Group Velocity and Velocity Factor

Phase velocity is the velocity of a constant phase point. For example, if you look at a point where the voltage or current wave crosses zero going in the positive voltage or current direction, it moves down the waveguide at the phase velocity. In a waveguide, the phase velocity is always greater than the speed of light c. It approaches c at very high frequency, and increases without bound as cutoff is approached.

The group velocity is the speed at which information can be moved. In other words, a change in the signal (e.g., turning it on or off or changing its amplitude) propagates at the group velocity. In a waveguide, the group velocity approaches c at very high frequency and 0 at cutoff.

Mathematically, vp = c/sqrt(1 - (f/fc)^2) vg = c * sqrt(1 - (f/fc)^2)

where vp is the phase velocity, vg is the group velocity, f is the frequency, and fc is the cutoff frequency. These equations are valid for TE and TM modes in hollow waveguides.

A medium in which the phase velocity varies with frequency is called a dispersive medium, and all hollow waveguides are in this category. Phase and group velocities are the same in non-dispersive media such as coaxial cable.

Kraus uses a caterpillar as an example: The humps on the caterpillar's back move at the phase velocity, but the caterpillar moves at the group velocity.

Roy Lewallen, W7EL

That's true for metal waveguides. Dielectric guides always have at least one mode at all frequencies, and vp

This is true for narrowband modulation, because it assumes that d(omega)/dk is constant. It also requires the approximation that the pulse shape propagates unchanged, which means that it works only for short distances. A very long guide with linear dispersion will produce the Fourier transform of the input. Various grandstanding academics have published papers claiming group velocities higher than c, but it always turns out to violate one or other of these conditions.

Cheers,

Phil Hobbs

Group velocity wants to describe a pulse containing more than one photon frequenccy.

In dispersive media the group velocity is a function of frequency of the photons that form a physical signal. So neighboured frequencies have a little different velocities. That is what is behind.

So group velocity as one uses the terminus in hard physical theory is nothing else as the true physical velocity at a certain frequency of the photonic carrier resp. field.

Group velocities of wave packets is something apart from that. If you have a carrier that containes a spectrum of frequencies it describes the broadening of the signal due to different carrier frequencies in which have different velocities. This definition is therefore unsharp and has only qualitative picturesque meaning !

So group velocity in a sharp sense is just the real velocity which with the field and the photons in move at and only at a certain frequency.

Josef Matz

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I was in a hurry this morning and didn't ask my main question. I think at this point I understand different frequencies travel at different speeds. Group Velocity vs Velocity Factor what is the difference? If vg = c * sqrt(1 - (f/fc)^2) hmm, maybe I should tell what I think I know. (I'm way over my head on this subject). If I generate a spark ( many frequencies) all these frequencies combine to make a waveshape, as the wave travels down the waveguide the waveshape changes because different frequencies are traveling at different speeds? Correct me as needed. So is 'group velocity' the velocity the peak of the signal as it travels down the waveguide? Forgive my ignorance but the formula vg = c * sqrt(1 - (f/fc)^2) doesn't work for me. (1-(f/fc)^2) is negative and I can't get the sqrt of a negative. What did I miss? Thanks, Mike

--- If you threw a stone into a pond, the group velocity of the wave generated would be how fast the ripples moved away from the point where the stone hit the water, while the phase velocity would be how long it took for a ripple peak to go from a peak to a trough and then back to a peak.

Likening it to a pair of scissors closing while cutting through a sheet of paper, the group velocity would be the speed of the ends of the blades approaching each other, while the phase velocity would be the speed of propagation of the cut as the blades sliced the paper apart.

-- JF

That should be:

vp = c/sqrt(1 - (fc/f)^2) vg = c * sqrt(1 - (fc/f)^2)

I apologize for the error.

Roy Lewallen, W7EL

Velocity factor is the ratio of the velocity of waves in the medium to the velocity of the speed of light in a vacuum. That is, VF = v/c, where v is the velocity in the medium, c is the speed of light in a vacuum, and VF is the velocity factor. I'm more familiar with its use in non-dispersive media, but there's no reason it couldn't also be used for dispersive media. If it is, then group velocity and phase velocity would each have a different velocity factor (and it would be greater than one for the phase velocity in a hollow waveguide), and it would also be a function of frequency.

I made an error and reversed f and fc in the equations. I've posted a correction.

Yes, that's correct. A dispersive medium distorts any waveshape except a pure sine wave. People used to working in the frequency domain often forget the vital importance of phase response in preserving waveshape integrity. I learned the hard way that microstrip line is dispersive when designing a delay line loss compensator for a high-speed sampling oscilloscope used for TDR, where very good waveshape integrity is essential.

Assuming that by "signal" you mean something other than a sine wave, it becomes a matter of definition depending on the application. When dealing with step functions, for example, the 50% point on the step is commonly used.

Nothing, it was I that missed my error in reversing f and fc. My apology. I've posted a correction.

Roy Lewallen, W7EL

Ok, guys you helped me get a better understanding without the math, that I wouldn't be able to do. The end point of this information is to initiate a spark pulse at one end of a waveguide and pickup the pulse two times. So I'll have two pickups separated by distance. The time between the two received pulses is distance times the velocity of the waveform. Hence all my questions about vf, vg and vp. Any added thoughts are appreciated. Thanks, Mike p.s. I have someone that can do all the maths, I just want a better understanding of what's happening.

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I have seen that at Nimtz too. He modifies the dispersion relation. Since i have not seen that before i just can say that light having this dispersion relation does not obey the wave equation but a more general equation which is called Klein - Gordon equation. So it is not elliptic polarized light. Such light can also move in homogene waves but with velocities less than c and be frozen at f =fc. For f >fc you get inhomogene waves and those can tunnel almost instant. Thats proof too.

But whats fc ? I dint know.

So i am going after that, my trial to find out something is that it light where the ellipse of elliptic polrized light rotates or in other words the field takes a screw. But ok i am not shure about that last now.

Josef