Just as water surface waves are found everywhere on the ocean's surface, internal waves are ubiquitous throughout the atmosphere. These waves constantly propagate and interact with other flows, but these interactions are difficult to observe due to inadequate current technology. Numerical simulations are often utilized in the study of internal waves. In this work, ray theory is used to numerically simulate the interaction of atmospheric internal waves with time-dependent and time-independent background flows, specifically the interaction of small-scale internal waves and large-scale inertial waves. Parameters such as initial wavenumbers and amplitudes of both small internal waves and inertial waves are determined that will cause the small waves to reach a turning point or critical level, or in the case of time-dependent flows, a wavenumber that reaches a critical value. Other parameters that may cause the waves to become unstable are included in the analysis, such as wave steepness and shear instability. These parameters are combined to determine the spectrum of waves that will experience instability during the interaction. Two principal interactions, small-scale internal waves propagating through an infinite wave train and small-scale internal waves propagating through an inertial wave packet, are simulated and compared. For the first interaction, the total frequency is conserved but is not for the latter. This deviance is measured and results show how it affects the outcome of the interaction. The interaction with an inertial wave packet compared to an inertial wave train results in a higher probability of reaching a Jones' critical level and a reduced probability of reaching a turning point, which is a better approximation of outcomes experienced by expected real atmospheric interactions.