We consider the adjustment of a compressible, rotating atmosphere to rapid, localized sources of hydrostatic and geostrophic imbalance. Such imbalances are generated in a linear model by arbitrary "injection" terms in the mass, momentum, and thermodynamic energy equations. These terms incorporate physical processes viewed as external to the dynamics of the larger scale flow. For example, heating by phase change or the convective redistribution of mass and momentum within a cumulus cloud are processes viewed as external to the dynamics of the mesoscale environment in which the convective cloud forms. Such processes generate a spectrum of acoustic, buoyancy, and Lamb waves that propagate into the surrounding environment. To first order, the waves interact with the environment by providing the displacements necessary to adjust the larger scale environment toward a new state of hydrostatic and geostrophic balance. The purpose of this paper is to demonstrate the dependence of the characteristics of the wave spectrum on the spatial and temporal scales as well as the physical source of the injection term. To achieve this goal, we express the solution of the linear model as a sum of spatially and temporally orthogonal structures. These structures are identified as either acoustic, Lamb, buoyancy, or steady modes. The energetics is partitioned similarly among the various wave and steady contributions. It is shown that acoustic waves are generated only when the characteristic duration of the forcing is shorter than or equal to the characteristic duration of the acoustic waves (i.e., approximately two minutes). However, Lamb waves - whose restoring mechanism is the same as that of the acoustic waves - may be generated by forcing of duration on the order of tens of minutes. In addition to examining the characteristics of the compressible wave spectrum, we determine the effect of the anelastic approximation on the characteristics of the buoyancy wave response.