The paper concerns the hydrodynamic analysis and optimization of an OWC spar buoy, possibly the simplest concept for a floating oscillating water column (OWC) wave energy converter. It is an axisymmetric device consisting basically of a submerged vertical tail-tube-fixed to an axisymmetric floater that oscillates essentially in heave. The air flow displaced by the water motion inside the tube drives a self-rectifying air turbine. The possible advantages of using a tube of non-uniform inner cross section are investigated theoretically and numerically, especially as a way of reducing the draught of the device without significantly impairing its power performance. The unsteady water flow in the tube is modelled as one-dimensional. The frequency-dependent hydrodynamic coefficients of the tube-floater pair were computed with a boundary-element code. A linear air turbine is assumed. The hydrodynamics of the wave energy absorption is analysed in the frequency domain, including the effect of air compressibility in the chamber; special attention is devoted to optimization. Numerical results are presented for device's performance in regular and irregular waves, including especially optimization of the tube geometry and of the turbine characteristic. Practical implications of these results are discussed.