When train speeds approach the wave propagation velocity of the supporting trackground system, track displacement amplitudes can increase significantly. This is undesirable and can cause both elevated maintenance and safety concerns. Thus, it is important to determine the speed at which these effects will occur, however modelling the problem in detail (e.g. using numerical techniques) can result in long run times. Therefore, as an alternative, this paper uses an analytical approach to investigate the dispersion characteristics of the track-ground system, and thus calculate the critical velocity. Unlike alternative approximate critical velocity calculation approaches, the proposed method is fully automated thus not requiring any post-processing to extract the soil dispersion curve. It has low computational requirements meaning it can be used quickly for large sections of track. After the model is outlined, it is validated against a larger, more comprehensive model and found to have high accuracy. A Monte Carlo analysis is then undertaken using 1000 random soil profiles. It is found that ballasted tracks typically have a lower critical velocity than slab tracks and that track height and soil saturation play an important role on the critical speed.