Structures subject to moving loads have several in rail, road and bridge engineering. When the velocity of the moving system approaches the critical velocity, then the induced vibrations are significantly augmented and safety and stability of the structure as well as of the moving system are compromised. The classical models predict the critical velocity much higher than the one observed in reality, because the wave propagation is restricted to the beam structure. But if the beam is supported by elastic continuum, then the waves can be dominant in the foundation and the interaction with the beam cannot be overlooked. This contribution analyses the critical velocity of an oscillator moving on a beam supported by a foundation of finite depth by semi-analytical methods.