The dynamic heave stability of an air cushion vehicle or hovercraft hovering over deep water without forward motion is investigated here analytically. The principal feature of the analysis is the modelling of the motion of the water surface beneath the cushion caused by fluctuations in the pressure of the cushion or cavity air. This surface motion interacts with the vehicle dynamics by modulating both the volume and exit flow area of the cushion. For analytical simplicity, the geometry chosen for study is a two-dimensional section of a rigid wall plenum chamber; this enables exploitation of classical linear wave formulas developed by Lamb for the surface motion generated by a spatially uniform surface pressure oscillating sinusoidally in time. To assess stability characteristics, the Nyquist criterion is applied to the linearized equations. Results are presented for two cases: one is representative of a small test vehicle, and the other of a large ise-breaking platform. They show that the water surface motion significantly affects stability through both of the proposed mechanisms, with cushion exit flow area modulation usually being more important. A feature of the results is that as the weight of a vehicle decreases many stability transitions occur. This suggests that simple guidelines for avoiding instability may not exist, so that stability augmentation devices may be required for vehicles designed to hover for extended periods over water.