Velocity of propagation in lossy media

[pboric]

regarding plane waves theory and their properties, for lossy media (conductors) in different books appear a formula that represent alfa y beta constants in the followig way:

alfa = beta = root square( pi*frequency*mu*sigma) valid for good
conductors (high loss material)

I have checked this formula in different books and it works well, and
it's used further to calculate skin effect as well

My question arises from one conclusion derived from the formula, that I
found in just a specific book (David Cheng's Fundamentals of
engineering electromagnetics).
In that book the author says:
velocity of propagation = omega / beta
For copper (good conductor):
sigma = 5.8 * 10 exp 7
mu = 4 *pi *10 exp -7
and therefore v = 720 m/sec. @ f = 3 Mhz.

so the velocity of propagation is << c

I am confused with the final result, because I've checked the formula, and the math in the example is right, but I know that in a copper transmission line, the velocity of propagation is about 2/3 c = 200,000 km/sec.
The formula appears in different books, but the specific example just
appears in Cheng's book
I think the application for the example is valid in a different situation,
but I can't figure out which

 

[marcusl]

I think beta here is what other books refer to as the wavenumber
k=2*pi/lambda
Since omega=2*pi*f, the formula is actually the familiar one
c=lambda*f

 

[pboric]

in lossy media, beta is the phase constant, and alfa is the attenuation constant.
the complex propagation constant gamma is defined = alfa + j * beta
in this case the wavenumber k is complex as well, and it is related to gamma by
gamma = j *k

for lossless media, alfa = 0 and there are no losses at all, just phase change (beta) and in that special case (alfa = 0), we get:
gamma = j*beta = j*k
so beta =k

 

[Meir Achuz]

Copper is not a "lossy" conductor".
In a lossy conductor the wave goes (almost) nowhere slowly.
That is why it is called skin depth.