We present the application of the recent physics-conforming COOL method [2,4] to the eigenvalue problem of a cylindrical waveguide filled with unmagnetized plasma. Using the Fourier transform only along the waveguide and not in poloidal direction, this is a relevant test case for a numerical discretization method in two dimensions (radial and poloidal). Analytically, the frequency spectrum consists of discrete electromagnetic parts and, depending on the electron density profile of the plasma, of infinitely degenerate and/or continuous, essentially electrostatic parts. If the plasma is absent, the latter reduces to the infinitely degenerate zero eigenvalue of electrostatics. A good discretization method for the Maxwell equations must reproduce these properties. It is shown here that the COOL method meets this demand properly and to very high precision.