# Talk:Dispersion relation

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## energy change per momentum change as defining principle for dispersion relations

The use of ${\displaystyle \partial {E}/\partial {p}}$ as a kickoff defining principle for wave dispersion seems narrowly applicable to energy and momenta of waves. As I understand it, dispersion refers to the separation of wave speeds according to fourier time-frequency. Waves are considered as longitudinal (e.g. sound, traffic) or transverse particle motion (e.g. water wave, seismic, string vibration, or beam bending), EM, or probability function. By referring to the separation of propagation speeds as dispersion allows directly connect dispersion to the dispersive PDE in maths, without need to formulate energy or momentum. The connection to energy and momentum needs to be given context where is can be applied: wave energy (only??). Perhaps the connection to ${\displaystyle \mathrm {d} \omega /\mathrm {d} k=\partial {E}/\partial {p}}$ should be moved further into the article.

The first example of kinetic energy of particles needs better specification. Are they identical non-interacting particles that travel along one direction, or is there one classical particle, and what of external forces? Does it really follow from ${\displaystyle E=p^{2}/2m}$ that the disperion relation for ${\displaystyle v=\mathrm {d} \omega /\mathrm {d} k}$ is quadratic? Michael J White 19:42, 28 January 2007 (UTC)

A long time ago, this article was a redirect to Group velocity, which is sufficient if you're only talking in terms of travelling waves. However in that case it is entirely separate with the usage in materials physics, for example in calculating the band structure of materials. For example, the effective mass of electrons in a conductors is directly calculated from the dispersion relation, but you would never figure that out from the Group velocity article, with reason. I wanted this article to give a basic definition with links to different articles for a more in-depth treatment.
As for the 'quadratic relation', I always worked with the energy-momentum relation ${\displaystyle E=p^{2}/2m}$ 'as being the' dispersion relation, but it is possible that some people call its derivative the dispersion relation. If you're sure, feel free to change it.UnHoly 08:35, 29 January 2007 (UTC)

## references and examples

Given the philosophy above of "basic definitions with links to more in-depth articles", I don't think it's incumbent on us to cite a lot of references here. I would be nice to see some primary references, as they get uncovered.

In my notes I have a bunch of other fun and simple dispersion relations, like that of an ideal string wave ω=k Sqrt[T/μ], a deep water wave ω=Sqrt[gk], and a de Broglie free matter wave. Would these also be nice to create short sections on? If so, we should probably put them in as subsections to a section specifically on "dispersion relation examples". Thermochap (talk) 14:37, 19 March 2008 (UTC)

As it is at the moment, the introduction of this article is gibberish to me. It says that the dispersion relation is the relation between the energy and the momentum of a system. And an example of the kinetic energy of massive particles, expressed in terms of momentum squared is given. Why is this called a dispersion relation?
Further it talks about particles instead of waves, which is also confusing in my opinion. Or is this article on (a specific set of) dispersion relations in quantum mechanics?
Frequency dispersion, e.g. as shown in for instance dispersion (optics) or dispersion (water waves) is clear, showing that components with different wavelengths travel at different phase speeds, so they disperse.
Further, "wave momentum" in e.g. water waves or acoustics is a tricky subject, about which a lot misunderstandings exist, see e.g. M. E. McIntyre (1981). "On the 'wave momentum' myth". Journal of Fluid Mechanics. 106: 331–347. doi:10.1017/S0022112081001626.. So, while it may be appropiate to define the dispersion relation in terms of momentum in the given example of massive particles, I do not like it as an introduction to a general "dispersion relation" article.
So, apart from examples for different dispersive phenomena, which would be nice, I would very much like the introduction of this article to be clarified. Crowsnest (talk) 16:16, 19 March 2008 (UTC)
Having seen the previous notes, I have already been tempted to clarify. The basic idea is that energy is generally proportional to ω while momentum is proportional to k. Hence E versus p looks just like ω versus k. Indeed this is a quantum mechanical connection, coming from de Broglie's p=h/λ, so that as a result it applies to all of these things i.e. waves and particles alike. At least this is true in detail for the examples I've investigated. A few words in the intro on their equivalence in this sense could probably help a lot, i.e. they aren't competing since (except for the factor of Planck's constant) they are one and the same. Thermochap (talk) 17:35, 19 March 2008 (UTC)
Thanks for explaining. In that case, for me an intro stating that the dispersion relation relates the phase speed and wavenumber, or frequency and wavenumber, would be much clearer. And some comments on why it is called dispersion relation. How do you think about moving the massive particle example to a new quantum mechanics section? In that section also the connection between the dispersion relationship and energy and momentum can be included. Crowsnest (talk) 18:24, 19 March 2008 (UTC)
Now that you mention it, the distinction has historical roots. I'll give it some thought. Since I work with electrons, there is no distinction between the particle and wave side even though in other application areas (like with water waves or with baseballs, respectively) one can sometimes ignore the particle or wave component. I'll also look into reasons for the nomenclature, and wouldn't mind a look at that J. Fluid Mech. paper if I can track it down. Thermochap (talk) 19:21, 19 March 2008 (UTC)
I extended the reference to the J. Fluid Mech. article above, so you may more easily track it down. Crowsnest (talk) 19:51, 19 March 2008 (UTC)

## Lead section and article name

The adaptations of Thermochap to the lead section change this article effectively into dispersion relation (quantum mechanics). As far as I am concerned, that is perfectly alright, but then the article should be renamed in such a way. Then, also a new overview article or disambiguation article dispersion relation will be needed, pointing to the relevant pages on dispersion relations in various fields of physics.
Some remarks:

• Ralph Kronig's name is spelled without Umlaut.
• According to WP:LEAD: "The lead should be able to stand alone as a concise overview of the article". And it should be quite brief. Both are missing at the moment. I suggest to create a "History" section, and move the historical info there (which is very interesting, in my opinion).
• As it is now, one may get the impression that this started with Kramers and Kronig, since there is no reference to earlier work on the dispersion relation. Perhaps from the point of view of quantum mechanics this may be true (I do not know). But for water waves, for instance, the dispersion relation was already determined by Laplace. And in the 19th century many discoveries related to dispersion, in water waves as well as many other fields of physics, were made by e.g. Lord Rayleigh, Lord Kelvin, George Biddell Airy and George Gabriel Stokes.

Crowsnest (talk) 17:04, 21 March 2008 (UTC)

Good suggestions. I think the article itself is still a work in progress, as is the overview. Folks probably had very good ideas about rainbows before 1927. Did they refer to some specific equations (e.g. for refractive index) as dispersion relations in those days? Even if not, some context on that work should probably be put in. Thermochap (talk) 18:03, 21 March 2008 (UTC)
On partitions of this article - The Kramers and Kronig stuff itself isn't really QM, since at least I think the early applications involved things like electrical circuits and light in dielectrics. Also, switching from KK's integrals to the question "how does omega depend on k" is still not QM. KK's integrals and ω(k) define two major threads, and hence possible sections in the article. The present sections fall mostly into the "how does omega depend on k" category. Although it's tough to argue that QM don't come in when one multiplies by Planck's constant, once done even that has applications to particle motions explored prior to QM, i.e. it includes classical and relativistic particles as well as quanta like photons, plasmons, phonons, etc. Hence I might be inclined to put detailed stuff on "applications to particles" as a subsection of the "how does omega depend on k" section. All of it is probably worth mentioning in the intro, simply to tell newcomers what they have to look forward to if they read a bit more. Thermochap (talk) 18:03, 21 March 2008 (UTC)
On umlauts, Toll's 1956 paper references Kronig with umlaut, and I think I've seen it that way in many other places as well. Hence I'm assuming that they just skipped it elsewhere on wikipedia so that it is easier to type in and search for, etc. Thermochap (talk) 18:21, 21 March 2008 (UTC)
As far as I can find, Kronig is without Umlaut. He was born in Dresden (Germany), from American parents. On the Dutch site of Delft University, where he was a professor since 1939 till his retirement, nothing is said about an Umlaut. Also googling on German language sites gives his name without Umlaut, so I am pretty sure it is without.
Further, the 1926 paper of Kronig is about QM and dispersion. The Kramers-Kronig relations are off course important outside QM for any causal physics system.
Crowsnest (talk) 00:53, 22 March 2008 (UTC)
Good work, and thanks for the heads up. I also found other references without the umlaut, including what looked like Kronig's signature on that stamp. Even if that umlaut was there at one time, my guess is he was fine with seeing it gone. Thermochap (talk) 01:01, 22 March 2008 (UTC)

## The diagram

My understanding of what Kramer et al. were doing suggests that a prism is not the appropriate way to illustrate this concept.

Kramer and Heisenberg discovered how to account for the dispersion of light that occurs when monochromatic light is directed into some medium that absorbs photons and subsequently radiates photons. If a high intensity photon is absorbed, a high intensity photon of the same frequency may be emitted, or two (or more) photons of lesser energy may be emitted depending on the history of the return of the electron to its equilibrium state.

Monochromatic light comes out of a prism as it went in. P0M (talk) 00:31, 22 June 2009 (UTC)

## Dispersion and propagation of general waveforms

A large part of this section can be condensed. When a progressive wave of permanent form is written as ${\displaystyle y(x,t)=f(\xi ),}$ with ${\displaystyle \xi =x-vt}$ the coordinate in a frame of reference where the wave is stationary — or as ${\displaystyle y(x,t)=f(\theta ),}$ with ${\displaystyle \theta =kx-\omega t}$ the wave phase and ${\displaystyle v=\omega /k}$ — the obvious way to decompose (if necessary) ${\displaystyle f=\sum c_{n}\varphi _{n}}$ will be in terms of functions ${\displaystyle \varphi _{n}}$ with also the argument either ${\displaystyle \xi }$ or ${\displaystyle \theta ,}$ so propagating with the phase speed ${\displaystyle v.}$ This was e.g. done by George Gabriel Stokes in 1847 for nonlinear water waves, in terms of (co)sines (Fourier series).

More interesting are the remarks with respect to nonlinearity (amplitude dispersion, e.g. Debnath, L. (2005), Nonlinear partial differential equations for scientists and engineers (2nd ed.), Birkhäuser, pp. 179–180, ISBN 0817643230). But the Okamoto reference (p. 263) is dubious in the present context, containing several errors with respect to conversions between wavenumber and wavelength. Better would be e.g. Craik, A.D.D. (1985), Wave interactions and fluid flows, Cambridge University Press, pp. 212–215, ISBN 0 521 36829 4. The Craik monograph (or review) also contains interesting stuff regarding the relation between the dispersion relation and stability (both for linear and nonlinear waves), and the relation between variational methods and dispersion relations (see also Whitham, Linear and nonlinear waves).

Best regards, Crowsnest (talk) 18:19, 9 July 2009 (UTC) P.S. The solitary wave has infinite wavelength, but it does have a characteristic width related to ${\displaystyle \beta ^{-1},}$ e.g. the half-width between the points where ${\displaystyle y={\tfrac {1}{2}}A.}$

There's really little or nothing in the section that relates to dispersion relations; it's about basis decomposition, dispersionless propagation, nonlinear systems, etc.; it's a chuck of material that was not suitable at Wavelength, and Brews is looking to put it somewhere. I agree with Crowsnest that it need a lot of condensing, particularly in not generalizing to basis functions besides sinusoids, which are the ones that relate to the dispersion relation, and it shouldn't focus on dispersionless and nonlinear stuff either, unless it's connected via sources to the topic of dispersion relation, which so far none of it is. Dicklyon (talk) 19:18, 9 July 2009 (UTC)
The purpose is to show how it happens that dispersion leads to distortion of the waveform, which is a general result of dispersion and fits very well in this article. There is no requirement to fit dispersion to sinusoids, although that is the commonly used approach. The digression is slight.
The introduction of nonlinearity is related because it points out that distortion may not occur even in the presence of dispersion under special circumstances.
Instead of blanket deletion, suggest amendments. Brews ohare (talk) 20:02, 9 July 2009 (UTC)
You copied it all from a place where you claimed a different purpose for it. My suggestion is that if you have something relevant, with sources that connect to the topic, then write it based on what the source says, not based on some old text you have hanging around that was rejected elsewhere. Dicklyon (talk) 20:25, 9 July 2009 (UTC)

The text is relevant; its history isn't. According to Dicklyon, this major effect of dispersion (distortion of waveforms)Fitts;Matsumoto; Daly;Forestieri;Goodman; Richardson; Chan is unrelated to the article on dispersion. As far as I can tell, there is no point in trying to pursue this matter with Dicklyon. Brews ohare (talk) 20:29, 9 July 2009 (UTC)

Right. Pursuing the matter of adding material that unsourced or irrelevant, that is not connected by sources to the topic, is not likely to pay off for you if I'm watching. If I made a mistake, and you did cite a source that mentions dispersion, please do point it out. Dicklyon (talk) 21:05, 9 July 2009 (UTC)
I see you went back and added some sources to your comment above. Why not add some text based on those, instead of the unsourced mess you added before? But be careful, because some authors, like Forestieri, use the term "distortion" for nonlinear effects (harmonic and intermodulation distortion), and not for dispersion effects, which are linear. Dicklyon (talk) 04:41, 10 July 2009 (UTC)
Why not add some text? Need you ask?? Brews ohare (talk) 13:02, 10 July 2009 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Dispersion relation/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 Article needs discussion of dispersion relations in quantum mechanics and solid state physics...

Last edited at 22:59, 10 October 2007 (UTC). Substituted at 13:30, 29 April 2016 (UTC)