The split cavity oscillator (SCO) is a resonant cavity device which can be used to modulate the current of a relativistic electron beam. The modulated electron beam can be used to produce high‐power microwaves. The SCO consists of a cylindrical cavity resonator which has been divided (split) into two identical regions (cavities) by a thin conducting foil (or screen) whose radius is less than the inner radius of the cylinder. The resulting gap between the foil perimeter and inner cylinder wall forms an annular slot which couples the cavities electromagnetically. The SCO supports standard transverse magnetic (TM) modes, in addition to split cavity TM modes. The latter will interact unstably with an electron beam, thereby producing a high degree of current modulation in a short distance. For the purpose of characterizing split cavity TM modes, and developing a design tool for SCOs, a cold dispersion relation (no electron beam) is derived by solving Maxwell’s equations for the fields in an equivalent periodic structure. The dispersion relation and field solutions are employed to numerically calculate eigenfrequencies and eigenfunctions for the SCO. The latter are used to develop predictive formula for the SCO oscillation frequency, and to gain some understanding of the interaction that takes place when an electron beam is present.