  Inertia-gravity waves are scattered by background flows as a result of
Doppler shift by a non-uniform velocity. In the WKB regime, the scattering
process reduces to a diffusion in spectral space. Other inhomogeneities the
waves encounter, such as density variations, also cause scattering and spectral
diffusion. We generalise the spectral diffusion equation to account for these
inhomogeneities. We apply the result to the rotating shallow water system, for
which height inhomogeneities arise from velocity inhomogeneities through
geostrophy, and to the Boussinesq system for which buoyancy inhomogeneities
arise similarly. We compare the contributions that height and buoyancy
variations make to the spectral diffusion with the contribution of the Doppler
shift. In both systems, we find regimes where all contributions are
significant. We support our findings with exact solutions of the diffusion
equation and with ray tracing simulations in the shallow water case.
