This paper presents an analytical solution to the ground vibration of a half-space subjected to sub- and super-critical moving loads oscillating with the frequency f0 and distributed in different forms. By applying the contour integral combined with the residue theorem, the approximate values of the integrals for the ground response are determined, with a rigorous rationale provided for the exclusive consideration of the Rayleigh waves on the ground. The validity of the present solutions is verified against existing solutions, including Auersch's. Through the parametric analysis, it is observed that the frequency f0 of dynamic loading of the vehicle exhibits significant influence on the critical speed, the magnitude of ground vibrations and their rate of attenuation. In addition, the effect of rail irregularity on the ground vibration was included in some applications of the present theory for high-speed trains under the sub-and super-critical speeds with respect to the Rayleigh waves speed. It was shown that track irregularity can largely amplify the velocity and acceleration responses of the ground, albeit exerting a comparably little effect on the ground displacement. Besides, resonant vibrations are likely to occur when the train is accelerated to the critical speed of the soil.