  Hydrodynamical simulations show that circumbinary disks become eccentric,
even when the binary is circular. Here we demonstrate that, in steady state,
the disk's eccentricity behaves as a long-lived free mode trapped by turning
points that naturally arise from a continuously truncated density profile.
Consequently, both the disk's precession rate and eccentricity profile may be
calculated via the simple linear theory for perturbed pressure-supported disks.
By formulating and solving the linear theory we find that (i) surprisingly, the
precession rate is roughly determined by the binary's quadrupole, even when the
quadrupole is very weak relative to pressure; (ii) the eccentricity profile is
largest near the inner edge of the disk, and falls exponentially outwards; and
(iii) the results from linear theory indeed agree with what is found in
simulations. Understanding the development of eccentric modes in circumbinary
disks is a crucial first step for understanding the long term (secular)
exchange of eccentricity, angular momentum and mass between the binary and the
gas. Potential applications include the search for a characteristic kinematic
signature in disks around candidate binaries and precession-induced modulation
of accretion over long timescales.
