Features of transition and Cherenkov radiations and the correlation between them

It is shown that transition radiation arising at the boundary of two media is being emitted as a Cherenkov one, if the phase velocity of transition radiation waves in the medium of transition radiation propagation becomes equal to the velocity of the moving radiating particle (the necessary condition for the Cherenkov radiation). The proof of this statement is based on the analysis of the transition radiation formation zone, which may become large enough and provide interference between the field of transition radiation and the own Coulomb field of the moving particle, in case when the Cherenkov radiation condition is fulfilled. As a result, the transition radiation field transforms into the Cherenkov field. The problem is considered for cases of both a waveguide and free space.