Introducing the periodic structure theory in the field of Solid-State Physics into track vibration reduction, this work analyzes the complex dispersion relation of the infinite periodic rail system equipped with a dynamic vibration absorber (DVA). For comparison, the uncontrolled rail model is also examined. All possible wave modes, including the propagative mode (P mode), purely evanescent mode (PE mode), evanescent edge mode (EE mode), and complex mode (C mode), in the considered periodic rail structures are analyzed. It is found that due to the elastic fastener, the P mode in the rail structure without control does not start from 0 Hz, but from a critical frequency, $f_{c1}$. Below this critical frequency, two pairs of C modes are observed. Above this critical frequency, these two pairs of C mode transition into a pair of PE mode and a pair of P mode. The PE mode transitions into a new pair of propagating waves (P mode-2) after the critical frequency, $f_{c2}$. At the ends of the first Brillouin zone, the P mode-1 transfers into the EE mode. Above the critical frequency $f_{c2}$, two P modes exist simultaneously. When they meet, one transfers into the other in the C mode. Unlike the EE mode, the C mode can be considered a transitional waveform between different propagative waves. Once the DVA is introduced, another new propagative wave (P mode) is observed. Both the PE mode and EE mode-1 are divided into two parts. For the first time, it is observed that the bound frequency of the band gap in a one-dimensional periodic structure falls within the first Brillouin Zone, not just at the two ends. The phase jump occurs at the DVA's local resonant frequency. The introduction of the DVA clearly enhances the Bragg scattering feature of the periodic rail structures.