Warped accretion discs are expected in many protostellar binary systems. In
this paper, we study the long-term evolution of disc warp and precession for
discs with dimensionless thickness $H/r$ larger than their viscosity parameter
$\alpha$, such that bending waves can propagate and dominate the warp
evolution. For small warps, these discs undergo approximately rigid-body
precession. We derive analytical expressions for the warp/twist profiles of the
disc and the alignment timescale for a variety of models. Applying our results
to circumbinary discs, we find that these discs align with the orbital plane of
the binary on a timescale comparable to the global precession time of the disc,
and typically much smaller than its viscous timescale. We discuss the
implications of our finding for the observations of misaligned circumbinary
discs (such as KH 15D) and circumbinary planetary systems (such as Kepler-413);
these observed misalignments provide useful constraints on the uncertain
aspects of the disc warp theory. On the other hand, we find that circumstellar
discs can maintain large misalignments with respect to the plane of the binary
companion over their entire lifetime. We estimate that inclination angles
larger than $\sim 20^\circ$ can be maintained for typical disc parameters.
Overall, our results suggest that while highly misaligned circumstellar discs
in binaries are expected to be common, such misalignments should be rare for
circumbinary discs. These expectations are consistent with current observations
of protoplanetary discs and exoplanets in binaries, and can be tested with
future observations.
