Unsteady components of the second-order axial velocity and temperature within a tapered pulse tube were obtained by using a novel hybrid method of solution which combines an analytical solution with a numerical solution. The effects of operating frequency, taper angle and cold eng temperature on the unsteady components of the second-order axial velocity and temperature were shown. The unsteady component of the second-order mass flux had the amplitude of the same order as the steady component when the velocities at the ends of the pulse tube have only first-order components