In a waveguide the phase velocity is not the energy velocity; this last one is always less than of speed of light. Therefor the physical low is always satisfied!

Best regards

In a waveguide the phase velocity is not the energy velocity; this last one is always less than of speed of light. Therefor the physical low is always satisfied!

Best regards

Actually neither the phase velocity nor the group velocity are good quantities to check whether the theory is causal or not. It only matters the limit of the dispersion relation in the high wave number regime. One can see this in a theorem in Landau's book on Electrodynamics of Continuous Media. 

Dear Oscar,

first, i apologize for the typo errors in my previous comment. However, I talked of energy speed; It was not casual. When it is not well- expressed by the eq. of the group velocity, the expressed concept does not change and the causality is satisfied.

Best Regards

Phase velocity is frequency times wavelength. In cylindrical waveguide wavelength at any frequency above cut-off is always higher than wavelength in vacuum thus phase velocity in waveguide is always higher than speed of light in vacuum. But phase velocity has no physical meaning. Waves travel with group velocity. You may look at these two papers to find out more detailed explanations.

https://www.researchgate.net/publication/3304391_How_fast_can_radio_messages_be_transmitted

https://www.researchgate.net/publication/236900204_Group_velocity

Regards

Article How Fast Can Radio Messages Be Transmitted?

Data Group velocity

Group velocity is a good concept, much more accurate than phase velocity, but I think the ultimate answer I think it is Landau-Lifshitz theorem on causality. The more I look at it the more beautiful I see it.

Regards, 

Group velocity inside a waveguide does not exceed the speed of light that I am aware of, but phase velocity certainly does.  Phase velocity is the speed that a portion of a wave with constant phase progresses over time.  If you have a single plane wave, this is simply the speed of the wave.  Now imagine you have two co-propagating waves interfering with each other at a slight angle apart from each other.  The interference of the two propagates a little faster than either of the waves.  Lastly, imagine two waves travelling in almost opposite directions, but off of 180 degrees by a small angle.  The speed of their interference will be incredibly fast and can easily exceed the speed of light.

You can analyze waveguides from a simple ray tracing perspective.  The fundamental mode is composed of two plane waves at a slight angle apart from each other bouncing off the walls of the waveguide.  Higher order modes bounce with a larger angle between them, sothe phase velocity of the higher order modes increases and can easily exceed the speed of light.  Overall, they are travelling slower due to the bounces, but their interference pattern propagates very fast.

The thing to keep in mind is that nothing is travelling faster than light other than the mathematically constructed interference pattern between two waves.  I know my answer was brief, but hopefully it makes some qualitative sense.

Phase velocity and group velocity are concepts within the scope of narrowband signals. When studying the broadband problem, only the propagation speed in the medium arises. For details see the article E. O. Schultz-DuBois "Sommerfeld Pre- and Postcursors in the contexts of waveguide transients" IEEE MTT, Vol 18, No. 8, pp. 455-460 August 1970.

Another way is to look at a pulse propagation in a waveguide. The pulse envelope moves at the group velocity, which is certainly slower than the speed of light. We argue that we need a pulse or something similar to carry information and thus the information is not moving faster than the speed of light.