Higher frequency bands, especially millimeter wave (mmWave), are a key enabler for future high speed railway (HSR) wireless communication systems. Hybrid analog and digital beamforming (HBF) schemes operating at mmWave frequency bands can exploit the sparse nature of the channel by using large scale antenna arrays (LSAs) to shape multiple beams that overcome the severe path loss. However, due to the long and straight geometric characteristic of the rail track, there is a significant variance in the train-to-ground communication rate when the train is near or far from the BS. To address this issue, a novel hybrid beamforming scheme is developed based on a non-uniform beam design. We first model and investigate different beam design schemes in terms of spectral efficiency. Then, an efficient straight route based adaptively analog beamforming scheme with the non-uniform beam is proposed for HSR, which is designed to resist path loss. Finally, an adaptive non-uniform HBF algorithm based on the orthogonal matching pursuit method (OMP) is developed to minimize the rate gap between different locations by selecting the appropriate predefined beams. Numerical results show that the proposed hybrid beamforming scheme can efficiently ensure signal coverage, and reduce the variance of the train-to-ground communication rate along a straight track route.