A computational investigation of flow structures of underexpanded supersonic free jet under periodic pressure perturbation condition has been carried out in the present study. Reynolds-averaged Navier–Stokes (RANS) equations with a single-equation Spalart–Allmaras (SA) turbulence model have been used in the computation. Static pressure at the nozzle inlet has been modelled with steady as well as time-dependent varying frequency sinusoidal profiles. Keeping the ambient conditions fixed, the NPR of the nozzle has been changed in the range between 2.2 and 3.2. In both steady and fluctuating cases, it is found that the shock cell structure and flow properties are significantly affected by variation of pressure. Lengths of shock cell and supersonic core are found to increase with the increase of inlet pressure. For fluctuating inlet pressure, the flow field is found to respond in a different manner depending on whether the NPR is increasing or decreasing. The frequency of oscillation of inlet pressure is also found to play significant roles. Hysteresis behaviour in several flow parameters has been identified. The maximum hysteresis of 28.7% occurs at the highest applied frequency of 100 Hz for first shock cell length. The generated thrust at different conditions does not show a hysteretic behaviour (below 1%) in oscillating NPR conditions except at high frequency where the maximum hysteresis is found to be 4.7%.