Wake-induced vibrations of a 2-DOF circular cylinder which is placed behind a stationary square cylinder with tandem arrangement are numerically investigated at low Reynolds numbers by using semi-implicit Characteristics-based split (CBS) finite element algorithm in this study. Numerical results demonstrate that the side length of the upstream square cylinder can significantly affect the characteristics of flow patterns, oscillation frequency, maximum amplitudes, X-Y trajectories, and hydrodynamic coefficients of the rear circular cylinder. The predominant vortex shedding patterns are 2S, 2P and 2T mode. In addition to the figure "8", "raindrop" and figure "dual-8" are observed in the X-Y vibrating trajectories for the circular cylinder. Finally, the interactions between cylinders are revealed according to the phase portrait of fluid force to displacement and the power spectral densities (PSD) features of the vibration amplitude on the rear circular cylinder and the instantaneous flow field, together with the wakeinduced vibration (WIV) mechanism underlying the oscillation characteristics of the circular cylinder behind a stationary square cylinder with different sizes.