This paper is concerned with the propagation of small amplitude gravity waves over a flow with non-uniform velocity distribution. For such a flow Burns derived a relation for the velocity of propagation in terms of the velocity distribution of the mean flow. This result is derived here in another way and some of its implications are discussed. It is shown that one of these is hardly acceptable physically. Burns's result holds only when a real value of the propagation velocity is assumed; the mentioned difficulties vanish if complex values are allowed for, implying damping or growth of the waves. Viscous effects which are the cause of damping or growth are important in the wall layer near the bottom and also at the critical depth, which is present when the wave speed is between zero and the fluid velocity at the free surface.

In § 2 the basic equations for the present problem are given. In § 3 exchange of momentum and energy between wave and primary flow is discussed. This is analogous to what happens at the critical height in a wind flow over wind-driven gravity waves. In § 4 the viscous effects at the bottom are included in the analysis and the complex equation for the propagation velocity is derived. Finally in § 5 illustrations of the theory are given for long waves over running flow and for the flow along a ship advancing in a wavy sea. In these examples a negative curvature of the mean velocity profile is shown to have a stabilizing effect.