  Gravitational waves are excellent tools to probe the foundations of General
Relativity in the strongly dynamical and non-linear regime. One such foundation
is Lorentz symmetry, which can be broken in the gravitational sector by the
existence of a preferred time direction, and thus, a preferred frame at each
spacetime point. This leads to a modification in the orbital decay rate of
binary systems, and also in the generation and chirping of their associated
gravitational waves. We here study whether waves emitted in the late,
quasi-circular inspiral of non-spinning, neutron star binaries can place
competitive constraints on two proxies of gravitational Lorentz-violation:
Einstein-\AE{}ther theory and khronometric gravity. We model the waves in the
small-coupling (or decoupling) limit and in the post-Newtonian approximation,
by perturbatively solving the field equations in small deformations from
General Relativity and in the small-velocity/weak-gravity approximation. We
assume a gravitational wave consistent with General Relativity has been
detected with second- and third-generation, ground-based detectors, and with
the proposed space-based mission, DECIGO, with and without coincident
electromagnetic counterparts. Without a counterpart, a detection consistent
with General Relativity of neutron star binaries can only place competitive
constraints on gravitational Lorentz violation when using future,
third-generation or space-based instruments. On the other hand, a single
counterpart is enough to place constraints that are 10 orders of magnitude more
stringent than current binary pulsar bounds, even when using second-generation
detectors. This is because Lorentz violation forces the group velocity of
gravitational waves to be different from that of light, and this difference can
be very accurately constrained with coincident observations.
