Description: We present theoretical work directed toward improving our understanding of the mesoscale influence of deep convection on its tropospheric environment through forced gravity waves. From the linear, hydrostatic, non-rotating, incompressible equations, we find a two-dimensional analytical solution to prescribed heating in a stratified atmosphere, which is upwardly radiating from the troposphere when the domain lid is sufficiently high. We interrogate the spatial and temporal sensitivity of both the vertical velocity and potential temperature to different heating functions, considering both the near-field and remote responses to steady and pulsed heating. We find that the mesoscale tropospheric response to convection is significantly dependent on the upward radiation characteristics of the gravity waves, which are in turn dependent upon the temporal and spatial structure of the source, and the assumed stratification. We find a 50% reduction in tropospherically averaged vertical velocity when moving from a trapped (i.e. low lid) to upwardly-radiating (i.e. high lid) solution, but even with maximal upward radiation, we still observe significant tropospheric vertical velocities in the far-field 4 hours after heating ends. We quantify the errors associated with coarsening a 10 km wide heating to a 100 km grid (in the way a General Circulation Model (GCM) would), observing a 20% reduction in vertical velocity. The implications of these results for the parameterisation of convection in low-resolution numerical models are quantified and it is shown that the smoothing of heating over a grid-box leads to significant in grid-box tendencies, due to the erroneous rate of transfer of compensating subsidence to neighbouring regions. Further, we explore a simple time-dependent heating parameterisation that minimises error in a parent GCM grid box, albeit at the expense of increased error in the neighbourhood.