The high frequency(HF) characteristics of the split cavity oscillator with three cavities are studied.Starting from homogeneous scalar Helmholtz's equations associated with Borgnis potential function in a cylindrical coordinate system

and based on the standing wave concept of slow-wave introduced in the paper

the analytic expressions of the dispersion relation and field distribution for azimuthally symmetric transverse magnetic modes in the split cavity oscillator with three cavities are derived

by using boundary conditions for Borgnis potential function in conjunction with field matching conditions at the common interface between the adjacent subregions.The resonance frequency of the split cavity oscillator with three cavities calculated by this analytic method is compared with that measured in experiments and it is found that they are in agreement quite well.