In a barotropic fluid, a free turbulent flow induces a fluctuating potential flow which is determined by the instantaneous flow near the edge of the turbulent flow. If the surrounding fluid is stably stratified, internal wave-motions are possible and, in general, wave-energy accumulates until it is sufficient to modify the turbulent flow. Here the growth of wave-motion from rest is examined with particular reference to the atmospheric problem of wave excitation by the surface boundary layer. Wind shear is supposed negligible outside the turbulent flow and the disturbances from the boundary layer are assumed to travel with a convection velocity V relative to the upper air. For times large compared with {−g/ρ(dρ/dz)}−½ (ρ is the potential density), most of the wave-energy resides in components of phase-velocity near the convection velocity. For a model atmosphere with increased stability above a tropopause, this resonance mechanism leads to the formation of wave-groups with crests at right-angles to the convection velocity and wavelengths near 2πV[−g/ρ(dρ/dz)]−½. Using likely values for the surface disturbances, wave-amplitudes of order 100 m can develop within several hours of the initiation of the boundary layer but sufficient amplitude to produce overturning or breaking is unlikely within a reasonable time.