amdx wrote:
Can someone explain how these two relate in a waveguide.
My limited understanding is, group velocity is slow near cutoff and
increases as frequency increases to almost c.
I don't know the difference between group velocity and phase velocity.
Thanks, Mike

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

Roy Lewallen wrote:
amdx wrote:
Can someone explain how these two relate in a waveguide.
My limited understanding is, group velocity is slow near cutoff and
increases as frequency increases to almost c.
I don't know the difference between group velocity and phase velocity.
Thanks, Mike

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.

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

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.

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

Correction:

Roy Lewallen wrote:

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

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