  We study the small vibrations of an axially travelling string with a
dashpoint damping at one end. The string is modelled by a wave equation in a
time-dependent interval with two endpoints moving at a constant speed $v$. For
the undamped case, we obtain a conserved functional equivalent to the energy of
the solution. We derive precise upper and lower estimates for the exponential
decay of the energy with explicit constants. These estimates do not seem to be
reported in the literature even for the non-travelling case $v=0$.
