A linear analysis for a gyrotron backward wave oscillator operating at a general transverse electric (TEln) mode of a cylindrical waveguide (slotted or uniform cross section) has been developed using a Maxwell–Vlasov equation under the tenuous beam approximation. The equation is solved by Laplace transformation to allow easy insertion of the boundary values. The start‐oscillations conditions are determined from an interference of the waveguide and beam modes with appropriate amplitudes and phases. The theory is used to determine the design parameters for stable operation of a second cyclotron harmonic gyropeniotron amplifier at 35 GHz in the &pgr; mode of a six vane magnetron‐type circuit with a 70 kV, 3 A beam.