients, allows us toobtain improved decay rates.

On the exponential time-decay for the one-dimensional wave equation with variable coefficients