A numerical investigation is conducted on the flow around and vibration response of an elastic square cylinder (side width ) in the wake of a stationary cylinder at Reynolds numbers of and 200 based on and the free-stream velocity. The downstream cylinder, referred to as the wake cylinder, is allowed to vibrate in the transverse direction only. The reduced velocity is varied from 1 to 30. Cylinder centre-to-centre spacing ratios of and 6 are considered. Simulations are also conducted for a single isolated cylinder, and the results are compared with those for the wake cylinder. The focus is given to vibration response, frequency response, fluctuating lift force, phase relationship between the lift and displacement, work done and the flow structure modification during the cylinder vibration. The results reveal that the dependence of the Strouhal number on can distinguish different branches more appropriately than that of the vibration amplitude on. The vibration response of the single cylinder at is characterized by the initial, lower and desynchronization branches. On the other hand, that at undergoes initial, lower and galloping branches. The galloping involves the characteristics of both the initial and the lower branches or the initial and the desynchronization branches depending on. For the wake cylinder, the gap flow has a significant impact on the vibration response, leading to (i) the absence of galloping at either and, (ii) the presence of an upper branch at, and (iii) an initial branch of different characteristics at,. The different facets are discussed in terms of wake structures, work done and phase lag between lift and displacement. © 2018 Cambridge University Press.