Energy decay estimates for an axially travelling string damped at one end

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 $v=0$.