  We consider the initial-value problem for the one-dimensional, time-dependent
wave equation with positive, Lipschitz continuous coefficients, which are
constant outside a bounded region. Under the assumption of compact support of
the initial data, we prove that the local energy decays exponentially fast in
time, and provide the explicit constant to which the solution converges for
large times. We give explicit estimates of the rate of this exponential decay
by two different techniques. The first one is based on the definition of a
modified, weighted local energy, with suitably constructed weights. The second
one is based on the integral formulation of the problem and, under a more
restrictive assumption on the variation of the coefficients, allows us to
obtain improved decay rates.
