Gyrotron traveling-wave tube (gyro-TWT) is capable of generating high-power microwave radiation in a millimeter wave range. It is one of the most promising candidates for the applications in the millimeter wave radar, communication systems, and environmental monitoring. The gyro-TWT can work at high frequency and produce high power output with high order modes. Although the high mode gyro-TWT can work at high frequency and produce high power output, the instability problem is a main factor to prevent the gyro-TWT performance from further improving and hinder this device from being put into the practical application. The earlier research of the instability primarily concentrated on the single-mode situation, which cannot be used to analyze the mutual effects between the other oscillation modes and the operating mode. Hence, it is important for academic study and engineering application to solve the mode competition problem. In this paper, based on lossy uniform/periodic dielectric-loaded metal cylindrical waveguide usually used in the international academic analysis and engineering research, a multimode steady-state beam wave interaction theory for gyro-TWT is established, which can consider the mutual effects between the other oscillation modes and the operating mode. As application examples, under the same condition of geometrical and electrical parameters, the theoretical results of the beam wave interaction for the TE01 fundamental mode gyro-TWTs are compared with the experimental results reported by NRL and IECAS for Ka band and those simulated with Magic code for W band in order to demonstrate the rationality of the theory. The results show that the theoretical results are in good agreement with the experimental and simulated ones. For the NRL design, when the velocity spread is 9.6%, the maximum output power from the theory is 127 kW at 34.09 GHz with a gain of 47.4 dB, an efficiency of 17.6%, and a -3 dB bandwidth of 1.01 GHz, and an NRL measured maximum output power is 130 kW at 34 GHz with a gain of 47.5 dB, an efficiency of 18% and a -3 dB bandwidth of 1.0 GHz. The maximum difference between the theory and the experiments occurs near the frequency of 34 GHz, the measured power by NRL is 127 kW and the calculated power is 118 kW, the relative error between the theory and the experiment is 8.5%. For the IECAS design, the simulated maximum output power from the theory is 113.73 kW at 33.85 GHz with a -3 dB bandwidth of 1.72 GHz when the velocity spread is 7%. The measured peak output power by IECAS is 110 kW at 33.88 GHz with a -3 dB bandwidth of 1.75 GHz. For a W band TE01 fundamental mode gyro-TWT design, the saturated output power is 112 kW at a frequency of 94.5 GHz with a gain of 34.28 dB and -3 dB bandwidth of about 4.1 GHz, and the saturated output power calculated with Magic code is 106.7 kW with a gain of 34.11 dB and 3 dB bandwidth of 3.9 GHz, the maximum relative errors between the theory and experiment are both about 5% for the output power and the bandwidth.