Abstract: First the geometrical factor is derived, then the integrodifferential equation of the instantaneous current and the beam kinetic energy is given, and the coupling coefficient of the gap of the coaxial cavity is obtained. With electron beam voltage of 600 kV, electron beam of 5 kA, and the gap voltage of the input cavity of 4.6, 32.7, 189 kV, and the number of the multi beam of 16, the fundamental harmonic current modulation coefficient versus distance z is calculated by the three dimensional particle-in-cell (PIC) simulation and the code of nonlinear theory respectively, the result of the theory and that of the simulation agree remarkably well. With gap voltage of 4.6 kV, the number of the multi beam of 8, 12, 16, and the ratio of the radius of solid beam and the channel of single electron beam of 6/15, 8/15, 10/15, the fundamental harmonic current modulation coefficient versus distance z is calculated by the code of nonlinear theory. With gap voltage of 4.6 kV, the fundamental harmonic current modulation coefficient and the kinetic versus the normalized time θ is calculated, normalization constant N(z) and the n-th harmonic current modulation coefficient C_n (n=1, 2, 4) versus distance z are calculated.