  For strongly repulsive bosons in one dimension, we provide detailed modeling
of the Bragg pulse used in quantum Newton's cradle-like settings or in Bragg
spectroscopy experiments. By employing the Fermi-Bose mapping for a finite
harmonically trapped gas and the Quench Action approach for a thermodynamic
system on a ring, we reconstruct the exact post-pulse many-body time evolution
of Lieb-Liniger gases in the Tonks-Girardeau limit, together with their
changing local density profile and momentum distribution. Our results display a
clear separation of timescales between rapid and trap-insensitive relaxation
immediately after the pulse, followed by slow in-trap periodic behaviour.
