Mathematical discussion is restricted to gravity waves of period short enough for the earth's rotation to be neglected, propagated horizontally through air whose horizontal velocity and static stability vary with height in the layers near the ground. The differential equation (15) is derived, describing how the vertical velocity of the wave motion varies with height for waves, of length 2 π/k, at rest relative to the coordinate system. Boundary conditions are then applied—at the ground where the motion is horizontal, and above where the kinetic energy is finite—to give the relationship betweenk and the velocity of the wave relative to the air above (−W).