The response of annular liquid membranes to sinusoidal mass flow rate fluctuations at the nozzle exit is analyzed as a function of the amplitude and frequency of the axial velocity fluctuations at the nozzle exit and thermodynamic compression of the gas enclosed by the membrane. It is shown that both the pressure of the gases enclosed by the annular membrane and the axial distance at which the annular membrane merges on the symmetry axis are periodic functions of time which have the same period as that of the mass flow rate fluctuations at the nozzle exit. They are also nearly sinusoidal functions of time for small amplitudes of the mass flow rate fluctuations at the nozzle exit, and exhibit delay and lag times with respect to the sinusoidal axial velocity fluctuations at the nozzle exit. Both the delay and the lag times are functions of the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit and the polytropic exponent. The amplitudes of both the pressure of the gases enclosed by the annular liquid membrane and the convergence length increase and decrease, resp., as the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit, resp., are increased. They also increase as the polytropic exponent is increased.