A resonator fiber-optic gyro (R-FOG) is a high-accuracy inertial rotation sensor based on the Sagnac effect. A fiber ring resonator is the core sensing element in the R-FOG. When the frequency of the fiber ring resonator input laser is swept linearly with time, ringing of the output resonance curve is observed. The output field of the fiber ring resonator is derived from the superposition of the light transmitted through the directional coupler directly and the multiple light components circulated in the fiber ring resonator when the frequency of the laser is swept. The amplitude and phase of the output field are analyzed, and it is found that the difference in time for different light components in the fiber ring resonator to reach a point of destructive interference causes the ringing phenomenon. Finally the ringing phenomenon is observed in experiments, and the experimental results agree with the theoretical analysis well.