I have posted this in some other physics forum, however, I have not yet gotten any definite answer. Hope some of you can help. I have been trying to understand why in a hollow metal tube, the Maxwell equation admit wave solution, and that they are always TE or TM. In all of the text books I read, they all say that the (E,B) solution for waveguide are: E(x,y,z,t)=E(x,y)exp(j(kz-wt)); and B(x,y,z,t)=B(x,y)exp(j(kz-wt)) (assuming the wave propagate in z-direction) While I totally agree that TM and TE waves satisfy the Maxwell Equation and the boundary conditions for waveguide in hollow perfect conductor, I have a few questions: 1) Each one of the TE and TM mode is a solution, but are they THE solutions?Can there be other form of solution differing from TE or TM mode? So far, in all the derivation I have seen, the TE/TM wave solution is resulted because it is assumed that E and B has the form E(x,y,z,t)=E(x,y)exp(j(kz-wt)) and B(x,y,z,t)=B(x,y)exp(j(kz-wt)). Why is it that apart from the sinusoidal variation, E and B only depends on x and y, but not on z? (i.e. why isn't E(x,y,z,t)=E(x,y,z)exp(j(kz-wt)) with the z dependency also in E(x,y,z)?) What will happen if I send out a very complex EM wave composed of a continuous spectrum of frequency inside a hollow tube? Will the wave traveling inside the waveguide necessarily become one of the TE or TM mode? 2) Why is the wave always TE or TM? I have seen a proof (in Griffiths) that TEM wave solution is impossible in hollow tube waveguide, but sending out a very narrow beam of TEM wave (e.g. visible light) into the waveguide parallel to the tube (i.e. sending it in the z direction) doesn't seem to violate any physics, and it seems to constitute a valid solution to the Maxwell equation. So, what is the gap in my reasoning here? Or is there any gap (hidden assumptions) in the proof that TEM wave is impossible? Please point out. Thank you.