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从圆柱坐标系下Ｂｏｒｇｎｉｓ位函数的齐次标量Ｈｅｌｍｈｏｌｔｚ方程出发，引人慢波驻波概念及其场表达式，利用Ｂｏｒｇｎｉｓ位函数的边界条件及相邻子区公共界面上的场匹配条件，导出了三腔渡越管振荡器内角向均匀ＴＭ模的色散关系及场分布的解析表达式。运用该解析法求得三腔渡越管振荡器的谐振频率与实验中测得的微波频率一致，求得的场分布与数值法得到的场分布十分吻合。
Starting from the homogeneous scalar Helmholtz equation of the Borgnis bit function in the cylinder coordinate system, the concept of the slow-wave standing wave and its field expression are introduced. By using the boundary conditions of the Borgnis bit function and the field matching conditions in the common interface of adjacent sub-regions, The dispersion relation and the analytical expression of the field distribution in a uniform TM mode of the triple cavity cross tube oscillator. Using this analytical method, the resonant frequency of the triple-cavity transitional tube oscillator is consistent with the microwave frequency measured in the experiment. The obtained field distribution is in good agreement with the field distribution obtained by the numerical method.