推导了带能散度的波导自由电子激光高、低两共振频率支的非定态增益解析公式．给出了近似显式公式，与数值模拟结果符合很好。从所得增益脉冲分布．估算了克服呆滞效应所需腔长失谐值，在时空框架内解释了自锁模机制。 The analytical solution of unsteady gain for the high and low resonance frequencies of waveguide free-electron laser with divergence was deduced. The approximate explicit formula is given, which is in good agreement with the numerical simulation results. Gain pulse distribution derived from. The detuning value of the cavity length needed to overcome the sluggish effect is estimated, and the self-mode-locking mechanism is explained within the space-time frame.