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 outward; 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.