Assuming a vacuum dielectric (for which air is a good approximation):

E/H = 377 ohms only in TEM mode waveguides. Can you explain under what
circumstances a TEM wave can have any other value?

At frequencies above cutoff in a TM mode waveguide, the intrinsic or
wave impedance (E/H) varies with frequency from zero at cutoff to 377
ohms at infinite frequency. Below cutoff the intrinsic impedance is
purely imaginary (reactive).

At frequencies above cutoff in a TE mode waveguide, the intrinsic
impedance varies with frequency from infinite at cutoff to 377 ohms at
infinite frequency. Below cutoff it's purely imaginary.

A good reference is Johnk, _Engineering Electromagnetic Fields and Waves_

Roy Lewallen, W7EL

Apterix80 wrote:
If possible i wish to know if in a waveguide it is E/H = Z=377
Ohm ,or ,as i suspect , the impedance migh be may quite very
different from such value , in a manner that the mode TEM can have
it at least E/H= 10000 Ohm too.

Thanks