The multimode beam wave interaction behavior in a three-cavity, W-band gyroklystron amplifier is studied using nonlinear, time-dependent, multimode analysis. The gyroklystron amplifier operates in the fundamental harmonic, higher order TE 021 mode. The presented theory is used to study the concept of mode competition. A peak output power of around 330 kW at 93.2 GHz, an electronic efficiency of 25%, and a saturated gain of 22.8 dB is obtained with a 75 kV, 17 A gyrating electron beam. The results obtained are found to be in good agreement with the time-independent single-mode analysis and the reported experimental results.Index Terms -Fast-wave device, microwave tube, millimeter wave high power amplifier, multimode analysis.