  Crustal quakes of highly magnetized neutron stars can disrupt their
magnetospheres, triggering energetic phenomena like X-ray and fast radio bursts
(FRBs). Understanding plasma wave dynamics in these extreme environments is
vital for predicting energy transport across scales to the radiation length.
This study models relativistic plasma wave interaction in magnetar
magnetospheres with force-free electrodynamics simulations. For propagation
along curved magnetic field lines, we observe the continuous conversion of
Alfv\'en waves (AWs) to fast magnetosonic (FMS) waves. The conversion
efficiency can be up to three times higher when counter-propagating AWs
interact in the equatorial region. Alfv\'en waves generate FMS waves of twice
their frequency during their first crossing of the magnetosphere. After the
initial transient burst of FMS waves, AWs convert to FMS waves periodically,
generating variations on timescales of the magnetospheric AW crossing time.
This decaying FMS wave tail carries a significant portion (half) of the total
energy emitted. Plastic damping of `bouncing' AWs by the magnetar crust has
minimal impact on the FMS efficiency. We discuss the implications of the
identified wave phenomena for magnetar observations. Outgoing FMS waves can
develop electric zones, potential sources of coherent radiation. Long
wavelength FMS waves could generate FRBs through reconnection beyond the light
cylinder.
