Lower and upper bounds of cutoff frequencies in metallic waveguides

It is shown how the upper and lower bounds of the cutoff frequencies of TE and TM modes in many waveguides bounded by metallic lossless walls and which may be hollow or comprise one or more inner conductors can be computed using two independent methods. The methods are applicable whenever the cross-section of the waveguide can be split up into several regions bounded by lines having a fixed coordinate. The theory is illustrated with reference to a rectangular coaxial line