The oscillating-water-column (OWC) wave energy converter with air turbine has been object of extensive research and development effort, including the deployment of floating and fixed-structure full-sized prototypes into the sea. It consists of a hollow (fixed or floating) structure, open to the sea below the water surface. Wave action alternately compresses and decompresses the air trapped above the inner water free-surface in a chamber, which forces air to flow through a turbine coupled to an electrical generator. The spring-like effect of air compressibility in the chamber is related to the density-pressure relationship. It is known to significantly affect the power performance of the full-sized converter, and is rarely accounted for in theoretical modelling, and even more rarely in physical model testing at reduced scale, as appears from the literature review. Three theoretical models of increasing complexity are analysed and compared: (i) the incompressible air model; (ii) the isentropic process model; (iii) and the (more difficult and rarely adopted) adiabatic non-isentropic process model in which losses due to the imperfectly efficient turbine are accounted for. The air is assumed as a perfect gas. The hydrodynamic modelling of wave energy absorption is based on linear water wave theory. The validity of the various simplifying assumptions, especially in the aero-thermodynamic domain, is illustrated by a case study with numerical results for a fixed-structure OWC equipped with a Wells turbine. Results are shown for regular and irregular waves, and for a theoretical simulation of model testing in wave tank at small scale.