Although monsoon depressions are a principal synoptic-scale element of the South Asian monsoon, producing extreme rainfall over India and surrounding regions, there exists no widely accepted mechanism explaining their occurrence. This study presents a hierarchy of numerical experiments aimed at finding such an explanation. Using a perturbation-basic state decomposition, we derive an anelastic system of equations that can represent disturbances growing in the complex, three-dimensional monsoon basic state. We find that modal solutions to these equations linearized about this basic state can explain many features of observed monsoon depressions, including their warm-over-cold core structure, westward propagation, and lower-tropospheric wind maximum. For the zonally symmetric case, these modes are barotropically unstable, drawing energy from the meridional shear of the monsoon trough. For the zonally varying basic state, modal solutions still derive energy from barotropic conversion, but fail to achieve positive net growth rates when dissipative processes are included. For the non-linear equation set, these modes can be excited by a heating impulse, and their energy then remains roughly constant over several days as barotropic energy transfers oppose dissipative losses. Our results support the idea that the general concept of barotropic instability can explain the structure, propagation, and geographic distribution of monsoon depressions, but not their rapid growth rates. We speculate that condensational heating coupled to these waves is needed to obtain a positive growth rate.