  We previously showed that a quantum quench in a one-dimensional translation
invariant system produces undamped oscillations of a local observable when the
post-quench state includes a single-quasiparticle mode and the observable
couples to that mode [J. Phys. A 47 (2014) 402001]. Here we consider quenches
that break initial translation invariance. Focusing on quenches performed only
on an interval of the whole system, we analytically determine the time
evolution of local observables, which occurs inside a truncated light cone
spreading away from the quenched interval as time increases. If the quench
excites a single-quasiparticle mode, oscillations with the frequency of the
quasiparticle mass stay undamped until a time increasing with the length of the
quenched interval, before eventually decaying as $t^{-1/2}$. The translation
invariant case with no damping is recovered as the length of the interval goes
to infinity.
