Wave-like motions within a low-level inversion capping a morning mixed layer are studied with two instrumented aircraft, one of which is equipped with a fast-response air sensing probe. Kelvin-Helmholtz waves and their different stages of development (growth, overturning and decay) are identified by means of spectral analyses of temperature and wind component records. By analysing energy conversion rate cospectra, it is found that mechanical production terms and buoyancy production terms, respectively positive and negative during growth stage, reverse signs when overturning occurs. These results and inspection of the temporal evolution of the vertical profiles of temperature suggest that the instability recurs until the initial surplus of shear is drained. Additionally, spectra computed at the top of the mixed layer are compared with those obtained within the underlying mixed layer. The results qualitativelyshow that the wind shear has a non-negligible effect on the entrainment of warm air through the mixed-layer top.