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The group velocity of a wave is the velocity with which the overall shape of the wave's amplitude (known as the envelope of the wave) propagates through space.
I have NO idea what this means! "overall shape of ...amplitude" ? I thought amplitude was a number (scalar). Even if its a vector, tensor or other function, the claim (implicit) that it has a "shape" which propagates is unhelpful gobble-de-gook.
What about talking about wave composition? or "repeating units" (as in poly-"mers")
Amplitude is indeed a scalar, but is it not necessarily constant. Consider shining a flashlight onto a wall. The brightness is high inside some circle, but outside the circle it is low and eventually vanishes. So if you were to plot the amplitude as a function of the position on the wall, it would look like some radially symmetric hill (high in the middle, low in the ends). That is what people mean when they talk about shape, except that they are talking about the shape as a function of time rather than a function of wall position. —Preceding unsigned comment added by 140.180.174.115 ( talk) 14:11, 8 December 2007 (UTC)
The article here claims that:
For light, group velocity and phase velocity are related by the formula
where:
but dispersion (optics) gives phase velocity:
and group velocity
which imply that:
Something is wrong here. i don't see why should be equal to one. Boud 13:52, 13 September 2005 (UTC)
See Talk:Phase_velocity#vg*vp = c*c. fgnievinski ( talk) 17:02, 23 May 2016 (UTC)
Leo-
For whom feels inclined, I think a section on discussing the numerous experiments that claim "faster than light" but end up measuring group velocity is warranted. It seems these keep cropping up. Cburnett 00:28, 18 August 2007 (UTC)
"However, superluminal communication is not possible in this case, since the signal velocity remains less than the speed of light."
The first paragraph culminating in this claim is illogical and self-contradictory. It amounts to predjudicing in favour of the theory of velocity-absolutism.
If the group velocity can propagate faster then light so can information and obviously so. Anyone who claims otherwise is being self-evidently idiotic.
The article starts off talking about how normal waves move. And then an inference is made that some implications don't hold when its waves of light. But the inference is arbitrary. And so the claim ought not be made. Since its fitting a square peg in a round hole in an ideologically motivated effort to support relativity.
Its an unnecessary importing of tribalism and politics into physics. —Preceding unsigned comment added by 122.148.183.191 ( talk) 02:21, August 26, 2007 (UTC)
It is you who is the self-evident idiot--obviously, the notion of a signal, as described by Brillouin in his famous book, must be a discontinuity in the light wave. And because discontinuities contain infinitely high frequency components, the discontinuity must travel at c (the speed of light in vacuum)--the material cannot respond to such high frequency content and is effectively transparent. Perhaps a little reading before posting a comment is in order, eh? —Preceding unsigned comment added by 128.112.50.49 ( talk) 01:18, 7 December 2007 (UTC)
To be honest I guess I understand the comment above in the sense that "physicists and scientists" sometimes basically copy paste arguments they have seen made by others without following whether there are any mistakes in the derivations. However, for that reason I consider that rather than politics it is the fact that the subject is hard enough to understand by itself thatleads people to make such claims. To be honest I even find hard to understand basic textbooks and I think it is due to the authors having done the same as others, basically copy paste an argument. In short, I think I also read in "Concepts of modern Physics" something about all this and then the author says that sending information via group velocity is imnpossible even though it is true that group velocity is faster than light. I will give it a look later to see what it actually says.
Although I do not have Brillouin's book to check the exact wording, it should be noted that a discontinuity in the light wave is not strictly necessary to transfer information; amplitude modulation, in fact, may be continuous, and it isn't the only possible example. In any case, "superluminal propagation" is somewhat a misnomer, since virtually all articles agree that no photon (during the experiment) travels faster than light, thus "superluminal group velocity" seems more correct and less confusing (but less used). Anyway, the natural question arises: does group velocity still have a physical meaning in these "special" experiments? If it has none, which seems to be the case, then we are ultimately talking about a superluminal mathematical abstraction, ending up with a result similar to phase velocity (which may be greater than c without generating problems). In any case, I think this part of the article should be upgraded in order to include recent information regarding this topic and, more importantly, to clarify things (going into the specific mathematical details, if needed). I know just a bit of this subject, not enough to fully write about it; someone with more knowledge in this field is needed to completely clarify things. — Preceding
unsigned comment added by
151.45.100.81 (
talk)
20:13, 9 April 2013 (UTC)
In the nice animated image, the red dot moves three times as fast as the green dot! Can someone produce a new image? (Could just change "twice" to "three times", but then it doesn't represent deep water waves!). The fact that it overtakes two green dots does not prove it goes twice as fast: it depends also on their relative densities (no. of spots in the image)! Simplifix ( talk) 08:52, 20 March 2008 (UTC)
Thanks for correcting it. It's a nice image. Simplifix ( talk) 11:19, 17 April 2008 (UTC)
I think there should be a reference for the sentence: "Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as molecules." in the section "Matter wave group velocity", because this is quite a strong statement. —Preceding unsigned comment added by Cholewa ( talk • contribs) 13:42, 17 April 2008 (UTC)
Great animation, eh? —Preceding unsigned comment added by 202.7.183.131 ( talk) 01:41, 6 October 2008 (UTC)
This article claims that the definition of group velocity is the partial derivative of angular frequency with respect to wave number. Is this actually a definition? If so, are there any sources which state this?
It seems to me that this is a useful relation, not a definition. I would expect the definition to read something like the definition given at the top of the article! -- sjl ( talk) 11:57, 16 March 2010 (UTC)
It's certainly not just a definition (although in practice it might be employed as though it were). Currently, the page on Wave sketches a general derivation of why (for a completely arbitrary wave) the velocity of an envelope works out equal to that partial derivative. This article would better explain its topic by including a section fleshing out that derivation a bit (and if possible, linking to a math page with detail on the trick involved). Cesiumfrog ( talk) 09:54, 21 March 2010 (UTC)
I removed the derivation section, since it is incorrect original research: if has narrow width, its Fourier transform is broad. Then, retaining only the leading term of the Taylor series of is very inaccurate. Vice versa, if is narrow-banded, is broad and not a wave at all. So please, add a -- preferably simpler -- mathematical description of the background of group velocity. Everybody may remove versions without adequate reliable sources. -- Crowsnest ( talk) 13:49, 15 January 2012 (UTC)
The article lists some formulas for group velocity:
For light, the refractive index n, vacuum wavelength λ0, and wavelength in the medium λ, are related by
with vp = ω/k the phase velocity. The group velocity, therefore, satisfies: |
Maybe you think there is an error? Well, here is python 2 or 3 code that checks all of these:
from __future__ import division, print_function
from math import pi
c = 3e8
def w_of_k(k):
return 5 + 6 * k + 7 * k**2 # a random function
def lam0_of_k(k):
w = w_of_k(k)
return 2*pi*c / w
def lam_of_k(k):
return 2*pi / k
def vp_of_k(k):
w = w_of_k(k)
return w / k
def n_of_k(k):
return c / vp_of_k(k)
k = 12.345 # a random value
w = w_of_k(k)
lam0 = lam0_of_k(k)
lam = lam_of_k(k)
vp = vp_of_k(k)
n = n_of_k(k)
dk = 1e-5
dw = w_of_k(k+dk) - w_of_k(k)
dlam0 = lam0_of_k(k+dk) - lam0_of_k(k)
dlam = lam_of_k(k+dk) - lam_of_k(k)
dvp = vp_of_k(k+dk) - vp_of_k(k)
dn = n_of_k(k+dk) - n_of_k(k)
print(dw / dk)
print(c / (n + w * (dn / dw)))
print(c / (n - lam0 * (dn / dlam0)))
print(vp * (1 + (lam / n) * (dn / dlam)))
print(vp - lam * (dvp / dlam))
print(vp + k * (dvp / dk))
This code will print the results of the six different formulas for vg. You can check for yourself that all six outputs approach the same limit (178.83 in this case) as dk gets smaller. If you change k, or if you change the dispersion formula, the six expressions will still all agree with each other -- I checked. (Of course I have checked the formulas analytically too, and I have checked them against references ... but I think this code above is the most fool-proof way to ensure that the formulas are free of typos.)
I hope that people do not change the formulas without double-checking them this way! -- Steve ( talk) 13:30, 26 September 2012 (UTC)
I moved some of the stuff about superluminal group velocity to another section since it's a quite messy stuff (not just superluminal, but always for lossy or gainful media). That said, the superluminal group velocity is associated with real effects, e.g., the arrival time of the peak of a pulse can be fast-forwarded to arrive sooner than light speed. I'm thinking of building a figure like http://www.sciencemag.org/content/326/5956/1074/F3.large.jpg (Figure 3 from the Boyd Science article), which illustrates nicely what is happening: we can have superluminal propagation of the peak of a pulse, however an actual signal such as a sharp leading edge cannot be made to move faster than light. -- Nanite ( talk) 10:53, 1 August 2015 (UTC)
Has anybody heard of the reverse being true?! (i.e. phase velocities traveling faster than c and group velocities [which carry the information and energy of the wave] being slower than c?) Perhaps research electromagnetic waves in plasmas where the group velocity goes to 0 but the phase velocity diverges to infinity... or plane electromagnetic wave pairs where it is possible to show superluminar phase velocities. Feel free to discuss this. Stephen. — Preceding unsigned comment added by Matter not ( talk • contribs) 23:42, 12 December 2015 (UTC)
https://yo###utu.be/CtFSovvN19g?t=4616 This professor seems to agree with me. (remove the ### from the link) — Preceding unsigned comment added by Matter not ( talk • contribs) 00:27, 13 December 2015 (UTC)
We already have Group delay and phase delay; Group velocity and phase velocity for the same reasons. fgnievinski ( talk) 17:18, 23 May 2016 (UTC)
A computer might be able to mathematically decompose the wave train into “individual wavelets of differing wavelengths traveling at different speeds,” but a person only discerns waves. I will rewrite that part of the lede to reduce the dubious content. Constant314 ( talk) 21:07, 22 April 2021 (UTC)