Calculating attenuation in waveguides below cut-off

The author derives the attenuation in a circular waveguide below cut-off with a finite metal conductivity for an arbitrary mode number. There are several potentially confusing points in this calculation particularly relating to the choice of signs, the fine details of the Bessel function behaviour and the definition of attenuation itself. A full derivation from first principles is presented during which any potential sources of confusion in the calculations are highlighted and clarified. Various closed-form approximations for the attenuation are discussed and are compared with full numerical calculations. The author concludes that errors introduced by these approximations can be quite significant (depending on the mode number) and recommends solving the transcendental equation directly for the attenuation rate.