The pressure oscillations of the solid rocket motor could reduce the performance of rocket motors and damage the payloads. Therefore, it is required to predict the conditions leading to turbulent transition from near-laminar to a turbulent flow in the vicinity of the propellant surface, and it is a problem of hydrodynamic instability. In this work, linear stability theory together with the spectral collocation method has been applied on the biglobal hydrodynamic stability analysis of the oscillatory Taylor-Culick flow. Setting the time scale of the basic flow and the hydrodynamic mode as the slow time and the fast time respectively, the linear stability equations are solved using the multiple time scale method. The biglobal hydrodynamic modes are obtained, and the results show that the maximum growth rate of the hydrodynamic mode and its corresponding frequency vary in one period of the basic flow. Furthermore, the parametric study, including the amplitude and the frequency of acoustic motions, is also carried out. The results show that an increase in the amplitude and the frequency of acoustic motions can both make the flow more unstable.