Talk:Wavelength/Archive 4

Question
What is the wavelength of this wave? What is the exact definion of wavelength? By Fourier Analysis? ––虞海 ( Yú Hǎi ) 17:37, 10 October 2010 (UTC)
 * Is that a meaningful question? What do the authorities define as "wavelength" ? --Wtshymanski (talk) 17:56, 10 October 2010 (UTC)


 * Please, per wp:talk page guidelines, take this to the wp:reference desk/science? Thanks. DVdm (talk) 18:09, 10 October 2010 (UTC)


 * It doesn't sound like a ref desk question to me, but rather a rhetorical question to see whether we have included a correct and working definition. I don't know of a definition based on Fourier analysis, but there are many alternatives, and maybe one of those, too.  Some definitions are predicated on the wave being periodic; others on it being sinusoidal.  The "distance between peaks or troughs" definition is usually adequate, and would give a sensible answer for the wave in question, but it may not be both precise and general enough to cover all things that people call wavelength.  Dicklyon (talk) 19:57, 10 October 2010 (UTC)


 * For slowly-varying and coherent wave trains a good definition is the one by Whitham (see e.g. his book Linear and nonlinear waves) through the definition of the wavenumber as the gradient of the carrier-wave phase &theta;(x,t): k=&nabla;&theta;, so &lambda;=2&pi;/|k|. The wave phase of the carrier wave is obtainable through the Hilbert transform of the band-pass filtered signal (removing nonlinear sub- and super-harmonics). -- Crowsnest (talk) 22:05, 10 October 2010 (UTC)


 * It's not even clear that this is a meaningful question. Not all waveforms have "a wavelength". General waveforms are composed of a spectrum of waves at different frequencies. One can only define an overall wavelength for a waveform in special cases.


 * Be sure to read Archive 2 of this talk page (link above). This kind of question has been discussed here before.--Srleffler (talk) 23:09, 10 October 2010 (UTC)


 * Ha! That's not likely to be a productive use of time.  I like the definition that Crowsnest came up with, though.  It works well for any wave that's remotely like sinusoidal.  Dicklyon (talk) 23:16, 10 October 2010 (UTC)

Prism and refraction
this edit removed the figure at the right with the explanation:
 * "Rm disputed image altogether. It is not really relevant to the topic here. We are more interested in change in velocity as a function of frequency than in change of direction."

The figure was part of the section discussing dispersion:
 * the relationship between ω and λ (or k) is called a dispersion relation.

So the topic is the change in the relation between ω and λ introduced by the refractive index of a medium, as indicated in the lower figure. The connection to the prism is via the Fresnel equations which explain that the angle of refraction varies with the refractive index, and thus, when n = n(&lambda;), different colors are refracted by different angles according to Snell's law:
 * $$\frac{\sin\theta_\mathrm{i}}{\sin\theta_\mathrm{t}} = \frac{n_2}{n_1} \ .$$

The inclusion of this point is of interest because the separation of colors using a prism is a well-known phenomenon, and its introduction here provides a useful connection for the reader to these topics. It is one of WP's most admired features that it serves to broaden the reader's concept of a topic by pointing out exactly such connections.

On this basis, I'd suggest the reintroduction of this figure with a better explanation and some links to the relevant WP articles on the related topics. Brews ohare (talk) 13:47, 2 April 2012 (UTC)

I have made an attempt at incorporating this suggestion. Brews ohare (talk) 18:32, 2 April 2012 (UTC)


 * If we are going to include an image of a prism dispersing light, it should be the one showing moving waves, because the topic of this article is wavelength, and the relevant effect is that dispersion causes waves with different wavelengths to move with different velocities in the medium. It is interesting that this is related to the angle of refraction, but the latter is not directly relevant to the topic of this article. --Srleffler (talk) 03:53, 3 April 2012 (UTC)
 * I removed the details about Snell's law and re-introduced links to the article on dispersion. The fact that prism dispersion is connected with change in wavelength in a medium is interesting and relevant. The details of how to calculate angle of dispersion in a prism are not relevant, and should be found in the linked articles if a reader is interested.
 * I restored the image that actually shows waves moving with different speeds in a prism, connected with dispersion of light in that prism, because that is the relevant phenomenon here.--Srleffler (talk) 04:54, 3 April 2012 (UTC)
 * I moved the discussion of refraction from an earlier section down to this one and modified the text a bit to fit it in more smoothly. Brews ohare (talk) 13:43, 3 April 2012 (UTC)

phase and group velocity
This topic may seem to be a digression in the article on wavelength. I am unsure how to handle it, but it shouldn't be ignored altogether. One aspect is shown in the figure: the wavelength of an envelope function differs from that of the constituents and moves at a different speed. Brews ohare (talk) 15:14, 3 April 2012 (UTC)
 * DickLyon: This material is not "off-topic bloat". It is relevant for several reasons. Perhaps the main reason is that it points out the wavelength of a combination waveform is not that of its constituents. Another reason is that the this section concerns effects of the dependence of speed of propagation upon wavelength, and this phenomena is one of those consequences. Brews ohare (talk) 18:02, 3 April 2012 (UTC)

As we discussed at length, years ago, the application of the term "wavelength" to the modulation is rare and unusual, dare I say idiosyncratic. And there are much better places to discuss phase velocity and group velocity than an article on wavelength, which already goes off on too many tangents. Dicklyon (talk) 20:04, 3 April 2012 (UTC)


 * Indeed "wavelength" is hardly ever used for this: I know it under the names group length, modulation length or envelope length. -- Crowsnest (talk) 20:53, 3 April 2012 (UTC)
 * Crowsnest: Thanks for those links that establish some terminology I was unaware of. It does seem, however, that if one has an envelope f that satisfies the normal definition of a periodic function, that is,
 * $$ f(\xi+\lambda)=f(\xi) \, $$
 * with &xi; = x-vt there is no doubt whatsoever that the normal definition of wavelength applies to this envelope function f, whatever name one may attach to the envelope itself. Don't you agree? Brews ohare (talk) 21:39, 3 April 2012 (UTC)


 * It's logical that the term could apply, but it's seldom or never used that way, so let's not. Dicklyon (talk) 00:27, 4 April 2012 (UTC)
 * The terms "group length", modulation length" and "envelope length" definitely are used to apply to the length of a wave packet, but I haven't found them used for a periodic envelope like that in the image above. In any event, it is not only "logical" to use the term wavelength in connection with a periodic envelope function, it is mathematically perfectly and completely correct according to the definition of a periodic function. Brews ohare (talk) 01:28, 4 April 2012 (UTC)
 * Wikipedia relies on sources, not logic. Our role is to report what is documented in reliable sources, not to synthesize our own knowledge, even when that knowledge follows logically from the source materials. See No original research and WP:SYNTHESIS for more on this.--Srleffler (talk) 03:10, 4 April 2012 (UTC)
 * That is of course absurd; if we didn't use logic, we'd be forced to make word for word copies of sources. --Wtshymanski (talk) 03:35, 4 April 2012 (UTC)
 * Brews is arguing that we should cover a usage of a term purely because it is a logical extension of the usual definition, despite admitting that he hasn't found any sources that use the term that way. This is pretty clearly not allowed by policy.--Srleffler (talk) 04:37, 4 April 2012 (UTC)

There is no "logical extension" of the definition of wavelength involved here. If a function f satisfies
 * $$ f(\xi+\lambda)=f(\xi) \, $$

then &lambda; is its wavelength. Period. The only point to discuss is whether periodic envelopes are worth mentioning. I'd guess that DickLyon and Srleffler would say "No, it is an uninteresting topic". Dismissing the matter on spurious grounds simply avoids the real basis for discussion. Brews ohare (talk) 05:46, 4 April 2012 (UTC)

Some references are: Stade and Holbrow et al. Brews ohare (talk) 15:07, 4 April 2012 (UTC)
 * Your first ref is not about waves, and the second is about sinusoidal waves; so what's your point? Dicklyon (talk) 15:33, 4 April 2012 (UTC)


 * Shouldn't all the discussion about propagation, dispersion, and other properties of waves be left to the article Wave? That would turn this article into a dictdef that could be moved to Wiktionary. --Wtshymanski (talk) 14:46, 4 April 2012 (UTC)
 * Wavelength is a property of waves, and it is sufficiently complicated to require its own article rather than loading down Wave. Brews ohare (talk) 15:07, 4 April 2012 (UTC)
 * We need the right compromise. There's a lot to say about wavelength, and we've pretty much said that and more.  Other stuff is better off in an article on waves.  Dicklyon (talk) 15:33, 4 April 2012 (UTC)
 * Dick: You have returned the discussion to the proper subject: is the treatment of periodic envelope functions "other stuff", or something that should be in the article? There is already a section Envelope_waves; perhaps this material should go there? Brews ohare (talk) 16:45, 4 April 2012 (UTC)
 * As you recall, we had a big to-do about that back in June/July 2009, before your year of topic-ban from physics and your year of block for continuing disruptions. I condensed what you had about envelopes and found the one source that connected that to "wavelength".  If there are more sources that connect envelope waves to the concept of wavelength, bring those forward for consideration.  Dicklyon (talk) 17:02, 4 April 2012 (UTC)

Dick: You digress. Past squabbles I suppose are meant to underline how difficult we are. Instead, we might focus upon the present: periodic envelope functions exist. They therefore have a wavelength. Is this a topic suitable for the section Envelope_waves? I don't think the question is one of "Do periodic envelopes exist?" nor "Do periodic envelopes have a wavelength?" Maybe the question is "Is it of interest that a composite of short-wavelength, fast-moving excitations can form a disturbance in a dispersive medium that has a longer wavelength and moves at a different speed?" Brews ohare (talk) 17:24, 4 April 2012 (UTC)
 * Brews, above you write "If a function f satisfies $$ f(\xi+\lambda)=f(\xi) \, $$ then &lambda; is its wavelength. Period." I disagree. Waves have wavelength, functions do not. A periodic envelope is not a wave, although a wave can have a periodic envelope. The wavelength of a wave with a periodic envelope is not the spatial period of the envelope function.--Srleffler (talk) 02:27, 5 April 2012 (UTC)
 * Srleffler, if I understand you, you would accept instead a statement: "If a function f satisfies f(&xi;+&lambda;) = f(&xi;) and this function describes a waveform with a wavelength &lambda;, then in the math describing this wave, the physical wavelength corresponds to the period of the describing periodic function." So the distinction here is one of semantics: whether a term described in physics as a "wavelength" has a mathematical analogue that might be called the wavelength of a function, or might be called something else. I have a feeling of vertigo here, of falling into some kind of debate over whether nature is imperfect and math is the more prefect Platonic reality.
 * Your second point is that the wavelength of a wave with an envelope is not the spatial period of the envelope. I suspect this is an exercise in semantics also. I suppose you might agree that the envelope is a physical item, and that an envelope can have a wavelength. That wavelength is not, of course, the wavelength of the component waves, if that is your object here. However, if the envelope is described by a periodic function, then the spatial period of that function represents the wavelength of the envelope in the mathematics. Brews ohare (talk) 05:12, 5 April 2012 (UTC)
 * Not clear what semantics you intend by "is a physical item", but it's very unusual to speak of the envelope as a wave or having a wavelength. I've found exactly one source that does so, and cited it (Denny).  And what it says about the envelope's velocity being determined by its wavelength is wrong, or at least seriously misleading, though the rest of its derivation of group velocity is pretty conventional.  If that's all we've got, I don't see a need to extrapolate the concept of wavelength to envelope functions.  Nobody does that.  Dicklyon (talk) 06:27, 5 April 2012 (UTC)


 * The most common use of the term "envelope" is to describe a wave packet, which of course has no wavelength, being a solitary propagating pulse. However, as shown in the image, envelopes can have a wavelength. Moreover, this particular example provides a vivid illustration of the fact that the envelope propagates at a different speed than its constituents. The group velocity is pointed out already in the section on Envelope_waves, and an illustration is worth 1000 words. It seems to me that "it is of interest that a composite of short-wavelength, fast-moving excitations can form a periodic disturbance in a dispersive medium that has a longer wavelength and moves at a different speed than its constituent waves." Don't you think some presentation of this matter could be constructed that would be acceptable to you? Brews ohare (talk) 15:09, 5 April 2012 (UTC)
 * Who are you quoting here? And why is this more interesting than the case of two irrationally related sinusoids forming a non-periodic disturbance? And anyway, the periodic disturbance doesn't propagate unchanged in a dispersive medium as your illustration shows; the envelope does, but that's not a disturbance.  Dicklyon (talk) 15:25, 5 April 2012 (UTC)
 * Dick: The subject of this article is wavelength. So an example of a periodic envelope that exhibits a wavelength is ipso facto more pertinent to this topic than a non-periodic disturbance. And a picture comparing group and phase velocity is more illuminating than a bare mention in words: "a composite of short-wavelength, fast-moving excitations can form a periodic disturbance in a dispersive medium that has a longer wavelength and moves at a different speed than its constituent waves"; although that sentence would be helpful too. Brews ohare (talk) 15:46, 5 April 2012 (UTC)
 * As you are more application oriented, maybe Figure 7.1 here is of interest? It is not an example of dispersive media, but it is an example of an envelope that has a wavelength. Brews ohare (talk) 16:06, 5 April 2012 (UTC) Another possible example which involves a dispersive and nonlinear medium is a train of solitons and here. In the ocean the lowest graph here may be of interest. I don't think these examples are for the article, just for illustration here.  Brews ohare (talk) 16:23, 5 April 2012 (UTC)
 * I discovered that the reference you found for the envelope discussion also uses the term wavelength for the envelope, so I made that observation in the envelope section. Brews ohare (talk) 18:43, 5 April 2012 (UTC)
 * As I said, that Denny ref is the only source I can find that associates the concept of "wavelength" with the envelope length. And some of what it says about it is wrong or misleading.  None of your other links go to pages with "wavelength" anywhere nearby.  So I think that even mentioning this concept is UNDUE weight.  Dicklyon (talk) 19:21, 5 April 2012 (UTC)
 * The misleading bit is "the envelope of modulation...moves at a speed that is determined by its own wavelength and period", which is either trivial or wrong. The speed is determined by the group velocity, or d\omega/dk, in the region of the two wavelengths, and is pretty much independent of the "wavelength" of the modulation envelope.  This is a very poor explanation all around, and not one the gives any weight to the idea that an envelope modulation is referred to as having a wavelength different from the mean wavelength of the underlying waves (as is done in talking about modulated radio and light waves, for example).  Dicklyon (talk) 19:33, 5 April 2012 (UTC)

I'll take a look for a better source. I was interested in pointing out wavelength of envelope as simply an example of the concept of wavelength, and not so much as a practical matter. I think that is a useful thing to do in driving home a concept. However, I have recently discovered there may be a very real application in what is called electric distance meters or EDMs, where a modulated light beam is used to measure distances in terms of the modulation length. An example discussion is here. What do you think about this? Brews ohare (talk) 19:38, 5 April 2012 (UTC)
 * I have replaced Denny. It would seem that there are many possible replacements. Brews ohare (talk) 21:40, 5 April 2012 (UTC)
 * I reverted you, because I do not see where the new reference applies the term 'wavelength' to the envelope. Did I miss it? A reference that uses the term "wavelength" in describing the envelope is crucial for including discussion of envelopes at all. Envelopes are barely worth mentioning at all in this article, and only because a tiny minority of authors describe the period of a periodic envelope as a "wavelength". --Srleffler (talk) 03:34, 6 April 2012 (UTC)
 * Srleffler: The reference uses the term wave number, I believe, related as everyone knows to the reciprocal of wavelength, and as pointed out in the reverted text. I changed the reference because Dick pointed out some infelicities in Denny's discussion of group velocity. However, if you prefer to leave Denny instead of accepting a more suitable source or looking for one yourself, well that makes clear your priorities, I guess. Brews ohare (talk) 05:24, 6 April 2012 (UTC)
 * Here is another possibility and here is a google book search for wavelength of a modulation envelope and here is one for modulation wavelength and here is one for envelope wavelength. Brews ohare (talk) 13:05, 6 April 2012 (UTC)

Mathematical representation
The article in its present form describes wavelength using a sine wave image and generalizes this simple case with the remark:
 * The concept can also be applied to periodic waves of non-sinusoidal shape

A more fundamental and rigorous approach would be to point out that a Fourier series assembled from sinusoidal functions of the form:
 * $$c_n = \cos \left(\frac{2\pi n}{\lambda}\xi \right) \ \, \ \ s_n=\sin \left( \frac{2\pi n}{\lambda }\xi \right) \ , $$

(n a positive integer) in the form:
 * $$f(\xi)=a_0 + \sum_{n>0} \left( a_n c_n + b_n s_n \right )$$

represents any (bounded and integrable) function in the interval −&lambda;/2 ≤ &xi; < &lambda;/2. This function has the property that it repeats periodically in &xi; as described by:
 * $$f(\xi+\lambda) = f(\xi)\, $$

where &lambda; is variously called the period or the wavelength of the function. By choosing
 * $$\xi = x-vt \, $$

where x is distance along an axis in space and t is time, the function f describes a waveform periodic in space with wavelength &lambda; propagating with time-invariant shape in the positive x direction with a velocity v.