A variational integral for the propagation coefficient of a cylindrical waveguide with imperfectly conducting walls

A variational integral for the complex propagation coefficient of a cylindrical waveguide having arbitrary cross-section and imperfectly conducting walls is derived. This integral is used, in particular, to determine the nature of the modes that may be propagated in a lossy rectangular waveguide. In certain conditions these stable modes are well described as simple linear combinations of a TEmn and TMmn mode. It is proposed that such combinations be called quasi-stable modes. The variational formulation shows that the two quasi-stable modes associated with the mode numbers m, n are the linear combinations that produce, respectively, maximum and minimum values of attenuation coefficient when calculated using the power-loss method.