The equilibrium state of weak horizontal temperature gradients in the tropics is maintained by the fast horizontal propagation of gravity waves (e.g. Bretherton and Smolarkiewicz, 1989), which redistributes convective heating as ‘compensating subsidence’ spreading radially away from the heat source. However, this picture of purely horizontal wave radiation is incomplete, because the tropical troposphere is not capped by a rigid lid, but by more stable region of fluid– a stratosphere. In this study, we use a simple model of the tropical atmosphere to elucidate why the rigid-lid approximation has proven so useful despite the lack of physical correspondence to the real tropical atmosphere, and to explain its limitations. First, we show that even a modest jump in stratification at the tropopause causes latent heating to project strongly onto the wavelengths twice the depth of the heating, which partially justifies the use of a rigid-lid approximation. However, we also suggest that the upward radiation of gravity waves out of the troposphere has an important influence on tropospheric dynamics, and this process is clearly neglected in the rigid-lid approximation. We show that gravity waves radiate upward on timescales relevant to tropospheric dynamics– in fact, an initially-localized first-baroclinic pulse of buoyancy spreads throughout the whole tropics in approximately 10 days. This provides a theoretical underpinning and possible alternate explanation for the 1-10 day damping timescale ubiquitous in studies of large-scale tropical dynamics, which is typically attributed to convective momentum transport.