Talk:Group velocity

"overall shape of ...amplitude"

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)

I think it should be "overall shape of ...amplitudes".--LungZeno (talk) 23:13, 18 January 2009 (UTC)

inconsistency with dispersion

The article here claims that:

For light, group velocity and phase velocity are related by the formula

$v_{g}v_{p}=c^{2}\,$

where:

v_p is the phase velocity

c is the speed of light in a vacuum.

but dispersion (optics) gives phase velocity:

$v={\frac {c}{n}}$

and group velocity

$v_{g}=c\left[n-\lambda {\frac {dn}{d\lambda }}\right]^{-1}$.

which imply that:

$v_{p}v_{g}=c^{2}\left[n(n-\lambda {\frac {dn}{d\lambda }})\right]^{-1}\not =c^{2}$.

Something is wrong here. i don't see why $\left[n(n-\lambda {\frac {dn}{d\lambda }})\right]$ should be equal to one. Boud 13:52, 13 September 2005 (UTC)

• I agree, it doesn't seem right. I've removed that equation, pending a cite. --Bob Mellish 22:05, 4 October 2005 (UTC)

• That formula is specific to (hollow metal-pipe) waveguides. --catslash 20:56, 3 September 2007 (UTC)

See Talk:Phase_velocity#vg*vp = c*c. fgnievinski (talk) 17:02, 23 May 2016 (UTC)

picture of group velocity

Leo-

Can someone please post a picture or a trailer where the group velocity is different from the phase velocity, like how the travelling wave would look in the transverse point of view?

Try http://krypton.mnsu.edu/~7364eb/Math113/groupvelocity.html unsigned

I tried to make a gif, but its a bit large. I think i need to cut out half the frames.. but i'm too lazy right now. Fresheneesz 03:34, 12 September 2006 (UTC)

"Faster than light"

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)

the animated graphic

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)

I see what you mean, and will work on it. Crowsnest (talk) 09:00, 20 March 2008 (UTC)

A new animation has been uploaded, to correct the problem. Thanks for pointing the error out! Crowsnest (talk) 09:19, 20 March 2008 (UTC)

Thanks for correcting it. It's a nice image. Simplifix (talk) 11:19, 17 April 2008 (UTC)

Thanks so much, it is a wonderful picture... it totally cleared every thing for me! ;) but why it should be exactly twice in deep water??? -sona-, 09/14/08

Missing reference for measurement of group velocity of matter wave functions

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)

I love the animation

Great animation, eh? —Preceding unsigned comment added by 202.7.183.131 (talk) 01:41, 6 October 2008 (UTC)

Definition, or just a formula?

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)

I reckon it's fairly common to use $\scriptstyle {\partial \omega /\partial k}$ as a definition (it's the first definition given in IEEE Dictionary of Electrical and Electronics Terms 6th ed. (though that's not a work I like much)). It follows on quite naturally from defining the phase velocity as $\scriptstyle {\omega /k}$, and is similar to defining the velocity of a particle as $\scriptstyle {\partial s/\partial t}$.

Some authors do start by defining it as the velocity of a wave packet, but this requires some explanations and caveats to make it precise, since wave packets of finite length will generally disperse and so will not have a definite velocity. Defining it as the velocity of amplitude modulation of a wave in the low-frequency limit of the modulating signal, would probably be sufficiently precise, but not very enlightening - and I haven't seen such a definition used. --catslash (talk) 22:41, 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)

Removed "Derivation section", for the time being

I removed the derivation section, since it is incorrect original research: if $\alpha (x)=\exp(-x^{2}/2\sigma ^{2})$ has narrow width, its Fourier transform $A(k)$ is broad. Then, retaining only the leading term of the Taylor series of $A(k)$ is very inaccurate. Vice versa, if $A(k)$ is narrow-banded, $\alpha (x)$ 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 error was introduced two days ago: [1]. I am putting back the section, with a citation (Griffiths quantum textbook) and with the error fixed.

I am familiar with only one other derivation ("the peak occurs where all the fourier components are in phase..." See Hecht optics textbook for example.) This has one line of algebra and no calculus, but relies on a very good intuitive background understanding of Fourier analysis. I don't think it's worth putting that one in the article. I like the derivation you deleted, I don't think you'll find a simpler one but you're welcome to look. :-) --Steve (talk) 15:50, 15 January 2012 (UTC)

It looks much better now. The other derivation you mention is the one based on the method of stationary phase. As far as I can recall (I don't have the book at hand) James Lighthill mentions at least 4 methods in his book "Waves in fluids". The simpler ones are the one for a bi-chromatic wave (two sines of slightly different frequency $\Delta \omega$ and wavenumber $\Delta k$), so $v_{g}=\Delta \omega /\Delta k.$

And the one defining angular frequency and wave number as the derivatives of the wave phase $\theta (x,t)$: $\alpha (x,t)=\Re \left\{a(x,t)\,\exp \left(-i\theta (x,t)\right)\right\}$ for a narrow banded signal $\alpha (x,t)$ and real-valued amplitude $a(x,t).$ Defining $\omega (x,t)=-\partial \theta /\partial t$ and $k(x,t)=+\partial \theta /\partial x,$ then $\partial k/\partial t+\partial \omega /\partial x=0$ (known as the conservation of wave crests). If $\omega$ and $k$ are related through the dispersion relation $\omega =\Omega (k)$, then

${\frac {\partial k}{\partial t}}+v_{g}\,{\frac {\partial k}{\partial x}}=0,$

with $v_{g}=\Omega '(k).$ -- Crowsnest (talk) 21:39, 15 January 2012 (UTC)

Formulas for group velocity

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 $\lambda _{0}={\frac {2\pi c}{\omega }},\;\;\lambda ={\frac {2\pi }{k}}={\frac {2\pi v_{p}}{\omega }},\;\;n={\frac {c}{v_{p}}}={\frac {\lambda _{0}}{\lambda }},$ with vp = ω/k the phase velocity. The group velocity, therefore, satisfies: $v_{g}={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}={\frac {c}{n-\lambda _{0}{\frac {\partial n}{\partial \lambda _{0}}}}}=v_{p}\left(1+{\frac {\lambda }{n}}{\frac {\partial n}{\partial \lambda }}\right)=v_{p}-\lambda {\frac {\partial v_{p}}{\partial \lambda }}=v_{p}+k{\frac {\partial v_{p}}{\partial k}}.$

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)

superluminal pulses

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)

Ooh, I would love to make an animation of what superluminal group velocity looks like. I put it on my to-do list, but it may be a long time before I get around to it :-D --Steve (talk) 12:36, 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)

@Matter not: exactly true! I love Lewin's lectures. :) Indeed, some dispersion relations can hit k=0 at nonzero frequency (e.g., with optical phonons and plasma; this is basically guaranteed to happen when permittivity passes through zero in any material). The group velocity there is zero yet phase velocity is infinite. It is perhaps a good idea to compare group vs phase velocities in the article (I thought there was already such a comparison but somehow it is missing?) --Nanite (talk) 06:43, 13 December 2015 (UTC)

@Nanite: Thanks for the reply. So you agree the article (which suggests the superluminal group velocities existing) is certainly incorrect and should be changed? The group velocity carries both the information and energy of the wave and cannot exceed c. Stephen

The section in the article is correct, clear, and referenced: Superluminal group velocity is possible. Information is limited by the front velocity which is less then c. As for energy flow, it flows at the group velocity only when there is not a lot of gain or loss. Superluminal group velocity only happens when there is a lot of gain or loss. --Steve (talk) 02:06, 16 December 2015 (UTC)

Merge

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)

Looks very dangerous to me, but if you are prepared to do the Herculean task satisfactorily, try it on your sandbox page and invite editors there, to vet it. I have, and you must have, witnessed merging disasters, and they are really hard to avoid. As a rule, WP readers go for short, self-contained pithy answers, not in-depth expositions of things they are not inclined to read seriously about, e.g. in books. There is redundant reduplications of the connection between the two (phase/group) in the respective articles, but this is good, not bad, as it helps the reader who is too lazy or inflexible to read and appreciate both. So, jamming more, rather than fewer, issues here would probably detract from the utility of the page, but of course, if you have gumption and time to do it, by all means, give it a try. Cuzkatzimhut (talk) 18:43, 23 May 2016 (UTC)

• Oppose. They are two different concepts, they don't exist only in opposition to each other. There is a lot to say about phase velocity that is not relevant to group velocity and vice-versa. Now, the articles are too short right now. They don't discuss every important aspect of the topics. If they did, it would be obvious to everyone that they shouldn't be merged, because we would see all the aspects of each topic that is not relevant to the other. If the articles were merged, it would basically prevent somebody from adding those aspects in the future. --Steve (talk) 00:05, 24 May 2016 (UTC)

• Weak oppose although different they're related and share media. Widefox; talk 01:35, 8 September 2016 (UTC)

Expanding water waves in a pond

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)