If I make a coaxial waveguide from the gap between two long, perfectly conducting, coaxial cylindrical surfaces, (e.g. perfect "coax" cable), then if I apply charges of $+q$ and $-q$ to the inner and outer conductors, the electric field in the gap will be transverse, lying completely within a perpendicular cross-section, and outside of the outer conductor the electric field would be zero.
If I then ran currents of $+I$ and and $-I$ in the inner and outer conductor, the magnetic field in the gap would also be transverse (in-plane) and outside would also be zero.
I believe that this answer says that this is sufficient to then conclude that at least the lowest order mode of AC (alternating current) propagation would be TEM (Transverse Electric and Magnetic fields). And I believe this could be shown by adding the appropriate $\exp(i(\omega t-kz))$ longitudinal terms and showing that it is still a solution of Maxwell's equations.
But I'm having difficulty understand if the answer is saying something more generalized than this? Is there something more there which is useful to the behavior waveguides made from generalized coaxial pairs than just "if the fields are each transverse in DC (direct current) then they'll also be transverse for AC"?
Please try to keep your answer somewhere near the level at which it is asked as much as possible. Thanks!