We present three-dimensional direct numerical simulations of internal waves excited by turbulent convection in a self-consistent, Boussinesq and Cartesian model of mixed convective and stably stratified fluids. We demonstrate that in the limit of large Rayleigh number ($Ra𝟄 [4\times 10^{7},10^{9}]$) and large stratification (Brunt–Väisälä frequencies$f_{N}≫ f_{c}$, where$f_{c}$is the convective frequency), simulations are in good agreement with a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region as described in Lecoanet & Quataert (Mon. Not. R. Astron. Soc., vol. 430 (3), 2013, pp. 2363–2376). Specifically, we demonstrate that the wave energy flux spectrum scales like$k_{\bot }^{4}\,f^{-13/2}$for weakly damped waves (with$k_{\bot }$and$f$the waves’ horizontal wavenumbers and frequencies, respectively), and that the total wave energy flux decays with$z$, the distance from the convective region, like$z^{-13/8}$.