The boundary layers associated with gravity waves in a fluid with a linear variation of density are discussed in order to examine the steady Eulerian velocities induced by the Reynolds stress. For the case of a standing wave, the induced steady motion is shown to decay in an outer boundary layer which represents a balance between buoyancy and diffusion when the wave slope is sufficiently small but when viscous decay effects are even smaller. When the wave slope is larger, it would appear that two outer regions must be considered. Results for progressive waves are discussed only briefly, as they are qualitatively similar to the surface wave case.