An analytical theory is developed for an oscillating water column (OWC) with a V-shaped channel to improve the pneumatic efficiency of wave energy extraction. An eigenfunction expansion method is used in a cylindrical coordinate system to investigate wave interaction with the OWC converter system. Auxiliary functions are introduced to capture the singular behaviours in the velocity field near the salient corners and cusped edges. Effects of the OWC dimensions, the opening angle and length of the V-shaped channel, as well as the incident wave direction, on the pneumatic efficiency of wave energy extraction are examined. Compared with a system without the V-shaped channel, our results show that the V-shaped channel can significantly increase the conversion efficiency and widen the range of wave frequency over which the OWC system can operate at a high efficiency. For typical coastal water depths, the OWC converter system can perform efficiently when the diameter of the OWC chamber is in the range of 1/5–1/2 times the water depth, the opening angle of the V-shaped channel is in the range of [π/2, 3π/4] and the length of the V-shaped channel is in the range of 1–1.5 times the water depth.