The mystery of group velocity

These two velocities are commonly used to describe the velocity of electromagnetic waves propagating on a transmission line.

Phase velocity is the propagation velocity of an equiphase wave surface of a single sinusoidal wave along a transmission line, and group velocity is the propagation velocity of the envelope of two sinusoidal waves with a slight difference in frequency superimposed. These velocities can be described with secondary constants of the transmission line as follows.

  Vp = w/b                                              (1)
  Vg = dw/db                                            (2)
	Vp = Phase velocity (m/s)
	Vg = Group velocity (m/s)
	w = 2*PI*f = Angular velocity  (rad/m)
	PI = 3.14159265.. (pi)
	f = Frequency (Hz)
	b = Phase constant of transmission line (rad/m)
	dw, db = differentials of w and b respectively

For a system with no dispersion, in other terms is lossless or distortionless, and the phase constant b is proportional to the frequency, the phase and group velocities match and become a constant independent of the frequency.

However, most textbooks (footnote 1) claim that

However, we discuss whether this is true or not.

We consider the group velocity in a very general transmission line to explain why this question arises.

The phase constant in a transmission line described by Heaviside's telegraphers' equation is as follows.

  b = sqrt(0.5*(sqrt((R^2+(w*L)^2)*(G^2+(w*C)^2)) + (w^2*L*C - RG)))    (3)
	R = Resistance of line (Ohm/m)
	L = Inductance of line (H/m)
	G = Conductance of line (S/m)
	C = Capacitance of line (F/m)
The frequency dependence of b is definitely not simple, but here we consider the high frequency range, or w*L/R >> 1 and w*C/G >> 1, which is the most practical regime. In this case, the approximation
   b ~ w*sqrt(L*C)*(1 + (R/2/w/L - G/2/w/C)^2)                          (4)
holds, and therefore the phase velocity becomes
  Vp = w/b ~ (1/sqrt(L*C)*(1 - (1/2)*(R/2/w/L - G/1/w/C)^2)             (5)
which is smaller than 1/sqrt(L*C) and approaches 1/sqrt(L*C) with an increase in frequency.

In a near-perfect transmission line that is non-magnetic and where the relative dielectric constant is 1,

  1/sqrt(L*C) = 1/sqrt(u0*e0) = c
	u0 = permeability of vacuum
	e0 = dielectric constant of vacuum
	c = speed of light of vacuum.
Since 1/sqrt(L*C) is the speed of light, phase velocity does not exceed the speed of light.

Next, we obtain the group velocity. It is cumbersome to give w as a function of b, and therefore we use the differential of the inverse function to obtain dw/db from db/dw. With this strategy, since

  db/dw  ~ sqrt(L*C)(1 - (1/2)*(R/2/w/L - G/1/w/C)^2)                   (6)
the group velocity is calculated as
  Vg = 1/(db/dw) ~ (1/sqrt(L*C)*(1 + (1/2)*(R/2/w/L - G/1/w/C)^2)       (7)
This is larger than 1/sqrt(L*C) and approaches 1/sqrt(L*C) with increase in frequency.

Therefore, in a near-perfect transmission line, at high frequencies, the group velocity exceeds the speed of light.

In summary, if group velocity is the propagation speed of energy and information, the special relativity theory can be disproved by using a very simple air-insulated line, but is this really the discovery of the century ?

Footnote 1

There are many, but here are some examples:

  Susumi Miwa,- High frequency electromagnetism
	(Tokyo Denki University Press) ISBN4--501-10530-5
  Page 139 - "Group velocity is known as the propagation velocity of signals and energy".

  Eiji Iwahashi - Theory on transmission Theory
	(Tokai University Press) ISBN4-486-01292-5
  Page  65- "Group velocity is the propagation velocity of information".

  Katsuhiro Tanaka,- Calculation methods on electromagnetism
	(Nihon Rikougaku Shuppankai) ISBN4-89019-129-1
  Page 175 - "The velocity that wave energy propagates in space is called group velocity".

  Shizuo Matsuda, - Basic theory of waveguide transmission 
	(Tokai University Press) ISBN4-486-01178-3
  Page 37 - "The velocity that wave energy or signals propagates in space is called group velocity".

  A.P. French, - MIT Physics Vibration and waves
	(Baifukan) ISBN4-563-02173-3, (Translated by Jun Hiramatsu, Seiichi Anfuku)
  Page 210 - "We also find that transportation of energy by waves happens at group velocity."

  Shuichi Takamura, - Introduction to electromagnetism for science and engineering 
	(Morikita Press) ISBN4-627-73441-7
  Page 161 - "In general, (phase velocity is) different from the velocity, namely group velocity,
	which energy in waves are carried.
Kouichi Hirabayashi, (C) 2006

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