Ocean waves are a huge and largely untapped resource of green energy. In order to extract energy from waves, a novel wave energy converter (WEC) consisting of a floating, hollow cylinder capped by a roof with a variable aperture is presented in this paper. The power take-off (PTO) system is composed of a linear generator attached to the seabed, driven by the heave motion of the floating cylinder through a tether line. The air pressure within the cylinder can be modified by adjusting the roof aperture. The hydrodynamic characteristics of this WEC are investigated through an analytical model based on potential flow theory, in which the wave diffraction/radiation problems are coupled with the air pressure fluctuation and PTO system. Analytical expressions are derived for the maximum power absorbed by the WEC under different optimization principles, revolving around the PTO damping, roof aperture damping and non-negative mooring stiffness. We find that the best power absorption is obtained when the aperture is either completely open or entirely closed, depending on the wave conditions. Intermediate values of the aperture are useful to minimize the heave motion and thus ensure survivability under extreme sea states.