A wide variety of propagating equatorial waves organize tropical convection. These waves produce large amplitude temperature and wind perturbations at the tropopause which propagate into the stratosphere. Spectra of lower stratospheric temperature over the "warm pool " region of the equatorial western Pacific in radiosonde data exhibit strong spectral peaks at frequencies corresponding to periods of around 45-50 days, as well as higher frequency peaks corresponding to convectively coupled Kelvin and mixed Rossby-gravity waves. We show that these perturbations are associated with vertically propagating energy forced by the convective heating envelope of the convectively coupled waves. These waves lead to temperature perturbations with downward phase propagation of large amplitude from around 50 hPa (20 km) down to around 250 hPa (11 km). Below that level phase propagation is upward, implying an upper tropospheric energy source region. The three dimensional structure of the waves is examined using NCEP and ECMWF reanalysis data. Radiosonde data are used as "reality check" anchor points to define the tropopause and stratospheric temperature signal. The association with the large scale diabatic heating signal is isolated by utilizing Outgoing Longwave Radiation (OLR) as a proxy for tropical convection. The maximum amplitudes of the stratospheric temperature and tropopause perturbations are not associated locally with the convection, but lead it by several days, and propagate horizontally at a similar phase speed as the convective envelope. The stratospheric QBO affects the extent of the vertical propagation of the waves by altering the basic state zonal flow, such that for Kelvin waves energy reaches only up to around the level of sign reversal of the wind when the QBO is characterized by westerlies above that level.