Resonance-frequency behaviour of a cavity with an evanescent section

The resonance frequency fr of a waveguide cavity, with an evanescent air gap of length d and symmetrically loaded with dielectric (length t), has two distinct types of behaviour that depend on t. If for d = 0, fr0 ¿ fr0 > fc0, where fca is the cutoff frequency of the air section, fr decreases for increasing d and approaches fca. However, if fr0 < fca, fr increases and approaches a constant value that depends on t.