Misplaced Pages

Talk:Wavelength: Difference between revisions

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editContent deleted Content addedVisualWikitext
Revision as of 14:26, 25 April 2012 editBrews ohare (talk | contribs)47,831 edits New section in article: Query to Dick← Previous edit Latest revision as of 15:17, 28 July 2024 edit undoJust plain Bill (talk | contribs)Extended confirmed users, Pending changes reviewers25,210 edits Reverted 1 edit by 2A02:8071:7150:4380:DD85:3ACD:D906:1BAB (talk): Talk pages are for discussing improvements to the article.Tags: Twinkle Undo 
(135 intermediate revisions by 27 users not shown)
Line 1: Line 1:
{{physics |class=B |importance=high}} {{WikiProject banner shell|class=B|vital=yes|1=
{{WikiProject Physics|importance=high}}
{{WikiProject Mathematics|importance=low}}
}}
{{Archive box| {{Archive box|
* ] <small>(2005—2008)</small> * ] <small>(2005—2008)</small>
* ] <small>(2009—)</small> * ] <small>(2009)</small>
* ] <small>(2009—2012)</small>
* ] <small>(2012)</small>
* ] <small>(2012—)</small>

}} }}


== More general waveforms wording ==
== Question ==


I removed the emphasis on the period ''T'' being the same at all points, since this might be misleading. While it is true that there is a period ''T'' that is common to all points, at ''some'' points the wave will repeat more than once in time ''T'', so the "period" as conventionally defined is shorter at these locations, by some integer factor. The source location for a periodic non-sinusoidal wave is one such location: the period there is shorter (possibly by a large factor) than the period at other locations. Since in many cases this shorter period of the source wave is known, the statement that the period ''T'' is the same at all points could mislead the reader into thinking that the period of the wave equals that of the source at all points, which is not true. A more extended discussion could clarify this, but it's probably better just not to get into it.
What is the wavelength of this wave? ] What is the ''exact'' definion of wavelength? By Fourier Analysis? ––] 17:37, 10 October 2010 (UTC)
: Is that a meaningful question? What do the authorities define as "wavelength" ? --] (]) 17:56, 10 October 2010 (UTC)


I also removed the reference to Fourier integrals. I didn't feel that it worked where it appeared in the paragraph. It broke the flow of concepts, making the paragraph less clear.--] (]) 01:40, 29 April 2012 (UTC)
::Please, per ], take this to the ]? Thanks. ] (]) 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. ] (]) 19:57, 10 October 2010 (UTC) :I added the mention of Fourier integral because it was part of what the source that I cited talk about, and to appease Brews a bit, but I don't mind it being gone. As for the period, I'm not sure I understand. How can the period anywhere be other than the period of the source, that is, the least common period of all the sinusoids that are propagating by the location? ] (]) 06:10, 29 April 2012 (UTC)
::I may have been mistaken about the period.--] (]) 06:57, 29 April 2012 (UTC)
:::It seems to me that the period is related in only a complicated fashion to the period of the source, involving the separation of the observation point from the source and also the dispersion relation. Maybe we need a source to tie this down? ] (]) 16:42, 30 April 2012 (UTC)
::::With the Fourier series decomposition, it's easy to see that the wave contains only harmonics of the source period. No new frequencies are added by propagation, even if there are reflecting ends, dispersion, or whatever. So you have harmonics of the period everywhere, and therefore the same period everywhere, no? ] (]) 22:59, 30 April 2012 (UTC)
:::::Yes. I had in mind that as the components got out of phase with one another the superposition would have a period that was longer than one cycle of the fundamental, but I see I was mistaken.--] (]) 03:17, 1 May 2012 (UTC)
{{outdent|8}}Using a Fourier series begs the question as it presumes a periodic result with the same period. We need a source here, not editors' speculation. ] (]) 08:18, 1 May 2012 (UTC)
To add to the speculation, and emphasize the need for an explanatory source, if the driver produces two sine waves close in frequency, the resulting periodic waveform has an envelope that oscillates at the beat frequency, which can be as low (or as long a wavelength) as one can imagine if the two frequencies are close together. That seems to suggest that the period of the waveform produced by the driver is less about the period of the driver than the beat frequency. ] (]) 09:00, 1 May 2012 (UTC)
amply demonstrates this point; see Figure 4.7.1 ] (]) 16:19, 1 May 2012 (UTC).
:Yes the period might be very long in the two-beating-frequencies case. But the source associates "wavelength" with the components, not with a long pattern. There's actually no "driver" or "source" that's relevant here, just periodic-in-time wave motion, which can be analyzed into harmonic frequency components. I don't see any speculation, but then again I don't see a source that says precisely what our text says, obvious though it is. ] (]) 00:48, 2 May 2012 (UTC)
:There is no "presumption", only ''definition''. The paragraph we are discussing is specifically about the important special case of waves that are periodic in time. There is no need to presume or speculate; periodicity in time is the specified initial condition. The only question is how the system evolves over time, and how it behaves at other spatial locations.
:Since the topic of discussion is waves that are periodic in time, the relevant period in the two-beating-waves case is the long period required for the full waveform to repeat. The Fourier series in this case is particularly simple, and the fact that the period of the wave as a whole is much longer than those of the nonzero Fourier components is not a problem. In my initial comments on this topic, I had presumed that there might be locations where the period of the combined wave might be short (like the periods of the components), but I was mistaken. The component waves maintain their frequencies as they propagate and there will be nowhere along their common path where they do not beat against one another, producing a waveform with the same, long, period.--] (]) 03:19, 2 May 2012 (UTC)
::Actually, it might be slightly more complicated than that in wave media with reflections that can make nulls for certain frequencies at certain locations. If one of the two components in the two-component beating pattern has a null, then the response at the location of the null will be just the other component, so it will have a short period there. However, what the article says is still true there: the wave is still periodic with the longer period T, even if also with some shorter period. Just to be sure, I have edited the text to try to make sure that the period T is stated as the period of the wave and can't be misunderstood as one of these possible shorter periods at a null in a corner case. ] (]) 03:39, 2 May 2012 (UTC)
:::A periodic disturbance in time at a particular location will result in a disturbance at all distances from the source when steady-state is reached. So a disturbance in time repeated with period ''T'' but of duration less than ''T'' will involve many frequencies, submultiples of ''T''. Nothing much changes as ''T'' changes, if the disturbance maintains the same form in time, but is simply spaced with larger "blank" periods in between. ''T'' is not a very useful parameter in describing matters, therefore, and it should not be framed as the key to analysis here. ] (]) 22:06, 3 May 2012 (UTC)
::::True, the periodicity of T is just there to make the Fourier series applicable. ] (]) 03:50, 4 May 2012 (UTC)


== More general waveforms references ==
::::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). -- ] (]) 22:05, 10 October 2010 (UTC)


The two references to the topic of general waveforms so far do not actually describe how these calculations are done, but provide only few words of description.
: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.


The stress upon a periodicity in time in the article in preference to the propagation of a waveform seems to me misguided. For example, if one makes the analogy with a performer blowing large soap bubbles in a park, the bubbles are launched as huge spheres, and as they are carried in the wind they enlarge and become ellipsoids. At a location near the launch one sees a periodic appearance of spheroids at the period of launch ''T''. At a remote position one sees a periodic appearance of enlarged ellipsoids with a period ''T''. Just how long the period ''T'' is between arrivals, or between repetitions of what happens periodically in time a a fixed location, as described by a Fourier series in time, is not so interesting as the process of transformation as the spheroids change to enlarged ellipsoids, that is, the propagation phenomenon. Changing the period and spacing the bubbles differently is not really essential.
:Be sure to read Archive 2 of this talk page (link above). This kind of question has been discussed here before.--] (]) 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. ] (]) 23:16, 10 October 2010 (UTC) So I think what is needed is a more interesting discussion with some more detailed references tying what happens to the dispersive nature of the medium. Emphasis upon the more-or-less incidental period between events is not the really interesting point. ] (]) 06:11, 2 May 2012 (UTC)


:I agree the section remains rather unsatisfying. I did my best to find a sensible way to incorporate the Fourier series into something to do with wavelength, cobbling what was there; but it's still a bit of a misfit. Most sources that talk about dispersion and Fourier analysis don't do in the context of periodic waves, and usually do it in terms of wavenumber, not wavelength. And their analysis doesn't usually conclude anything related to wavelength or to repetition in space. Probably we should just simplify the section, since there are better articles for covering these other concepts of waves in linear dispersive media. As for the concept of wavelength being applied to other than approximately sinusoidal waves, it's unusual at best; discussing it can easily be misleading, or spiral into contradictions, like when you get into claims that it's well-defined for arbitrary periodic functions. ] (]) 05:58, 3 May 2012 (UTC)
== Prism and refraction ==
::Hi Dick: Quite possibly the easiest approach is to use wavevector, or maybe to use a simple example instead of trying the general case. As you know, however, I do not agree in the slightest that application of wavelength to a periodic wave in space of general form is in any way misleading, although the occurrence of such waves in nature is not general, but restricted to particular media. From a conceptual point of view, wavelength is what Fourier series is about in space, with the simple interpretation of the general argument &xi; as ''x'' instead of angle, or time. From this stance, as I have pointed out by direct quotations from at least three sources, ''many'' authors do exactly that. ] (]) 21:09, 3 May 2012 (UTC)
:::Not that many authors do that. And none of them seem to reveal any reason for doing a sinusoidal decomposition of the spatial pattern. At least in the case of the dispersive linear system there's a reason to decompose into sinusoids. ] (]) 03:52, 4 May 2012 (UTC)
::::Hi Dick: Perhaps it is just argumentative, but here's a question: why do authors use Fourier series when argument &xi; is interpreted as time, or angle, or whatever? Why is it automatically a wasted effort only when &xi; is interpreted as a spatial variable? Why is period ''T'' more significant as a time period than &lambda; is a a spatial period? Could it be that in fact there is ''no'' difference at all? ] (]) 20:40, 5 May 2012 (UTC)
:::::It's not unusual to use wavenumber (or reciprocal wavelength) in a Fourier transform, as a way to get a sinusoidal decomposition, especially for wave packets. But periodic-in-space waves are a relatively rare corner, seldom encountered where a sinusoidal decomposition would be helpful. If they're also periodic in time at the same time, in a linear system, the system is nondispersive, and has a trivial wave equation, for which a sinusoidal decomposition is not needed; it adds nothing to the understanding of the system. If they're in a nonlinear system, the sinusoidal decomposition is not particularly illuminating either. ] (]) 22:27, 5 May 2012 (UTC)
::::Here's another question: when an oboe plays a note, it sounds different than when a violin plays the same note. Could it be that the difference can be expressed as a difference in the Fourier series expressing the wavelengths of vibration supported by the general waveform in the oboe's air column for that note compared to the wavelengths present in the general waveform on the violin string when the same note is present? Would that be an interesting enough example of Fourier expansion of the general waveform in space to warrant mentioning the use of Fourier series for spatial analysis of waveforms? Or, perhaps, an article ] is needed? ] (]) 20:52, 5 May 2012 (UTC)
:::::The sound difference has much more to do with the waveform in air; this is what propagates and carries the pattern. The physics within the instrument gives rise to different modes, and to a temporal near-periodicity from how the signal interacts with the reed or the bow, but I haven't seen an analysis like you're describing, where the composite signal is analyzed into space-domain sinusoids. Of course it could be done. ] (]) 22:27, 5 May 2012 (UTC)
:::::Once it's left the instrument the waveform is not uniform, i.e. there is no 'general waveform', even allowing for distance attenuation. The sound you hear an inch from a source is very different from the sound you hear three feet or thirty feet from it: the best example is the human voice as we're most familiar with it: often you can tell how far away someone is quite accurately by the sound of their voice. This is less noticeable with an instrument because its pure note dominates and so it depends on the particular frequency of the instrument. And of course in most cases performers don't want you to hear different sounds depending on how far away you are, and will go to great lengths to minimise effects of distance (modifying the building to compensate for example).--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 22:57, 5 May 2012 (UTC)
::::To pursue this matter further, harmonics are produced on the guitar by deliberately , a direct application of . ] (]) 21:05, 5 May 2012 (UTC)
:::::That doesn't produce harmonics, but kills the fundamental and certain other harmonics. Yes, it is described in terms of wavelength on the string, or the modes of the instrument. But not in terms of a sinusoidal decomposition of a periodic-in-space pattern. ] (]) 22:27, 5 May 2012 (UTC)
::Dick & Blackburne: You might find some interest in reading a or two on this subject instead of relying on your recollections. Something interesting can be done here. ] (]) 20:51, 10 May 2012 (UTC)
:::I have plenty of good books on the physics and psychophysics of music and musical instruments; it's not clear what you see as relevant in the page you've linked in that college physics text. ] (]) 21:24, 10 May 2012 (UTC)
::::Dick: Figures 14.26 and 14.27 of compare waveforms for a pure note on a tuning fork with the same note on a clarinet and a flute. The point, of course, is that the characteristic voice of the instrument is expressed in its peculiar waveform, which is in each case periodic with the same wavelength but of different shape. Accordingly, the differences between voices is sought in the different harmonics of the fundamental found in each. This difference can be expressed in time or in space, although the latter requires some expression of the characteristics of the medium, which cannot be unduly dispersive. Obviously, instruments usually operate in air, and the slight dispersion of sound in air is no impediment to applying a spatial analysis. The design of an instrument is perhaps even more clearly related to wavelength, as the ''dimensions'' of the instrument determine how an excitation of a particular spatial mode will be related to its various harmonics; for example, how the standing wave on a violin string is connected to the various resonances of the instrument. ] (]) 13:04, 15 May 2012 (UTC)


==Removal of Fourier series in time section==
]
I went ahead and removed the unsatisfying paragraph about the Fourier series and periodic-in-time waves, as it was not very useful, nor very germane to the topic. ] (]) 04:04, 15 May 2012 (UTC)
removed the figure at the right with the explanation:
:This removal was a good step: it tried to introduce Fourier series using the topic of Fourier series in time, applied to waveforms that have no identifiable wavelength in space. The door is now open to introduce Fourier series in a context appropriate to the subject of wavelength, that is, the context of spatially periodic general waveforms which, of course, always have an identifiable wavelength. That such waveforms can be and are analyzed using Fourier series is well documented, and the objection that such waveforms are not necessarily found in general media restricts its applicability in general, but doesn't mean it deserves no mention here. ] (]) 12:34, 15 May 2012 (UTC)
:"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."
:I'd suggest a reconsideration of the text below:

The figure was part of the section discussing dispersion:
:the relationship between ω and λ (or ''k'') is called a ].
]
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 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 ]:
:<math>\frac{\sin\theta_\mathrm{i}}{\sin\theta_\mathrm{t}} = \frac{n_2}{n_1} \ .</math>
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. ] (]) 13:47, 2 April 2012 (UTC)

I have made an attempt at incorporating this suggestion. ] (]) 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. --] (]) 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.--] (]) 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. ] (]) 13:43, 3 April 2012 (UTC)

== phase and group velocity ==

], and the green dots propagate with the ].]]
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. ] (]) 15:14, 3 April 2012 (UTC)
:DickLyon: 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. ] (]) 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. ] (]) 20:04, 3 April 2012 (UTC)

:Indeed "wavelength" is ] for this: I know it under the names , or . -- ] (]) 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,
:::<math> f(\xi+\lambda)=f(\xi) \ , </math>
::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? ] (]) 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. ] (]) 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. ] (]) 01:28, 4 April 2012 (UTC)
:::::Misplaced Pages 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 ] and ] for more on this.--] (]) 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. --] (]) 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.--] (]) 04:37, 4 April 2012 (UTC)
{{outdent|8}} There is no "logical extension" of the definition of wavelength involved here. If a function ''f'' satisfies
:::<math> f(\xi+\lambda)=f(\xi) \ , </math>
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. ] (]) 05:46, 4 April 2012 (UTC)

Some references are: and ] (]) 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? ] (]) 15:33, 4 April 2012 (UTC)

:{{outdent}} Shouldn't all the discussion about propagation, dispersion, and other properties of waves be left to the article ]? That would turn this article into a dictdef that could be moved to Wiktionary. --] (]) 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 ]. ] (]) 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. ] (]) 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 ]; perhaps this material should go there? ] (]) 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. ] (]) 17:02, 4 April 2012 (UTC)
{{outdent|8}}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 ]? 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?" ] (]) 17:24, 4 April 2012 (UTC)
:Brews, above you write "If a function ''f'' satisfies <math> f(\xi+\lambda)=f(\xi) \ , </math> 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.--] (]) 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. ] (]) 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. ] (]) 06:27, 5 April 2012 (UTC)
], and the green dots propagate with the ].]]
::::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 ], 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? ] (]) 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. ] (]) 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. ] (]) 15:46, 5 April 2012 (UTC)
::::::As you are more application oriented, maybe here is of interest? It is not an example of dispersive media, but it is an example of an envelope that has a wavelength. ] (]) 16:06, 5 April 2012 (UTC) Another possible example which involves a dispersive and nonlinear medium is a and . In the ocean may be of interest. I don't think these examples are for the article, just for illustration here. ] (]) 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 ]. ] (]) 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. ] (]) 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). ] (]) 19:33, 5 April 2012 (UTC)
{{outdent|8}}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 . What do you think about this? ] (]) 19:38, 5 April 2012 (UTC)
:I have replaced Denny. It would seem that there are many possible replacements. ] (]) 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". --] (]) 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. ] (]) 05:24, 6 April 2012 (UTC)
::: and is a google book search for wavelength of a modulation envelope and is one for modulation wavelength and is one for envelope wavelength. ] (]) 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 ] assembled from sinusoidal functions of the form:
:<math>c_n = \cos \left(\frac{2\pi n}{\lambda}\xi \right) \ \ , \ \ s_n=\sin \left( \frac{2\pi n}{\lambda }\xi \right) \ , </math>
(''n'' a positive integer) in the form:
:<math>f(\xi)=a_0 + \sum_{n>0} \left( a_n c_n + b_n s_n \right )</math>
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:
:<math>f(\xi+\lambda) = f(\xi)\ , </math>
where &lambda; is variously called the ''period'' or the ''wavelength'' of the function. By choosing
:<math>\xi = x-vt \ , </math>
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''.
====References====
*{{cite book |title=Fourier Analysis |author=Eric Stade |url=http://books.google.com/books?id=gMPVFRHfgGYC&pg=PA1&dq=definition+wavelength+period+%22Fourier+series%22&hl=en&sa=X&ei=BWF8T57QIbLZiQLq_JHGDQ&ved=0CDsQ6AEwAA#v=onepage&q=definition%20wavelength%20period%20%22Fourier%20series%22&f=false |page=3 |publisher=John Wiley & Sons |year=2011 |isbn=1118165519}}
*{{cite book |author=Gerald B Folland |title=Fourier Analysis and Its Applications |chapter=Chapter 2: Fourier Series |pages=18 ''ff'' |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PA18 |pages=18 ''ff'' |year=2009 |isbn=0821847902 |publisher=American Mathematical Society}}
*{{cite book |title=Music and sound |author=Llewelyn Southworth Lloyd |url=http://books.google.com/books?id=LxTwmfDvTr4C&pg=PA156&dq=%22any+periodic+vibration+of+wave-length+%22&hl=en&sa=X&ei=4YZ8T4z7BqaMigLRj5GfDQ&ved=0CD8Q6AEwAA#v=onepage&q=%22any%20periodic%20vibration%20of%20wave-length%20%22&f=false |page=156 |isbn=0836951883 |year=1937 |publisher=Ayer Publishing}}
*{{cite book |url=http://books.google.com/books?id=ZIZmyOG-DxwC&pg=PA205&dq=%22is+a+periodic+function+of+wavelength%22&hl=en&sa=X&ei=Ooh8T9T1F6ThiALYmsW_DQ&ved=0CDYQ6AEwAA#v=onepage&q=%22is%20a%20periodic%20function%20of%20wavelength%22&f=false |title=Schaum's outline of theory and problems of optics |author=Eugene Hecht |page=205 |publisher=McGraw-Hill Professional |year=1975 |isbn=0070277303}}
*{{cite book |title=An Introduction to Mineral Sciences |page=65 |quote=''Fourier analysis'' is a mathematical method of expressing any periodic function with wavelength &lambda; as a sum of sinusoidal functions whose wavelengths are integral fractions of &lambda; (''i.e.'' &lambda;, &lambda;/2, &lambda;/3, ''etc.) |url=http://www.amazon.com/Introduction-Mineral-Sciences-Andrew-Putnis/dp/0521429471#reader_0521429471 |isbn=0521429471 |year=1992 |publisher=Cambridge University Press |author=Andrew Putnis}}

I don't think this description need appear in the introduction, but perhaps in the section ]. It should appear in ] because it is a general concept of wavelength for waves of arbitrary shape, and makes the connection to ], which is an important idea in the theory of waves that the reader should become aware of. ] (]) 16:28, 4 April 2012 (UTC)

:The first hardly mentions waves, and in the one example where it does, it has the wavelength in milliseconds – hardly a useful source for wavelength. The second doesn't appear to mention wavelength at all. Let's not make this an article on Fourier series or periodic functions. ] (]) 17:09, 4 April 2012 (UTC)
::Apparently the ''mention'' of Fourier series makes this ''about'' Fourier series. That is like saying the mention of Hilbert space makes an article on quantum mechanics about Hilbert space. I think you can address this suggestion more seriously. ] (]) 17:28, 4 April 2012 (UTC)
::Although it is difficult to take seriously your suggestion that Fourier series are unrelated to wavelength, I've added <s>two</s> three more sources that make this point verbatim. ] (]) 17:50, 4 April 2012 (UTC)

:Brews, I don't see any benefit at all in introducing the treatment you describe to this article. It adds a bunch of math that conveys no relevant information that is not already covered in the article.--] (]) 02:34, 5 April 2012 (UTC)

::Agreed. Any complete transform, orthogonal or otherwise, would give the same result, which is that a periodic function can be decomposed as a sum. I can't see what introducing Fourier transforms or series here does for the concept of wavelength, or why this transform is more interesting than, say, a Haar transform, or polynomials, or sinc functions, or wavelets. It's not clear why Brews call this "A more fundamental and rigorous approach". Of course, I understand that sinusoids, being the eigenfunctions of continuous-time linear systems, do have a special role to play, especially in analyzing dispersion, or how the shape of a wave changes with location. That, however, it a topic incompatible with this assumption of periodic functions what propagate while holding their shape. The sinusoidal decomposition would be useful in an article on wave propagation and dispersion, but I can't see how it's helpful here. ] (]) 04:24, 5 April 2012 (UTC)
:::Dicklyon and Srleffler: I believe you both misunderstand the purpose of mentioning Fourier series here. It introduces the mathematical background for expressing a general periodic waveform as a superposition of simple sinusoids. It is the mathematical extension of the remark in the introduction that "The concept can also be applied to periodic waves of non-sinusoidal shape". It is most simply applicable in non-dispersive media, as the Fourier series in its simplest form describes propagation of a wave of fixed shape .
:::It seems to me that these simple remarks can assist readers to understand the use of Fourier series in the general application of the concept of wavelength, and lead them to the relevant WP articles on this topic. Such connections are a major part of the value of WP, as is attested to in almost every appraisal.
:::Your remarks indicate a bias against mathematical explanation. That is your personal right, of course, but it should not be imposed upon every reader of WP.
:::So I suggest that you provide some sensible objections to this well-sourced and pertinent addition to the article.] (]) 15:00, 5 April 2012 (UTC)
::::I think you're still very confused about the physical/mathematical purpose of decomposing a periodic wave into sinusoidal components. Nothing in what you've said takes any advantage of the components being sinusoidal, and in a nondispersive medium, there is no advantage that I'm aware of, since all the components have the same velocity and you might as well just propagate the original waveform. I have no bias against using a decomposition into eigenfunctions to help analyze a linear system, but that's not what you're doing. ] (]) 15:22, 5 April 2012 (UTC)
:::::OK, Dick. I see that you have a completely different orientation to this topic than I intend. Maybe I can explain better. The subject of this article is wavelength, and I wish to stick very closely to that topic. The introduction says, as mentioned above, "The concept can also be applied to periodic waves of non-sinusoidal shape". I pose this question: how would you support this statement mathematically? Suppose for the moment that you want to undertake that support. The most general "periodic wave of non-sinusoidal shape" is a Fourier series. As stated by {{cite book |title=An Introduction to Mineral Sciences |page=65 |quote=''Fourier analysis'' is a mathematical method of expressing any periodic function with wavelength &lambda; as a sum of sinusoidal functions whose wavelengths are integral fractions of &lambda; (''i.e.'' &lambda;, &lambda;/2, &lambda;/3, ''etc.) |url=http://www.amazon.com/Introduction-Mineral-Sciences-Andrew-Putnis/dp/0521429471#reader_0521429471 |isbn=0521429471 |year=1992 |publisher=Cambridge University Press |author=Andrew Putnis}}. That is the point I want to drive home to the reader. ] (]) 15:30, 5 April 2012 (UTC)
::::::Even if it were useful to support the statement mathematically, it is not clear to me that you have done so. As I see it, the application of "wavelength" to the spatial period of a non-sinusoidal wave is purely a matter of definition. It doesn't really matter whether one says that the distance over which the wave repeats is its "wavelength", or that the "wavelength" of the wave is equal to the wavelength of the lowest-order component in its Fourier series. These are the same thing; neither is more fundamental than the other. The only difference between them is that the Fourier-based explanation is harder to understand. Fourier expansions can be very useful, and are often helpful in understanding the fundamental physics of a system. In this case, however, the expansion is not useful, and does not make the physics any clearer.--] (]) 03:12, 6 April 2012 (UTC)
:::::::Srleffler: It is now apparent that you do not understand the relation between math and the physics it describes. ] (]) 05:37, 6 April 2012 (UTC)
::::::And it's not clear what point you're trying to drive home to the reader, or how a sinusoidal decomposition helps. Your statement that "The most general 'periodic wave of non-sinusoidal shape' is a Fourier series" is complete hogwash; it's no more general than any other description of a periodic wave, which was my point in mention some of the other possible decompositions. If there's something special about sinusoids, you need to understand what that is. Fourier used them because they were the eigensolutions of the system he was analyzing, and that's the kind of physical situation where they become helpful. They are very helpful indeed for analyzing waves in continuous linear systems that are dispersive, but mentioning them as you did there gives no connection to that. And you can't even start to do a Fourier series of a periodic wave until after you've identified the period, so it seems all the more pointless. As for the application of the concept of wavelength to waves of nonsinusoidal shape, even that is a rather minority usage; usually it's applied only to waves that are at least locally nearly sinusoidal, or have one crest and one trough per cycle, so there's an obvious meaning to distance between adjacent crests or troughs. The extension to arbitrary periodic waves doesn't include the usual not-quite periodic case, but opens up a can of worms in terms of different periods (and near periods), in way that is very seldom encountered in the literature; just ref 18. ] (]) 04:43, 6 April 2012 (UTC)
:::::::Your view that it is "hogwash" to say a Fourier series can represent the most general (bounded, integrable) periodic function means to me that either (i) you misread the statement as some kind of claim that ''only'' a Fourier series can do this, which was never said, or (ii) you have no idea what is going on. I'll adopt the first view, just for the sake of things. The rest of your discussion about why exactly Fourier did what he did and where eigenfunctions are helpful is not part of the discussion. Dick, you are so anxious to argue applications that you cannot get the subject of the discussion straight. The object is to provide ''mathematical'' support to the statement ""The concept can also be applied to periodic waves of non-sinusoidal shape". There is ''nothing'' in this statement of simple fact or its mathematical support that relates to your comments. ] (]) 05:37, 6 April 2012 (UTC)
::::::::The statement I objected to was ""The most general 'periodic wave of non-sinusoidal shape' is a Fourier series", not "a Fourier series can represent the most general (bounded, integrable) periodic function". And as Srleffler pointed out, this mathematical complication is in no way providing support for the statement that "The concept can also be applied to periodic waves of non-sinusoidal shape". I admit to being completely baffled by what your point is in introducing sinusoidal decomposition of periodic functions here. ] (]) 06:04, 6 April 2012 (UTC)
{{outdent|9}}Dick: Your bafflement is due to your fixation on the wrong subject. The matter is logically as follows:
#The statement is made in the introduction that "The concept can also be applied to periodic waves of non-sinusoidal shape".
#The observation now can be made that any periodic wave can be represented by a Fourier series.
#The wavelength of a wave represented by a Fourier series is that of the leading sinusoidal term.
#QED

There is no need to inquire into the apparatus of ], what to do with quasi-periodic functions, or any of the other issues you have brought up. ] (]) 12:40, 6 April 2012 (UTC)

:This explanation of your idea avoids some of the difficulties encountered above. The problem with it is that nothing is proved; there is no "QED". Step 3 is simply a ''definition'' of the term "wavelength", as applied to nonsinusoidal waves. Defining the wavelength of such waves as the wavelength of the leading sinusoidal term is not better than defining it as the distance over which the function repeats in space; they are fundamentally the same thing. Your formalism doesn't provide any useful "mathematical support". It adds complexity, without being any more rigorous or any more useful.--] (]) 04:44, 7 April 2012 (UTC)
::Srleffler: Of course they are the same thing; that is the whole point. That is the QED. The Fourier series shows a mathematical mechanism ''in principle'' for establishing the wavelength of an arbitrary periodic waveform. ] (]) 15:43, 7 April 2012 (UTC)
:::And it would be equally true and meaningless to use square waves, or triangle waves, or sawtooth waves. Of course, there are infinitely many periodic waveforms that would be missing a fundamental component (different ones for the different basis functions, though); so you could analyze it with different basis functions and hope to find at least one with a fundamental. To be sure to get one, you could use the function itself as the fundamental basis function. Gee, math is fun. ] (]) 04:59, 7 April 2012 (UTC)
::::Dick: of course Fourier series apply to square waves, or triangle waves, ''etc''. That is what is meant by saying Fourier series applies to an ''arbitrary periodic function''. And, again, there is ''no claim for uniqueness''; we need only ''one'' way to establish the point, even though many ways may exist. ] (]) 15:43, 7 April 2012 (UTC)
:::::Sorry if I was unclear. It's not that square waves can be decomposed into sinusoids. I'm saying you could equally well decompose your periodic function into square waves, or various other basis functions (even if they're not orthogonal, so not the usual generalization of Fourier transforms). In all such cases though, you need to know the period before you start, so how it has anything to do with "establishing the wavelength of an arbitrary periodic waveform" remains elusive. ] (]) 15:49, 7 April 2012 (UTC)
::::::Dick: You are thinking in a practical manner again. The abstract notion here is to demonstrate that the thing can be done, not necessarily how to go about it. ] (]) 15:54, 7 April 2012 (UTC)
::::::Actually, as Srleffler and the Fourier theorem itself state, as a practical matter, if one knows f(&xi;+&lambda;)=f(&xi;) you already know the wavelength if you can establish &lambda; as being the shortest such length; there may be anomalous cases where you aren't sure, and matching a Fourier representation to the function could help you decide what &lambda; was. That seems most likely to arise when trying to approximate some experimental data with a background noise level. The technical point, however, is that the approach exists in principle, regardless of its practicality, and shows a &lambda; can be found for an arbitrary periodic function. ] (]) 16:18, 7 April 2012 (UTC)
{{outdent|8}}Dick and Srleffler: Aside from providing a general technique to establish wavelength, there is value in pointing out the connection of wavelength to Fourier series for no other reason than Fourier series are a very fundamental part of functional analysis, and the connection of wavelength to this seminal apparatus has value for the reader all by itself. ] (]) 15:54, 7 April 2012 (UTC)

== Reference to Fourier series ==

{{rfc|sci|rfcid=DDA12B5}}
Comment is sought as to whether a reference to Fourier series is appropriate under the heading ]. Some editors appear to find connecting Fourier series to wavelength is a digression. The text in question is provided below. ] (]) 20:09, 20 April 2012 (UTC)

===Text referring to Fourier series===
This text is to appear following the equation defining the wavelength of a periodic function in the section ], just before the header ]:
{| cellpadding="2" style="border: 1px solid darkgray; background:#E6F2CE;" align="center" {| cellpadding="2" style="border: 1px solid darkgray; background:#E6F2CE;" align="center"
|The wavelength, say &lambda;, of a general spatially periodic waveform is the spatial interval in which one cycle of the function repeats itself. Sinusoidal waves with wavelengths related to &lambda; can superimpose to create this spatially periodic waveform. Such a superposition of sinusoids is mathematically described as a ], and is simply a summation of the sinusoidally varying component waves: |The wavelength, say &lambda;, of a general spatially periodic waveform is the spatial interval in which one cycle of the function repeats itself. Sinusoidal waves with wavelengths related to &lambda; can superimpose to create this spatially periodic waveform. Such a superposition of sinusoids is mathematically described as a ], and is simply a summation of the sinusoidally varying component waves:
|-
|.. "''Fourier analysis'' is a mathematical method of expressing any periodic function with wavelength &lambda; as a sum of sinusoidal functions whose wavelengths are integral fractions of &lambda; (''i.e.'' &lambda;, &lambda;/2, &lambda;/3, ''etc.)"<ref name=Putnis1 group = Note/>
|- |-
|.. "''Fourier's theorem'' states that a function ''f(x)'' of spatial period &lambda;, can be synthesized as a sum of harmonic functions whose wavelengths are integral submultiples of &lambda; (''i.e.'' &lambda;, &lambda;/2, &lambda;/3, ''etc.'')."<ref name=Schaum group=Note/> |.. "''Fourier's theorem'' states that a function ''f(x)'' of spatial period &lambda;, can be synthesized as a sum of harmonic functions whose wavelengths are integral submultiples of &lambda; (''i.e.'' &lambda;, &lambda;/2, &lambda;/3, ''etc.'')."<ref name=Schaum group=Note/>
Line 167: Line 66:
|'''References''' |'''References'''
{{Reflist |group=Note|refs= {{Reflist |group=Note|refs=

<ref name=Putnis1 group = Note>
Quotation from {{cite book |title=An Introduction to Mineral Sciences |page=65 |url=http://books.google.com/books?id=yMGzmOqYescC&pg=PA65&dq=%22Fourier+analysis+is+a+mathematical+method+of+expressing+any+periodic+function+with+wavelength%22&hl=en&sa=X&ei=XimUT6WQKcnciAL3qJkY&ved=0CDIQ6AEwAA#v=onepage&q=%22Fourier%20analysis%20is%20a%20mathematical%20method%20of%20expressing%20any%20periodic%20function%20with%20wavelength%22&f=false |isbn=0521429471 |year=1992 |publisher=Cambridge University Press |author=Andrew Putnis}}
</ref>

<ref name=Schaum group =Note> <ref name=Schaum group =Note>
{{cite book |title=Schaum's Outline of Theory and Problems of Optics |publisher=McGraw-Hill Professional |url=http://books.google.com/books?id=ZIZmyOG-DxwC&pg=PA205&dq=%22can+be+synthesized+as+a+sum+of+harmonic+functions+whose+wavelengths+are+integral+submultiples%22&hl=en&sa=X&ei=7rCVT4jJCYKpiQL2iuWICg&ved=0CDIQ6AEwAA#v=onepage&q=%22can%20be%20synthesized%20as%20a%20sum%20of%20harmonic%20functions%20whose%20wavelengths%20are%20integral%20submultiples%22&f=false |page=205 |author=Eugene Hecht |year=1975 |isbn=0070277303}} {{cite book |title=Schaum's Outline of Theory and Problems of Optics |publisher=McGraw-Hill Professional |url=http://books.google.com/books?id=ZIZmyOG-DxwC&pg=PA205&dq=%22can+be+synthesized+as+a+sum+of+harmonic+functions+whose+wavelengths+are+integral+submultiples%22&hl=en&sa=X&ei=7rCVT4jJCYKpiQL2iuWICg&ved=0CDIQ6AEwAA#v=onepage&q=%22can%20be%20synthesized%20as%20a%20sum%20of%20harmonic%20functions%20whose%20wavelengths%20are%20integral%20submultiples%22&f=false |page=205 |author=Eugene Hecht |year=1975 |isbn=0070277303}}
</ref> </ref>
}} }}
|} ] (]) 12:41, 15 May 2012 (UTC)
|}


: Brews, drop it. You've ], you've ] on this, neither time did you convince other editors. Proposing yet another variation on it after failing to convince other editors multiple times is simply disruptive.--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 16:18, 15 May 2012 (UTC)
===Author's remarks===
::I agree. This is a dead issue. Brews, ].--] (]) 17:03, 15 May 2012 (UTC)
:This text states the specific connection between wavelength and Fourier series as applied to spatial periodicity. It alerts readers to this connection between wavelength and some very important mathematics.
:::Srleffler: As Blackburne has not advanced any actual argument against inclusion of this text, your "agreement" with him is only as to his cheer-leading and not about any "agreement" upon substance. Your own comments regarding the RfC on the above text were as follows:
::::"The introduction of Fourier series appears neither to make the concept of wavelength clearer nor to provide a better fundamental definition of wavelength of a general periodic wave. The current proposed text is admittedly better than the many attempts prior to this RFC, in that it doesn't belabour the issue and focuses most directly on the connection between the two topics.
:Some discussion appears on this Talk page regarding a previous suggestion involving a more elaborate discussion of Fourier series presented ]. That discussion is not pertinent in the present case because what we have here is a much more curtailed description. ] (]) 13:47, 22 April 2012 (UTC)
::::I do see the appeal in trying to replace the definition of wavelength of a non-sinusoidal wave in terms of the wave's period of repetition with a definition that is tied directly to sinusoidal waves. I'm partial to this for the same reason that I was originally opposed to applying the term "wavelength" to non-sinuoidal waves at all. It's not clear to me that there is a non-negligible set of readers for whom this treatment would be beneficial, however."--Srleffler (talk) 18:13, 22 April 2012 (UTC)
:::As I understand these points you raise, your objection to including this text is that it does not clarify the concept of ''wavelength''. However, that is not the purpose of this text. What this text aims to do is to alert readers that there is a connection of spatially periodic waveforms (waveforms with a ''wavelength'') to Fourier series. That connection is undeniable.
:::Your further objection is that nobody cares anyway. Inasmuch as several sources mention this connection, and indeed elaborate upon it at length, your opinion is not universal. ] (]) 19:16, 15 May 2012 (UTC)
::::You have raised this issue before, multiple times. We have spent far more time discussing it than it was worth. No further discussion of this issue is merited. Please stop trying to disrupt the editing process by repeatedly bringing forward the same issues over and over again with only slight variations. --] (]) 03:45, 16 May 2012 (UTC)
:::I think, Srleffler, that your objections are in fact against an earlier proposal to introduce Fourier series as a ''definition'' of wavelength, which is not proposed here. You may have a different opinion about the present proposal. ] (]) 19:31, 15 May 2012 (UTC)


::::The truth is that few sources make that connection, and they don't take it anywhere useful. You have gone back to a formulation that would be just as true and useful if triangle waves were used instead of sine waves; that is, not useful at all, since the sinusoidal components provide no help in analyzing such a situation, where the medium is either nondispersive or nonlinear. The Schaum's Outline book that you cite introduces the Fourier series there only as a step toward getting a Fourier transform, to get a way to represent waves that are NOT period in space, which is useful; and it says it's more common to do it in terms of k than lambda, which is true, so it's not very related to wavelength. And your statement that "The wavelength, say &lambda;, of a general spatially periodic waveform is the spatial interval in which one cycle of the function repeats itself" is contrary to typical usage of the term "wavelength" (that is, for the local wavelength of approximately sinusoidal waves) and is not supported by the source; in fact, your source defines the term "wavelength" only with respect to sinusoidal components, and applies it only fleetingly to a spatially periodic function. The text (if you can call it that) is also flaky in that when it introduces sinusoids in section 1.3 it completely misses their point, again saying something that would be just as true with triangle waves or square waves or a variety of other basis sets. We have been through all this many times. The objections of numerous editors are in the record if you'd like to review them further. ] (]) 23:29, 15 May 2012 (UTC)
===Comments===
:::::Dick: You are missing the point here. There is no attempt to propose that ] is the one and only way to expand an arbitrary function in terms of other functions, which might fall under the rubric of ]. The point here is simply to make the connection of a spatially periodic function of general form that satisfies ''f(x+&lambda;)=f(x)'' to the Fourier series. A Fourier series, as you must know, inevitably results in a periodic function throughout space. Fourier series is, moreover, a very well known and important aspect of mathematical analysis, and a link to make readers aware of the connection is just an ordinary use of an aside that widens the reader's appreciation of the topic ''wavelength'' and its connection to the mathematical analysis of periodicity. The text contains a ''direct quote'' from a textbook, and virtually the same language occurs in other as well: "Fourier's theorem states that any periodic function f(x) can be expressed as the sum of a series of sinusoidal functions which have wavelengths that are integral fractions of the wavelength λ of f(x)"
"It is noteworthy that the wavelength of a general periodic waveform is related mathematically to its ] expression..." is very odd. First of all, the source does not support a claim that "it is noteworthy"; that just sets off BS detectors. Second, the relationship of the wavelength (or the period) of a general periodic function to it Fourier series is simply that you need to know the former to compute the latter. How does this help explain or understand wavelength? And if he had used the word "period" like everyone else, would we even be discussing it? No. The only reason this guy cares about Fourier analysis of things with wavelengths is that he's doing X-Ray crystallography, and I don't think it's appropriate to get into that here; maybe a link from the crystals section would be OK. ] (]) 01:57, 21 April 2012 (UTC)
:::::There is nothing misleading or inappropriate here, as you well know, and your unsupported assertions to the contrary do not reflect well upon your understanding of the subject, nor indeed, upon your appreciation of one of the major benefits of WP: helping readers widen their awareness of a topic. ] (]) 14:11, 16 May 2012 (UTC)
:Dick, the source is cited as the origin of the quotation. I've moved the footnote to make this clearer.
::::::There is no need in Fourier series to ''define'' what wave length is. Hence one should not tell about this in the ''definition''. On the other hand, there is nothing wrong to mention Fourier series somewhere in the article. Bringing that kind of dispute to Arbcom seems incredibly strange to me. ] (]) 04:35, 17 May 2012 (UTC)
:This author's use of "wavelength" in the quotation is entirely appropriate and accurate and often used. Ordinarily (although not invariably) ''period'' is taken to refer to periodicity in time or with respect to some general variable, say &xi; or &theta;, while ''wavelength'' invariably refers to periodicity in space. Of course, periodicity can be expressed in many different ways, but ''wavelength'' is the subject here, and ''wavelength'' is used to define a periodic function in the subsection ] where this text referring to Fourier series is proposed to be placed. The mathematical definition is stated in the article as ''f''(''x−vt+&lambda;'')=''f''(''x−vt''). So I don't accept your stance that "wavelength" is being awkwardly squeezed into the discussion by using an uncommon usage of terminology.
:As for the importance of this topic, the entire subject of waveforms periodic in space with a certain wavelength is inextricably connected to Fourier series. Perhaps you can suggest some more innocuous wording to replace "noteworthy" that you would find acceptable? ] (]) 15:25, 21 April 2012 (UTC) :::::::I had already crafted a paragraph to say what could sensibly be said about Fourier series, applied to periodic-in-time waves, but nobody much liked it and it wasn't particularly relevant to wavelength, so I took it out; nobody objected to that. ] (]) 05:37, 17 May 2012 (UTC)
::I'll wait until we get other comments. This is supposed to be a RFC, not us arguing some more. ] (]) 15:46, 21 April 2012 (UTC) ::::::::So, what is exactly the problem with describing non-sinusoidal waves using Fourier series? I do not see any problems. But probably this belongs to other articles about waves. ] (]) 04:14, 18 May 2012 (UTC)
:::I removed "noteworthy" as unnecessary. ] (]) 16:07, 21 April 2012 (UTC)


== Introduction edit ==
: I agree with ]. Fourier analysis is a tool for mathematical study of periodic functions. It has nothing in particular to do with wavelength. You can use Fourier analysis on a purely time-based phenomenon or an abstract function, neither of which has a wavelength. The Fourier series does not help readers understand waves or wavelength. Quite the opposite. And the source for the quote is a narrow technical one inappropriate for a mathematical article (and a commercial link is entirely inappropriate for a reference – let readers find that via the ISBN if they want to).--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 19:56, 21 April 2012 (UTC)
::John, you are making a statement logically equivalent to this: A knife is a cutting implement so it is irrelevant to point out under "weapon" that a knife can be used this way. Likewise, Fourier analysis can be viewed as a general discussion of periodicity in any variable. So you claim in particular, and as is the topic here, it is irrelevant to point out its use to analyze spatial variation. ] (]) 20:44, 21 April 2012 (UTC)
:::You can reinterpret my statement as you like; my point is I agree with ] and now ] that it doesn't belong.--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 21:08, 21 April 2012 (UTC)
::::John, no, the point is you "agree" with these editors but offer no reason for doing so, as I have pointed out. ] (]) 00:57, 22 April 2012 (UTC)
::::My reasons are as you seem to have trouble finding them.--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 11:04, 22 April 2012 (UTC)
:::::Right, John. As to your comments that you have linked with caustic observation about myself, I see again that you propose as a "reason" for your supposed "agreement" with DickLyon, that because Fourier series have wide application to periodicity in general, therefore my proposed text stating the specific connection between wavelength and Fourier series as applied to spatial periodicity is neither useful nor appropriate. I think you can see your position is neither logical nor sensitive to the key role of WP in broadening a reader's grasp of the context of a topic they are reading about. ] (]) 13:39, 22 April 2012 (UTC)


The sentence "For example, in sinusoidal waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength" should be edited to mention that this is true for a particle on the surface, particles below the surface moving in smaller circles. <span style="font-size: smaller;" class="autosigned">— Preceding ] comment added by ] (]) 17:30, 1 March 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
*'''Oppose''' It has nothing in particular to add to the topic of the article and thus strays off topic. ] (]) 20:24, 21 April 2012 (UTC)
:Fixed.--] (]) 00:14, 2 March 2013 (UTC)
::IRWolfie: How exactly is it straying off topic to point out the connection between wavelength and Fourier series for spatially periodic functions? Are you claiming the connection is not there, or that the role of wavelength does not enter the construction of a Fourier series for a spatially varying function? Either claim would directly contradict the cited sources (or in fact, any sources). ] (]) 20:44, 21 April 2012 (UTC)
::The point is that ''wavelength'' is defined in general in terms of a general, spatially periodic waveform, and such waveforms are described by Fourier series having a particular relation to the wavelength of the waveform. That is a nontrivial observation and connects the topic of wavelength to the very important context of Fourier series. A sentence or two about this is not a serious expansion of the article and is a small addition to provide this important connection for the topic of wavelength. ] (]) 01:35, 22 April 2012 (UTC)
:::Regarding your first paragraph: the fact that wavelength plays a role in the construction of Fourier series does not help your argument. The problem is that the connection goes the wrong way. Wavelength is certainly relevant to Fourier series, but that does not imply that Fourier series are relevant to an article on wavelength. Relevance is not always bidirectional.
:::Regarding the second paragraph: It is not clear to me that this observation is "nontrivial". --] (]) 16:00, 22 April 2012 (UTC)
::::I agree that backing Fourier series into a discussion of wavelength goes the wrong way around. A "See also" entry, or an inline wikilink such as the one in the section on wave packets, should suffice. __ ] (]) 16:17, 22 April 2012 (UTC)
:::::''Hi Srleffler'': I am happy that you see wavelength plays a role in Fourier series for spatially periodic functions. Your general approach to rejection of this observation in ] is that somehow the importance of wavelength to Fourier series does not support its mention here because Fourier series is "not relevant to wavelength". That stance seems strange to me. You point out that this type of objection is not universally applicable, so for instance a ''pumpkin'' might be referred to in an article on ''vegetables'', and ''vice versa''. Perhaps you could explain further why the connection of wavelength to the sub-domain of Fourier series for spatially varying functions is of no interest to those interested in wavelength? Aren't readers of ] entitled to know about this connection to a truly gigantic area of mathematics? ] (]) 16:25, 22 April 2012 (UTC)
::::::This is a general issue that comes up now and again in discussions of appropriate content for Misplaced Pages articles. The fact that topic A is of great importance to field B justifies mentioning A in the article on B. It ''does not follow'', however, that topic B deserves a mention in the article on A. Usually, B ''will'' be worth mentioning but for other reasons, not merely because A is important to B. Cases where there is strong one-way relevance are not uncommon, but are certainly not the majority. As a result, when they arise it may provoke a talk page discussion such as we have been having.
::::::The question we need to settle here is whether Fourier series have some relevance in the discussion of wavelength, regardless of the latter's importance to the former. --] (]) 18:13, 22 April 2012 (UTC)
:::::''Just plain Bill'': The suggested text is not " backing Fourier series into a discussion of wavelength". The idea of wavelength is introduced in the article via the periodicity condition ''f''(''x−vt+&lambda;'')=''f''(''x−vt''). That is exactly and inescapably the subject of Fourier series. Pointing out that connection in a sourced sentence is hardly a large cost to this article in terms of added space. ] (]) 16:25, 22 April 2012 (UTC)
{{od|5}}The suggested text has been through at least three iterations since that green box appeared above. To a reader passing by, it is no longer apparent which version was being discussed at various times. I would prefer to see new suggestions given their own space where they were introduced, without obliterating the previous one(s).


== Citation to "Subquantum Kinetics" pseudoscience ==
Wavelength may be helpful in understanding some applications of Fourier series. Fourier analysis is not necessarily helpful in understanding wavelength. That's the asymmetry I meant by "backing into." Space on the page is one thing, but introducing tangential material is, IMO, a distraction to the reader. Sourced, you say... it is our job to be selective about which sources are relevant to the subject. __ ] (]) 17:29, 22 April 2012 (UTC)
:That's my concern too. The introduction of Fourier series appears neither to make the concept of wavelength clearer nor to provide a better fundamental definition of wavelength of a general periodic wave. The current proposed text is admittedly better than the many attempts prior to this RFC, in that it doesn't belabour the issue and focuses most directly on the connection between the two topics.
:I do see the appeal in trying to ''replace'' the definition of wavelength of a non-sinusoidal wave in terms of the wave's period of repetition with a definition that is tied directly to sinusoidal waves. I'm partial to this for the same reason that I was originally opposed to applying the term "wavelength" to non-sinuoidal waves at all. It's not clear to me that there is a non-negligible set of readers for whom this treatment would be beneficial, however.--] (]) 18:13, 22 April 2012 (UTC)
::It's hard to use Fourier series to define wavelength, just as it's hard to use it to define ], since there may not by a sinusoidal component with wavelength matching the wavelength that you're trying to define, and because you need to know the wavelength before you can do a Fourier series. But we could perhaps connect it in by noting that non-sinusoidal waves are sometimes analyzed in terms of superimposed sinusoids, each of which has its own well-defined wavelength. The current "The wavelength of a general periodic waveform is related mathematically to its Fourier series..." completely misses the point, and provides no motivation for the claimed relationship, and no real role for the series, or the decomposition that it induces. The X-ray crystallography source isn't clear on this either, and in that field it would be much more fruitful to use wavevectors, since the wavelength formulation doesn't extend to the multi-dimensional case very naturally. If we had a source about decomposing waves into sinusoidal components for the purpose of easier analysis, that might make sense somewhere here; but it wouldn't necessarily be tied to perioidic waves, since its applicability is broader than that; so it might need to be Fourier transforms instead of series. Like Brews, many authors have forgotten what's special about sine waves that motivates such decompositions (for example, : "Sinusoidal waves are important because they occur in many physical situations..." -- seriously?). ] (]) 18:37, 22 April 2012 (UTC)
:::I am happy to see this conversation turning to real issues. It seems there are several points raised that I'd like to try to summarize and respond to, hopefully in a constructive manner:
::# Fourier series for spatially periodic waveforms ''employ'' wavelength, but do not illuminate the concept of ''wavelength''. That seems to me a valid observation. I'd argue, however, that because wavelength shows up prominently in such Fourier series, it is reasonable to point to this fact here. Gasoline may not be illuminated by its role in the internal combustion engine, but it can be of interest in an article on gasoline that it is used in internal combustion engines, and ''vice versa''.
::# DickLyon's remark: "you need to know the wavelength before you can do a Fourier series". This comment relates to using Fourier series to ''determine'' wavelength. That is an interesting exercise, but it is not the proposed reason for inserting this brief reference to Fourier series here. It seems to me that "using Fourier series" to determine the wavelength of a complicated repeating waveform is most likely to be a practical undertaking in the theory of separating a signal from background, where one might fit some experimental data with &lambda; as a variable parameter and adjust it for a best fit. However, that or other practical instances where one would use Fourier series to ''determine'' wavelength, seems beyond the intended scope of simply pointing out that Fourier series use &lambda;.
::# DickLyon's remark: "it would be much more fruitful to use wavevectors, since the wavelength formulation doesn't extend to the multi-dimensional case very naturally." This article is about ''wavelength'', so it is a digression to ask whether Fourier series are more easily generalized using different concepts.
::# DickLyon's remark: "its applicability is broader than that; so it might need to be Fourier transforms instead of series." If one wishes to discuss non-periodic waveforms, the ''wavelength'' does not come up because its underlying definition is ''f''(''x−vt+&lambda;'')=''f''(''x−vt''), which does not apply except to periodic functions. Later in the article wave packets are described, but the text about Fourier series suggested here is proposed for insertion in the section on general ''periodic'' waveforms.
::# DickLyon's remark: "many authors have forgotten what's special about sine waves that motivates such decompositions". If there is something particular that should be said about &lambda; in this regard, that is an argument for adding to the proposed text, not for eliminating it. For example, one could digress to suggest that different musical instruments playing the "same" note have different voices because their Fourier series have different terms.
::] (]) 21:47, 22 April 2012 (UTC)
'''Weak Support''' I support mentioning Fourier series in passing, since I think it's a related concept. However I agree that it's useless for explaining what "wavelength" is to someone that doesn't know. If you understand Fourier analysis, you know what wavelength is. But the converse isn't true, so it's useful to have a mention in this article so that someone learning about wavelength here can go on and learn about Fourier. <small>'''<span style="color:Olive">Waleswatcher</span>''' ]</small> 18:20, 22 April 2012 (UTC)


Why on earth is there a citation to "Paul A. LaViolette (2003). Subquantum Kinetics: A Systems Approach to Physics and Cosmology" (citation for "the notion of a wavelength also may be applied to these wave packets"). That book is utterly pseudoscientific (and the author is known for some way out fringe ideas), and does not belong in a science article in my opinion. Aren't there better references to use for wave packets? ] (]) 23:42, 1 January 2015 (UTC)
:'''Oppose:''' Just because wavelength and Fourier series relate to waves, doesn't imply there is any important connection, even ''mentioning'' the Fourier series '''really isn't essential''' at all. This conversation on including something non-essential to an article is also ''fairly silly'', for such a ''petty issue''. Why include it? Adding it to the "see also" section is plenty if you're that desperate to mention it in passing. That bunch of maths given right at the top is needless for understanding wavelength.
:Fixed. Thanks for pointing that out.--] (]) 05:01, 2 January 2015 (UTC)


== First use of lambda ==
:A much simpler mathematical approach to understanding how the wavelength is associated with the phase of a wave is the fractions of a wave cycle:


When was lambda first used for wavelength? I noticed Fresnel used λ in 1819 in 'Memoire on the diffraction of light' and Herschel used λ = v T in 1828 in 'On the Theory of Light'. ] (]) 09:49, 23 October 2015 (UTC)
::<math>\frac{x}{\lambda} = \frac{\phi}{2\pi} = \frac{\tau}{T} </math>


How to calculate wavelength ] (]) 18:40, 11 March 2020 (UTC)
:where ''x'' is a spatial length, <math>\phi</math> is a phase angle, <math>\tau</math> is a time lag/lead. This formula is not in the article, though would be more instructive than a Fourier series becuae it states the angular and time analogues to wavelength. Even so this is not essential for inclusion. Another thing not in the article is that wavelength can be calculated by:


== Vacuum wavelength ==
::<math>\lambda = \frac{L}{N} </math>


Presently the wikilink ] goes to this article, but it is not clarified. I would suggest adding something like the following blurb.
:where ''N'' is the number of wave cycles passing through two fixed points of seperation ''L''. More intuitive than the Fourier series, becuase this formula actually says what wavelength '''is'''?? Obviously not used in practice, its only for theoretical interest and definitely not essential. <span style="font-family:'TW Cen MT';">] ] ]</span> 23:38, 22 April 2012 (UTC)
{{quote|text=

When light passes between different materials, the wavelength changes although the frequency stays the same. In the field of ], it is rare to refer to the invariant frequency of light, but instead to refer to the ''vacuum wavelength'' of light, which is ] divided by the light frequency. This convention is commonly used even when describing light inside materials where the actual wavelength of the light is not the same as the vacuum wavelength.
*'''Comment'''. The proposed text seems very strange. The wording of "The wavelength of a general periodic waveform is related mathematically to its Fourier series expression as a summation of sinusoidally varying waves" is too mysterious. ("related mathematically"? how?) As others have noted, it also emphasizes sinusoidal basis functions, which have nothing at all to do with the wavelength. I think a better alternative would be simply something like "The process of ] allows any periodic waveform to be expanded as a superposition of given basis waveforms whose wavelengths divide that of the original waveform." No real opinion on whether that's worth including though. ] (]) 00:22, 23 April 2012 (UTC)

::To talk about such other decompositions, as Brews did, goes even further off topic. There's a reason why sine waves are special here, and to talk about the decomposition while not saying the reason just misses the boat. ] (]) 02:39, 23 April 2012 (UTC)
:::''Sławomir Biały'': Your "better alternative" is a paraphrase of the quote provided. Is it worth including? Of course it is: the whole subject of the mathematics of periodic waveforms is exactly the subject of Fourier series, and periodic waveforms form the definition of ''wavelength'' itself. Why wouldn't this connection be pointed out? What is the cost here? Is it that it takes too much room? Nonsense. Is it that the reader already knows about the connection? Maybe in some cases, but certainly not all. Adding the proposed sentence simply is a pointer to wider horizons. There is ''no cost'' to including the proposed text. ] (]) 04:08, 23 April 2012 (UTC)

:::''DickLyon'': There is no reference to "other decompositions". There was originally, because you raised this issue, but it is there no more. And what is "the reason" for Fourier series that you allude to but never state explicitly? ] (]) 04:08, 23 April 2012 (UTC)
::::I raised it because what you said about sinusoids would have been equally true and applicable with other decompositions. The point was to illustrate how irrelevant the Fourier series was in your statement connecting it to wavelength, not to encourage you to generalize it. ] (]) 06:18, 23 April 2012 (UTC)
:::(@Dicklyon's original reply) I think your reply nicely summarizes some of my original reservations as well, that I was not able to clearly articulate. ] (]) 11:53, 23 April 2012 (UTC)
::::Sławomir Biały: It is unfortunate that you prefer to accolade the misconceptions about Fourier series entertained by Dick Lyon rather than address them. "Other decompositions" are not able to represent a periodic function throughput its domain, so Fourier series are special. Moreover, this topic of uniqueness really doesn't matter to the discussion, as the real issue is making this helpful connection between wavelength and the famous, powerful, and seminal apparatus of the Fourier series. ] (]) 13:08, 23 April 2012 (UTC)
:::::''"Other decompositions" are not able to represent a periodic function throughput its domain, so Fourier series are special.'' &larr; That's just wrong. ] (]) 16:43, 23 April 2012 (UTC)
:'''Oppose'''. I agree with most of the comment above. FWIW, I might phrase the alternative this way: "Any periodic function, with repetition interval λ, can be represented by a mathematically equivalent sum of sinusoids with repetition intervals λ/k, for k=1,2,...,∞. (See ])" But it would still seem incongruous in the proposed location. A footnote would be a little better. --] (]) 01:52, 23 April 2012 (UTC)
::I would object to putting odd content into footnotes. Brews has a long history of doing that, and I'm going to object if anyone tries it here. ] (]) 02:39, 23 April 2012 (UTC)

:::''Bob K'': Why is it "incongruous" or as DIck says "odd content" to point out the mathematical apparatus known as Fourier series that is the <u>''entire formal treatment''</u> of arbitrary periodic waveforms, when it is placed in a subsection about arbitrary periodic waveforms? Shouldn't the unknowing reader be made aware of this mathematical apparatus, and its connection to ''wavelength''? What is the ''cost'' of including such a pointer? I see none at all: it is all upside and no downside. ] (]) 04:08, 23 April 2012 (UTC)

:::::The incongruity is that the statement is an abrupt leap into the realm of frequency distributions without an explanation of why that might be useful. We need a seque. For instance: "Two periodic functions with the same wavelength are often compared in terms of their other frequency content." That opens another issue, which is ] or ]? Personally, I would rather see the continuous transform of one cycle of the waveform. The ] provides only discrete samples of that transform. But if I am now straying off topic, I think that only strengthens my original point about incongruity. --] (]) 12:29, 23 April 2012 (UTC)

::::::Bob K: A segue, or maybe a new subsection could lead to an elaboration of the significance of the terms in a Fourier series and what they tell us about the complex waveform they represent. I'd have no problem with that discussion, which could be an elaboration about how wavelength connects to harmonics, and what they mean physically, or how they distinguish an oboe from a violin. However, DickLyon would never accept this as a valuable subtopic.
::::::For the proposed text, however, a much more limited purpose is entertained. It is simply ''to point out'' that wavelength is a fundamental concept in the powerful, widely used, and seminal apparatus of Fourier series. The insertion is a ''heads up'', not a discussion. It connects Fourier series to wavelength simply and directly through a quotation. The insertion is proposed to appear in the subsection on arbitrary periodic waveforms, which is the subject of Fourier series, so further introduction seems unnecessary to me. ] (]) 13:33, 23 April 2012 (UTC)

:::::::I just don't find that approach interesting enough to get my vote. A more interesting approach is to <u>start</u> with a problem like analyzing the difference between an oboe and a violin, playing the same chord (if that's the right word... I'm not a musician). Point out that they would have the same wavelength, but different waveforms. That leads to the subject of harmonics and harmonic distributions and to Fourier series, if you want to take it that far. That said, this might not be the most appropriate place in Misplaced Pages for that information. And it might already exist someplace else that can simply be Wikilinked. --] (]) 00:56, 24 April 2012 (UTC)

::::No meaningful connection between wavelength and sinusoidal Fourier series components has yet been mentioned. ] (]) 04:14, 23 April 2012 (UTC)
:::::Dick: It is pointed out that the sinusoids entering the Fourier series have wavelengths that are integral fractions of the wavelength of the periodic waveform. That is mathematically meaningful. If you were to allow further digression, the subject of harmonics could be discussed, which would be physically meaningful as well. The topic of determining wavelength for a waveform in a noisy background by fitting it to the best-wavelength Fourier series could be examined. Perhaps you have some additional insights that you would be happy to see added? ] (]) 05:15, 23 April 2012 (UTC)

===Summary===
To this point, objections to including the ] take on two forms: (i) Fourier series do not illuminate the concept of ''wavelength'', a statement I agree with, and (ii) Fourier series is a "strange", "incongruous" and "odd" addition to an article on wavelength.

No-one seems brave enough to tackle the point here: the formal apparatus for dealing with arbitrary waveforms periodic in space <u>''is''</u> the Fourier series, no less. Indicating this point may make some readers of this subsection on arbitrary periodic waveforms aware of this hugely significant mathematical framework.

Addressing the connection to Fourier series involves a minor addition of a sourced one-line quotation used to associate Fourier series with wavelength, the topic of the article. It is a <u>''no-cost addition''</u> to the article. Its accuracy is not in question. Its value to some is not in question. It is of very minor length. So what is the problem here? ] (]) 04:35, 23 April 2012 (UTC)

:The problem is that you haven't yet made a sensible connection. If your assertion is true that ''the formal apparatus for dealing with arbitrary waveforms periodic in space <u>''is''</u> the Fourier series, no less'', then show us that with a source that says why, and maybe we'll be getting close to a useful connection. Actually, it's not true in general, but if you find the domain within which it is sort of true, you'll be on a good path to understanding how sinusoids relate to wavelength, and then maybe be able to say something sensible. ] (]) 06:16, 23 April 2012 (UTC)
::''DickLyon'': Your claim that "no sensible connection" has been made between Fourier series and wavelength seems to ignore the quote provided:
::::"Fourier analysis is a mathematical method of expressing any periodic function with wavelength λ as a sum of sinusoidal functions whose wavelengths are integral fractions of λ (i.e. λ, λ/2, λ/3, etc.)"
::Doesn't this statement make the connection between Fourier series and wavelength? Of course it does. ] (]) 12:55, 23 April 2012 (UTC)

::Dick, apparently you doubt the assertion that ''the formal apparatus for dealing with arbitrary waveforms periodic in space <u>''is''</u> the Fourier series''. It is completely true for arbitrary spatially periodic waveforms. I know you like to think of it as a subset of various methods for series expansions, which it is, but it is the subset that deals with arbitrary spatially periodic waveforms: ''none'' of the other ] deals with an arbitrary spatially periodic waveform over its entire domain. If you wish to take issue with this observation, please let me know. And please avoid commenting upon my abilities and good sense. ] (]) 12:33, 23 April 2012 (UTC)
::To repeat what was said before:
:::] include, for example, ] and ].<ref name=Folland group=Item/> However, most often these generalized series represent the function over a finite interval, say one wavelength, and do not represent the periodic function throughout its entire ].<ref name=orthogonal group=Item/> The length of the chosen interval appears in the analysis, but the concept of ''wavelength'', or spatial periodicity, is not fundamental to these generalized Fourier series.

::'''Items'''
{{Reflist |group=Item|refs=
<ref name=Folland group=Item>
{{cite book |author=Gerald B Folland |title=Fourier Analysis and Its Applications |chapter=Contents |pages=ix ''ff'' |url=http://books.google.com/books?id=ix2iCQ-o9x4C&pg=PR9 |year=2009 |isbn=0821847902 |publisher=American Mathematical Society |edition = reprint of Wadsworth & Brooks/Cole 1992 }}
</ref>

<ref name=orthogonal group=Item>
] expressions most often are based upon ]s defined over a finite interval, for example, solutions to the ] in an interval , so these expansions do not represent the periodic function outside this selected interval.
</ref>
}} }}
(More or less?)
:::] (]) 12:44, 23 April 2012 (UTC)
--] (]) 16:40, 21 July 2016 (UTC)

:Seems like a good idea.--] (]) 03:26, 22 July 2016 (UTC)
===The point===
The question of uniqueness of Fourier series in describing arbitrary spatially periodic waveforms is a nicety. Uniqueness establishes that other decompositions of a waveform do not employ the notion of ''wavelength'', or spatial periodicity of the arbitrary but periodic waveform. The real point, however, is that ] is a huge topic, a powerful apparatus, and it uses ]. The reader should be alerted to this connection. That "heads-up" is the point of the ]. ] (]) 14:22, 23 April 2012 (UTC)
:{{ec}}And Fourier series also uses trigonometry. And calculus. And infinite series. And basic arithmetic. And real numbers. Does that mean (as it's "a huge topic, a powerful apparatus") that Fourier series should be discussed in all those articles? No. But they are much more important to an essentially mathematical topic, not least as ] do not depend on wavelength at all.
:And please stop editing and re-editing your own comments. Use the 'Show preview' button to get it right first time, write what you mean to write once and wait for replies. It creates work for other editors if they find the post they are replying to have changed (often multiple times) in the few minutes after they were posted.--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 14:55, 23 April 2012 (UTC)
::Blackburne, what you fail to recognize is that wavelength and spatial periodicity basically are the same concept, and also the fundamental concept underlying Fourier series. That makes it useful to point out this connection, while your topics do not qualify. So sorry that my editing to get the wording right has proved so distracting to your thought. ] (]) 15:26, 23 April 2012 (UTC)
::BTW, Blackburne, your observation that "] do not depend on wavelength at all" is a flat contradiction of the quotation in the ].
:::Perhaps you could point out which part of ] uses the wavelength? Or where in that article it is mentioned? It isn't mentioned because Fourier series do not depend at all on wavelength.--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 15:47, 23 April 2012 (UTC)
::::Blackburne, what you still fail to recognize is that the topic here is ''periodicity in space'', as that is where wavelength has a role. It happens that Fourier series apply to a function periodic in ''any'' variable. However, when applied to spatial periodicity, the Fourier series directly incorporates wavelength as pointed out in the quotation in ]. Please try to adopt the appropriate context. ] (]) 15:54, 23 April 2012 (UTC)

:::::What about time period and ] of wave oscillations? And about ]? Why not include the Fourier series there also for the hell of it, because waves are also periodic in time and phase angle???

:::::If you actually think about the reader (more likely to be someone less interested in maths, but would like to know about wavelength), rather than ''assuming'' they will switch on to the Fourier series when they read "a superposition of waves ''each with wavelength'' (really!) can be superimposed to form a wave with periodicity in space, can be mathematically analysed as a Fourier series", isn't it more likely that they would like to find out what the wavelength '''is''' and what '''its for''' and '''why its measured''' (etc)?? will they not be thinking "so a Fourier series describes periodic functions and waves can also be described by periodic functions... isn't that a description of superposition?" (or words to that effect) ?? It serves the reader no purpose - again adding a '''see also''' link or template will indicate "for further descriptions of (spatial) periodicity and wavelength see the main articles ]" (or words to effect).

:::::How many editors have said the same thing? Isn't it obvious? '''Stop accusing other editors''' of "failing to understand" - ''you'' are the one who seems to fail to understand the blatant consensus against adding a fluttery statement on the Fourier series, and are too ignorant of the reasons they provide. <span style="font-family:'TW Cen MT';">] ] ]</span> 16:42, 23 April 2012 (UTC)

::::::F = q(E+v×B): You bring up the topic of including a heads-up to Fourier series in other articles. That may be appropriate in some cases and in others not. As with these other topics, its appropriateness in the context of wavelength should be judged on its own merits.
::::::You also suggest that not all readers will find a heads-up to Fourier series is of interest, but of course, some will. There is benefit in presenting this heads-up for those that do find value in it. All it costs is a few lines that the disinterested can skip, if they like.
::::::I'd agree that many objections to this insertion have been registered. Some objections are based upon misconceptions about the nature of Fourier series, and can be discounted. The remaining objection is the one you raise: Fourier series is a "strange", "incongruous" and "odd" addition to an article on wavelength. You call it a "fluttery" statement, which I can only assume means something similar.
::::::All the ] is, actually, is a heads-up pointing to Fourier series for spatially periodic waveforms. It's brief, it's accurate, and it's helpful to some. There is no downside to its inclusion. ] (]) 17:11, 23 April 2012 (UTC)

:::::::Perhaps we can compromise on this: part of your current proposal
::::::::''"summation of sinusoidally varying waves with wavelengths related to the wavelength of the periodic waveform itself:"''
:::::::is quite a mouthfull and not much help, a typical reader could get lost. Perhaps change the statement to:
::::::::''"Sinusiodal waves can ] and create a spatially periodic ], which can be mathematically described by a ]. The wavelength in this case corresponds to the (spatial) period in the Fourier series, i.e. the spatial interval for which one cycle of the function repeats itself."''

:::::::from a ''physical'' perspective (or words to that effect). That way the series is mentioned in passing with no obfuscation. Perhaps the best ''mathematical'' perspective (if preferred by others) would be ]'s version above. I would simply prefer to add

::::::::<nowiki>{{Hatnote|For further descriptions of periodic functions in a more general context to wavelength and phase, see also ].}}</nowiki>

:::::::at the top of the ] section though, since this is more of a sidenote with no interruption of continuity, but will not do so. Not fussed about the reference... <span style="font-family:'TW Cen MT';">] ] ]</span> 17:55, 23 April 2012 (UTC)
::::::::F = q(E+v×B): I have taken a whack at rewording along these lines. ] (]) 19:28, 23 April 2012 (UTC)

:::::::::Hrm, your version gives the explanation in 3 sentences, mine in 2, and seems more wordy than necessary. All we need to include Fourier's series is state superposition and representation by the series. Also adding two (very much) identical quotations will not amplify any meaning, I don't think there is any need for the quotations but you may as well keep the references for inline citations.

:::::::::Anyway I have made my points, and will no longer participate in this (can't contribute much more anyway - exams). Feel free to compromise further with the others... <span style="font-family:'TW Cen MT';">] ] ]</span> 20:31, 23 April 2012 (UTC)

::::::::::Where are these new versions you guys are comparing? And do they say why it would be more useful to use sine waves than, say, triangle waves, for such a decomposition? If not, then it's still quite pointless, isn't it? ] (]) 23:22, 23 April 2012 (UTC)
:::::::::::Dick: I rewrote the contents of the "green box".
:::::::::::What are the properties of your "triangle waves"? Are your "triangle waves" defined over all space and do they exhibit periodicity with a certain wavelength themselves? Are series expansions using your triangle waves automatically periodic with wavelength &lambda;? I imagine you could construct such a basis set, but it would be a fringe topic compared to Fourier series, and I wouldn't be surprised if to prove their completeness you'd have to reduce them to superpositions of sinsuoids and resort to Fourier's theorem. ] (]) 00:35, 24 April 2012 (UTC)
::::::::::::Triangles work in all respects you've mentioned just as well as sines do. It's not about what's mainstream and what's fringe, it's about what makes sine waves special, and I've told you at least four time in this talk page already. Here, read up: some clues: , , , , , . ] (]) 01:23, 24 April 2012 (UTC)

===DickLyon's sources===
Dick Lyon has undertaken to illustrate "what makes sine waves special" by referring to several sources. I have provided a bit more about these sources below. They fall into two groups: those that repeat what is already in the ], and those that require a considerable expansion of this proposal to go into nuances not so far entertained as a possible insert.

1. : Quote: "there is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave that can be propagated without a change of form; and even in the exceptional cases where the velocity is independent of wavelength, no generality is really lost by this procedure, because in accordance with Fourier's theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wavelengths in harmonical progression."

The contribution of this source to the discussion of the ] is only to reinforce what is said in the two quotations. It suggests that in addition to this, more could be said about the propagation of waves in dispersive media, where only a wave of a given wavelength propagates unchanged. If the proposed text is to be expanded upon, that point could be added.

2. Quote: "any periodic function can be represented as the superposition of harmonic terms of frequencies ''f<sub>1</sub>'', ''f<sub>2</sub> = 2 f<sub>1</sub>'', ''f<sub>3</sub>=3f<sub>1</sub>''... where ''f<sub>1</sub>''=1/&lambda;"

The contribution of this source to the discussion of the ] is only to repeat what is said there already.

3. Quote: "By superposition of the fundamental solutions one can usually construct a formal solution... The fundamental solutions describe in relation to one or several variables, sinusoidal functions with frequencies that are an integer multiple of a fundamental frequency. This fundamental frequency already emerges when one calculates the eigenvalues."

This general discussion is what was already alluded to in the above recapitulation of material regarding ]. The example provided later in this section is confined to the finite interval , and does not apply to a waveform periodic throughout all space. If we were allowed to expand the discussion, such matters could be explored.

4. This text describes propagation in nonlinear media. While this is an interesting subject, it is not germane to this discussion which concerns simply ''mentioning'' Fourier series and its connection to wavelength.

5. This text discusses Fourier series as one example of eigenfunction methods. Again, this matter has been dealt with in the above recapitulation of ]. If DickLyon's point is that Fourier series is one type of eigenfunction expansion, fine. But that point does not illustrate how special Fourier series are, but buries their individuality by suggesting they are ''merely'' one of the many finite-interval approaches. Fourier series representation are special because, in contrast, they apply to the entire domain of the periodic function , not just to some finite interval.

6. This text describes the ], which is again a wonderful subject that has nothing to do with the present discussion.

In summary, this blizzard of sources adds no new considerations concerning addition of the proposed text, although they provide some avenues to expand upon it. ] (]) 03:13, 24 April 2012 (UTC)

===New section in article===

I've rewritten the article section to try to motivate the sinusoidal decomposition; now there's a reason to mention Fourier series. Feel free to revert if anyone objects. ] (]) 01:44, 24 April 2012 (UTC)
:Dick has identified the need to rewrite this section, and has made a first attempt to do so.
:From this revision it now is clear what Dick has been alluding to all along about the "special nature" of the sinusoidal wave: it propagates with unchanging waveform in a dispersive medium. This is the observation made by . This fact is made a segue to ], introduced as a handy way to handle propagation of a disturbance propagating in a dispersive medium by treating it as a superposition of waves of a single frequency, so each component propagates unchanged, even though the shape of the propagating disturbance varies in time and in space, and has no associated wavelength. Dick uses the wording "linear medium", in preference to "dispersive medium", a poor choice as the topic "non-linear" media (also discussed) is not in contrast to "linear" medium in this sense of the word.

:Now, this property of sine waves in dispersive media is an interesting point to raise and it does introduce the notion of Fourier series, more properly, the Fourier transform because periodicity of the waveform is not part of this discussion.

:However, this singular property of sinusoids is a point contradicted later in Dick's revision when it is pointed out that periodic waveforms that are not sinusoids also can propagate in some media and they do have a wavelength. The discussion of wavelength for general waveforms has been emasculated by eliminating the generalized definition of wavelength for an arbitrary waveform ''f(x−vt+λ)=f(x−vt)'', the historical and mathematical foundation of Fourier series. The removal of the general definition eliminates this simple segue to the ].

:The new version is a start. However, this section remains a disjointed assembly of disconnected topics. If the general definition of wavelength is not restored, it remains unclear how the ] might fit into the ultimate revision. ] (]) 12:22, 24 April 2012 (UTC)
::I've rearranged Dick's revision, changed some wording, and inserted a version of the proposed text. I think the result is less disjointed. ] (]) 13:58, 24 April 2012 (UTC)

:::I believe you added several kilobytes and destroyed the point of it. I'll wait and see if anyone else cares to comment. ] (]) 15:38, 24 April 2012 (UTC)
::::Dick: I am alarmed by your statement that this version has destroyed the point of yours. IMO it is mainly a rearrangement of your own wording, plus the restoration of the periodicity condition that you removed. I added the historical remark of the end paragraph. ] (]) 16:40, 24 April 2012 (UTC)
:::::The consensus above is against adding the digression on Fourier series, however worded, so that should not be there. Other than that it has destroyed the logic and structure of the section changing it from something that read well to a mess, for no good reason (at least no reason given in the overly terse edit summaries).--<small>]</small><sup>]</sup><sub style="margin-left:-2.0ex;">]</sub> 17:20, 24 April 2012 (UTC)
::::::Blackburne: I have made a conscientious effort in my rewrite as explained carefully in my critique of DickLyon's initial effort immediately above. The logic of the section is exactly as Dick wrote it. The reorganization mainly changes things to put each topic in one place instead of having them scattered about. Beyond that I restored the periodicity definition that was there originally, and changed or added a few sources. As for the connection to Fourier series: I believe that has to be assessed in the new context. ] (]) 17:31, 24 April 2012 (UTC)
{{od}} Brews, thanks for trying, but it's hard to make a sensible rewrite when you still don't quite understand it and you have some favorite ideas you want to cram into it. I may take another stab at it. Some problems:
* 1. The opening generality "In general, the propagation of a disturbance takes a different form in different media, for example, media where the velocity of propagation depends upon wavelength (dispersive media) or upon the amplitude of the wave (non-linear media)" makes the next sentence "The wavelength is functionally related to frequency by the physics of the medium" meaningless or false.
* 2. The sentence "The same is true for a waveform that is composed as a combination of many sinusoids, all of the same wavelength but differing in amplitude, or phase, or both" has no useful role there, just a distractor.
* 3. "For example, sinusoids are simple solutions (eigensolutions or eigenfunctions)" got disconnected from "propagates with no shape change" which is essentially the definition of eigenfunctions
* 4. "Just which wavelengths can contribute to a waveform is decided by boundary conditions. (See the section on standing waves.)" is an unnecessary tangent
* 5. "Under special circumstances, waves other than sinusoids propagate with unchanging shape and constant velocity, called traveling waves" is a complete mixup of unrelated concepts.
* 6. "Periodic waves have a well-defined wavelength" is false in general; that's why you find this application of "wavelength" to arbitrary periodic shapes so rarely in source.
* 7. The sentence "For a fixed-shape waveform that also is periodic, its wavelength λ is defined mathematically in one spatial dimension x by the definition of periodicity" and following formula do not define the wavelength, since the formula would also be satisfied by all integer multiples of the wavelength.
* 8. The reintroduction of Fourier series as a way to analyze spatially periodic waves again completely misses the point of why one would want to decompose waves into sinusoids, and repeats a statement that would be no less true or relevant if done with triangle waves, and is backed up by a source that is equally clueless. If you apply Fourier series to time-periodic waves, though, it's easy to see how it is useful in linear media, and not just in the rare corner case of non-dispersive media; since you didn't get that, you destroyed it.
] (]) 21:41, 24 April 2012 (UTC)

:Dick: To avoid repeating each of your points, I have numbered them.

:1. The dispersion relation definitely is related to the physics of the medium. A common example is the ] in crystals where the propagation of various vibrational modes, for example, is determined by solving the problem of crystal vibrations (the medium). The dispersion relation for an acoustic mode in, say BaTiO<sub>3</sub> is different in another medium, say GaAs. I think you are aware of this matter, so I guess your statement has some other meaning that you might try to explain further.

:2. This "distractor" came directly following the self-same segue from the source you provided to Lord Rayleigh. If it didn't distract him, I can live with it.

:3. The definition of an eigenfunction is that it satisfies an operator equation with an eigenvalue, as in ''O f = &lambda; f''. It is not defined in terms of propagation.

:4. Perhaps so. It seemed to me to answer a question that pops up in this context, and it connects this section to the previous one. In any event, as a general matter, eigenvalues cannot be determined without boundary conditions, and if the boundary conditions change so do the eigenvalues and the eigenfunctions.

:5. There is no mix-up here. It's not a mix-up, but a shared property. Following your lead (and Lord Rayleigh's) sine waves are interesting because they propagate unchanged in certain media. By the same token, then, traveling waves are interesting because they also propagate unchanged. It seems that cnoidal waves actually are superposed to form more complex traveling waves, a clumsy parallel to sines and cosines. However, I think this comparison is seen more accurately as a segue than as a deep parallel.

:6. A periodic wave, as opposed to some approximation of a periodic wave, always has a wavelength as defined by ''f(x-vt+&lambda;)=f(x-vt)''. If the wave doesn't satisfy this periodicity condition it is not a periodic wave. At least, this is the case in one dimension. Maybe you have something else in mind?

:7. Fine, we'll make it the least value of &lambda; for which this is the case.

:8. This is a really interesting remark, and one we might focus upon as it seems to me to be the real cause of our failing to understand each other. To begin, I suppose we are talking about the last few lines beginning with a reference to Joseph Fourier. If that is the case, these words can be interpreted as no more than historical background, as they make no claims about the applicability of Fourier series. As an historical note, you might object that it is a digression, but that's about it.

:To pursue the matter further, you say: "If you apply Fourier series to time-periodic waves, though, it's easy to see how it is useful in linear media, and not just in the rare corner case of non-dispersive media; since you didn't get that, you destroyed it." Now, IMO this point was already belabored in the opening paragraphs where it was pointed out that sine waves propagate with fixed form enabling the application of the ] to the propagation of arbitrary waveforms. These words were cribbed from your first draft, except the more accurate Fourier ''transform'' is alluded to because the context is general waveforms and not spatially periodic ones. Are you clear that Fourier ''series'' represent periodic functions with the property ''f(&xi;) = f(&xi;+&lambda;)'' for ''all'' values of the argument &xi; and not restricted to representing ''f'' only for &xi; sitting inside some selected interval ? In particular, we can use &xi;=''x-vt''.

:I await your further comments, Dick. ] (]) 23:32, 24 April 2012 (UTC)

::For reference, here is the section as I had it, which I think was much more correct, and not subject to the "problems" you are imagining:

----

The concept of wavelength is most often applied to sinusoidal, or nearly sinusoidal, waves, because in a linear system the sinusoid is the unique shape that propagates with no shape change – just a phase change and potentially an amplitude change. The wavelength (or alternatively ] or ]) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. That is, sinusoids are the simplest solutions (eigensolutions or eigenfunctions) of ]s that describe linear physical media, and the frequencies and wavelengths are related by the ]s of the equations or those media. From these simple ] solutions, more complex solutions can be built up by ].

In the special case of dispersion-free and uniform media, waves other than sinusoids propagate with unchanging shape and constant velocity. In certain circumstances, waves of unchanghing shape can also occur in nonlinear media; for example, the figure shows ocean waves in shallow water that have sharper crests and flatter troughs than those of a sinusoid. Large-amplitude ]s with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.<ref>{{cite book
| title = Global environment remote sensing
| author = Ken'ichi Okamoto
| publisher = IOS Press
| year = 2001
| isbn = 9781586031015
| page = 263
| url = http://books.google.com/books?id=tXQy5JdQyZoC&pg=PA263&dq=wave-length++non-sinusoidal
}}</ref> An example is the ], a periodic traveling wave named because it is described by the ] of ''m''-th order, usually denoted as {{nowrap|''cn''(''x''; ''m'')}}.<ref name=Kundo>
{{cite book
|title=Tsunami and Nonlinear Waves
|author=Roger Grimshaw
|editor=Anjan Kundu
|url=http://books.google.com/books?id=2Dtgq-1CGWIC&pg=PA52
|pages=52 ''ff''
|chapter=Solitary waves propagating over variable topography
|isbn=364209032X
|year=2007
|publisher=Springer
}}</ref>

]

If a traveling wave has a fixed shape that repeats, it is a ''periodic wave''.<ref name=McPherson>
{{cite book
|title=Introduction to Macromolecular Crystallography
|author=Alexander McPherson |url=http://books.google.com/books?id=o7sXm2GSr9IC&pg=PA77
|page=77
|chapter=Waves and their properties
|isbn=0470185902
|year=2009
|edition=2
|publisher=Wiley
}}</ref> Such waves may have a well-defined wavelength even though they are not sinusoidal. As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform. Mathematically, the amplitude ''f'' of an unchanging waveform moving with a velocity ''v'' can be expressed as {{nowrap|''f''(''x'' − ''vt'')}}, with ''x'' = position and ''t'' = time. The amplitude at location {{nowrap|''x'' + Δ''x''}} at time {{nowrap|''t'' + Δ''t''}} is the same as that at location ''x'' at time ''t'', if Δ''x'' and Δ''t'' are related by {{nowrap|1=Δ''x'' = ''v''Δ''t''}}.

In more general linear media (that is, ] media), a wave that is periodic in time will not necessarily repeat in space, so may not have a well-defined wavelength. Such waves are typically analyzed into sinusoidal waves via the ], so that the different propagation speeds and wavelengths of their different frequency components can be separately handled. To an observer at a fixed location, the amplitude varies in time and repeats itself with a certain ''period'', ''T''. During every period, an integer number of each component wavelength of the wave passes the observer, but with different relative phases at different observer locations.


== Lede doesn't provide a concise definition ==
----


The first sentence of the lede implies that wavelength is a characteristic of a sine wave only. Later in the lede it says: oh by the way, it's a characteristic of any periodic wave. The lede should be written so that it is in compliance with ] (''...If its subject is definable, then the first sentence should give a concise definition''). The first sentence should be something like, "In physics, a wavelength is the distance between any two successive parts of a periodic wave that are in phase, i.e., that are at idential points of its cycle." <b>]&nbsp;(]•])</b> 06:12, 23 March 2018 (UTC)
::I don't have time to try to explain it to you again. It may not be perfect, but what it says about Fourier series is much more sensible than anything you've come up with. ] (]) 05:24, 25 April 2012 (UTC)
:That's better , thank you. <b>]&nbsp;(]•])</b> 15:54, 7 April 2018 (UTC)
:::Dick: You haven't tried to explain as far as I can see. You simply assert (based upon your personal opinion alone) that my rewrite has problems, which it does not, and that yours is better, which it is not. The way forward is to address the responses I've made to your points above, and come to agreement.
:::Your paragraph repeated by you above contains nothing of value that is not in my rewrite, and my rewrite avoids several problems of wording in your paragraph, as noted earlier, but not responded to by you. You may be too busy to engage, and if so that is unfortunate. But simply reasserting the past formulation without addressing comments upon it is a waste of what little time you have to spend. ] (]) 05:40, 25 April 2012 (UTC)
:::It may be that a difficulty with getting across what you want to say stems from using the wrong subsection to explain it: instead of the ] the topic of dispersion and the special advantages of sine waves might go better in the ] where dispersion is described. What do you think? ] (]) 14:26, 25 April 2012 (UTC)

Latest revision as of 15:17, 28 July 2024

This  level-4 vital article is rated B-class on Misplaced Pages's content assessment scale.
It is of interest to the following WikiProjects:
WikiProject iconPhysics High‑importance
WikiProject iconThis article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.PhysicsWikipedia:WikiProject PhysicsTemplate:WikiProject Physicsphysics
HighThis article has been rated as High-importance on the project's importance scale.
WikiProject iconMathematics Low‑priority
WikiProject iconThis article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics
LowThis article has been rated as Low-priority on the project's priority scale.
Archiving icon
Archives

More general waveforms wording

I removed the emphasis on the period T being the same at all points, since this might be misleading. While it is true that there is a period T that is common to all points, at some points the wave will repeat more than once in time T, so the "period" as conventionally defined is shorter at these locations, by some integer factor. The source location for a periodic non-sinusoidal wave is one such location: the period there is shorter (possibly by a large factor) than the period at other locations. Since in many cases this shorter period of the source wave is known, the statement that the period T is the same at all points could mislead the reader into thinking that the period of the wave equals that of the source at all points, which is not true. A more extended discussion could clarify this, but it's probably better just not to get into it.

I also removed the reference to Fourier integrals. I didn't feel that it worked where it appeared in the paragraph. It broke the flow of concepts, making the paragraph less clear.--Srleffler (talk) 01:40, 29 April 2012 (UTC)

I added the mention of Fourier integral because it was part of what the source that I cited talk about, and to appease Brews a bit, but I don't mind it being gone. As for the period, I'm not sure I understand. How can the period anywhere be other than the period of the source, that is, the least common period of all the sinusoids that are propagating by the location? Dicklyon (talk) 06:10, 29 April 2012 (UTC)
I may have been mistaken about the period.--Srleffler (talk) 06:57, 29 April 2012 (UTC)
It seems to me that the period is related in only a complicated fashion to the period of the source, involving the separation of the observation point from the source and also the dispersion relation. Maybe we need a source to tie this down? Brews ohare (talk) 16:42, 30 April 2012 (UTC)
With the Fourier series decomposition, it's easy to see that the wave contains only harmonics of the source period. No new frequencies are added by propagation, even if there are reflecting ends, dispersion, or whatever. So you have harmonics of the period everywhere, and therefore the same period everywhere, no? Dicklyon (talk) 22:59, 30 April 2012 (UTC)
Yes. I had in mind that as the components got out of phase with one another the superposition would have a period that was longer than one cycle of the fundamental, but I see I was mistaken.--Srleffler (talk) 03:17, 1 May 2012 (UTC)

Using a Fourier series begs the question as it presumes a periodic result with the same period. We need a source here, not editors' speculation. Brews ohare (talk) 08:18, 1 May 2012 (UTC)

To add to the speculation, and emphasize the need for an explanatory source, if the driver produces two sine waves close in frequency, the resulting periodic waveform has an envelope that oscillates at the beat frequency, which can be as low (or as long a wavelength) as one can imagine if the two frequencies are close together. That seems to suggest that the period of the waveform produced by the driver is less about the period of the driver than the beat frequency. Brews ohare (talk) 09:00, 1 May 2012 (UTC) DickLyon's source amply demonstrates this point; see Figure 4.7.1 Brews ohare (talk) 16:19, 1 May 2012 (UTC).

Yes the period might be very long in the two-beating-frequencies case. But the source associates "wavelength" with the components, not with a long pattern. There's actually no "driver" or "source" that's relevant here, just periodic-in-time wave motion, which can be analyzed into harmonic frequency components. I don't see any speculation, but then again I don't see a source that says precisely what our text says, obvious though it is. Dicklyon (talk) 00:48, 2 May 2012 (UTC)
There is no "presumption", only definition. The paragraph we are discussing is specifically about the important special case of waves that are periodic in time. There is no need to presume or speculate; periodicity in time is the specified initial condition. The only question is how the system evolves over time, and how it behaves at other spatial locations.
Since the topic of discussion is waves that are periodic in time, the relevant period in the two-beating-waves case is the long period required for the full waveform to repeat. The Fourier series in this case is particularly simple, and the fact that the period of the wave as a whole is much longer than those of the nonzero Fourier components is not a problem. In my initial comments on this topic, I had presumed that there might be locations where the period of the combined wave might be short (like the periods of the components), but I was mistaken. The component waves maintain their frequencies as they propagate and there will be nowhere along their common path where they do not beat against one another, producing a waveform with the same, long, period.--Srleffler (talk) 03:19, 2 May 2012 (UTC)
Actually, it might be slightly more complicated than that in wave media with reflections that can make nulls for certain frequencies at certain locations. If one of the two components in the two-component beating pattern has a null, then the response at the location of the null will be just the other component, so it will have a short period there. However, what the article says is still true there: the wave is still periodic with the longer period T, even if also with some shorter period. Just to be sure, I have edited the text to try to make sure that the period T is stated as the period of the wave and can't be misunderstood as one of these possible shorter periods at a null in a corner case. Dicklyon (talk) 03:39, 2 May 2012 (UTC)
A periodic disturbance in time at a particular location will result in a disturbance at all distances from the source when steady-state is reached. So a disturbance in time repeated with period T but of duration less than T will involve many frequencies, submultiples of T. Nothing much changes as T changes, if the disturbance maintains the same form in time, but is simply spaced with larger "blank" periods in between. T is not a very useful parameter in describing matters, therefore, and it should not be framed as the key to analysis here. Brews ohare (talk) 22:06, 3 May 2012 (UTC)
True, the periodicity of T is just there to make the Fourier series applicable. Dicklyon (talk) 03:50, 4 May 2012 (UTC)

More general waveforms references

The two references to the topic of general waveforms so far do not actually describe how these calculations are done, but provide only few words of description.

The stress upon a periodicity in time in the article in preference to the propagation of a waveform seems to me misguided. For example, if one makes the analogy with a performer blowing large soap bubbles in a park, the bubbles are launched as huge spheres, and as they are carried in the wind they enlarge and become ellipsoids. At a location near the launch one sees a periodic appearance of spheroids at the period of launch T. At a remote position one sees a periodic appearance of enlarged ellipsoids with a period T. Just how long the period T is between arrivals, or between repetitions of what happens periodically in time a a fixed location, as described by a Fourier series in time, is not so interesting as the process of transformation as the spheroids change to enlarged ellipsoids, that is, the propagation phenomenon. Changing the period and spacing the bubbles differently is not really essential.

So I think what is needed is a more interesting discussion with some more detailed references tying what happens to the dispersive nature of the medium. Emphasis upon the more-or-less incidental period between events is not the really interesting point. Brews ohare (talk) 06:11, 2 May 2012 (UTC)

I agree the section remains rather unsatisfying. I did my best to find a sensible way to incorporate the Fourier series into something to do with wavelength, cobbling what was there; but it's still a bit of a misfit. Most sources that talk about dispersion and Fourier analysis don't do in the context of periodic waves, and usually do it in terms of wavenumber, not wavelength. And their analysis doesn't usually conclude anything related to wavelength or to repetition in space. Probably we should just simplify the section, since there are better articles for covering these other concepts of waves in linear dispersive media. As for the concept of wavelength being applied to other than approximately sinusoidal waves, it's unusual at best; discussing it can easily be misleading, or spiral into contradictions, like when you get into claims that it's well-defined for arbitrary periodic functions. Dicklyon (talk) 05:58, 3 May 2012 (UTC)
Hi Dick: Quite possibly the easiest approach is to use wavevector, or maybe to use a simple example instead of trying the general case. As you know, however, I do not agree in the slightest that application of wavelength to a periodic wave in space of general form is in any way misleading, although the occurrence of such waves in nature is not general, but restricted to particular media. From a conceptual point of view, wavelength is what Fourier series is about in space, with the simple interpretation of the general argument ξ as x instead of angle, or time. From this stance, as I have pointed out by direct quotations from at least three sources, many authors do exactly that. Brews ohare (talk) 21:09, 3 May 2012 (UTC)
Not that many authors do that. And none of them seem to reveal any reason for doing a sinusoidal decomposition of the spatial pattern. At least in the case of the dispersive linear system there's a reason to decompose into sinusoids. Dicklyon (talk) 03:52, 4 May 2012 (UTC)
Hi Dick: Perhaps it is just argumentative, but here's a question: why do authors use Fourier series when argument ξ is interpreted as time, or angle, or whatever? Why is it automatically a wasted effort only when ξ is interpreted as a spatial variable? Why is period T more significant as a time period than λ is a a spatial period? Could it be that in fact there is no difference at all? Brews ohare (talk) 20:40, 5 May 2012 (UTC)
It's not unusual to use wavenumber (or reciprocal wavelength) in a Fourier transform, as a way to get a sinusoidal decomposition, especially for wave packets. But periodic-in-space waves are a relatively rare corner, seldom encountered where a sinusoidal decomposition would be helpful. If they're also periodic in time at the same time, in a linear system, the system is nondispersive, and has a trivial wave equation, for which a sinusoidal decomposition is not needed; it adds nothing to the understanding of the system. If they're in a nonlinear system, the sinusoidal decomposition is not particularly illuminating either. Dicklyon (talk) 22:27, 5 May 2012 (UTC)
Here's another question: when an oboe plays a note, it sounds different than when a violin plays the same note. Could it be that the difference can be expressed as a difference in the Fourier series expressing the wavelengths of vibration supported by the general waveform in the oboe's air column for that note compared to the wavelengths present in the general waveform on the violin string when the same note is present? Would that be an interesting enough example of Fourier expansion of the general waveform in space to warrant mentioning the use of Fourier series for spatial analysis of waveforms? Or, perhaps, an article Wavelength (music) is needed? Brews ohare (talk) 20:52, 5 May 2012 (UTC)
The sound difference has much more to do with the waveform in air; this is what propagates and carries the pattern. The physics within the instrument gives rise to different modes, and to a temporal near-periodicity from how the signal interacts with the reed or the bow, but I haven't seen an analysis like you're describing, where the composite signal is analyzed into space-domain sinusoids. Of course it could be done. Dicklyon (talk) 22:27, 5 May 2012 (UTC)
Once it's left the instrument the waveform is not uniform, i.e. there is no 'general waveform', even allowing for distance attenuation. The sound you hear an inch from a source is very different from the sound you hear three feet or thirty feet from it: the best example is the human voice as we're most familiar with it: often you can tell how far away someone is quite accurately by the sound of their voice. This is less noticeable with an instrument because its pure note dominates and so it depends on the particular frequency of the instrument. And of course in most cases performers don't want you to hear different sounds depending on how far away you are, and will go to great lengths to minimise effects of distance (modifying the building to compensate for example).--JohnBlackburnedeeds 22:57, 5 May 2012 (UTC)
To pursue this matter further, harmonics are produced on the guitar by deliberately introducing a node on the guitar string, a direct application of wavelength considerations. Brews ohare (talk) 21:05, 5 May 2012 (UTC)
That doesn't produce harmonics, but kills the fundamental and certain other harmonics. Yes, it is described in terms of wavelength on the string, or the modes of the instrument. But not in terms of a sinusoidal decomposition of a periodic-in-space pattern. Dicklyon (talk) 22:27, 5 May 2012 (UTC)
Dick & Blackburne: You might find some interest in reading a source or two on this subject instead of relying on your recollections. Something interesting can be done here. Brews ohare (talk) 20:51, 10 May 2012 (UTC)
I have plenty of good books on the physics and psychophysics of music and musical instruments; it's not clear what you see as relevant in the page you've linked in that college physics text. Dicklyon (talk) 21:24, 10 May 2012 (UTC)
Dick: Figures 14.26 and 14.27 of this source compare waveforms for a pure note on a tuning fork with the same note on a clarinet and a flute. The point, of course, is that the characteristic voice of the instrument is expressed in its peculiar waveform, which is in each case periodic with the same wavelength but of different shape. Accordingly, the differences between voices is sought in the different harmonics of the fundamental found in each. This difference can be expressed in time or in space, although the latter requires some expression of the characteristics of the medium, which cannot be unduly dispersive. Obviously, instruments usually operate in air, and the slight dispersion of sound in air is no impediment to applying a spatial analysis. The design of an instrument is perhaps even more clearly related to wavelength, as the dimensions of the instrument determine how an excitation of a particular spatial mode will be related to its various harmonics; for example, how the standing wave on a violin string is connected to the various resonances of the instrument. Brews ohare (talk) 13:04, 15 May 2012 (UTC)

Removal of Fourier series in time section

I went ahead and removed the unsatisfying paragraph about the Fourier series and periodic-in-time waves, as it was not very useful, nor very germane to the topic. Dicklyon (talk) 04:04, 15 May 2012 (UTC)

This removal was a good step: it tried to introduce Fourier series using the topic of Fourier series in time, applied to waveforms that have no identifiable wavelength in space. The door is now open to introduce Fourier series in a context appropriate to the subject of wavelength, that is, the context of spatially periodic general waveforms which, of course, always have an identifiable wavelength. That such waveforms can be and are analyzed using Fourier series is well documented, and the objection that such waveforms are not necessarily found in general media restricts its applicability in general, but doesn't mean it deserves no mention here. Brews ohare (talk) 12:34, 15 May 2012 (UTC)
I'd suggest a reconsideration of the text below:
The wavelength, say λ, of a general spatially periodic waveform is the spatial interval in which one cycle of the function repeats itself. Sinusoidal waves with wavelengths related to λ can superimpose to create this spatially periodic waveform. Such a superposition of sinusoids is mathematically described as a Fourier series, and is simply a summation of the sinusoidally varying component waves:
.. "Fourier's theorem states that a function f(x) of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral submultiples of λ (i.e. λ, λ/2, λ/3, etc.)."
References
  1. Eugene Hecht (1975). Schaum's Outline of Theory and Problems of Optics. McGraw-Hill Professional. p. 205. ISBN 0070277303.

Brews ohare (talk) 12:41, 15 May 2012 (UTC)

Brews, drop it. You've proposed this once, you've had your RfC on this, neither time did you convince other editors. Proposing yet another variation on it after failing to convince other editors multiple times is simply disruptive.--JohnBlackburnedeeds 16:18, 15 May 2012 (UTC)
I agree. This is a dead issue. Brews, drop the stick and move away from the horse.--Srleffler (talk) 17:03, 15 May 2012 (UTC)
Srleffler: As Blackburne has not advanced any actual argument against inclusion of this text, your "agreement" with him is only as to his cheer-leading and not about any "agreement" upon substance. Your own comments regarding the RfC on the above text were as follows:
"The introduction of Fourier series appears neither to make the concept of wavelength clearer nor to provide a better fundamental definition of wavelength of a general periodic wave. The current proposed text is admittedly better than the many attempts prior to this RFC, in that it doesn't belabour the issue and focuses most directly on the connection between the two topics.
I do see the appeal in trying to replace the definition of wavelength of a non-sinusoidal wave in terms of the wave's period of repetition with a definition that is tied directly to sinusoidal waves. I'm partial to this for the same reason that I was originally opposed to applying the term "wavelength" to non-sinuoidal waves at all. It's not clear to me that there is a non-negligible set of readers for whom this treatment would be beneficial, however."--Srleffler (talk) 18:13, 22 April 2012 (UTC)
As I understand these points you raise, your objection to including this text is that it does not clarify the concept of wavelength. However, that is not the purpose of this text. What this text aims to do is to alert readers that there is a connection of spatially periodic waveforms (waveforms with a wavelength) to Fourier series. That connection is undeniable.
Your further objection is that nobody cares anyway. Inasmuch as several sources mention this connection, and indeed elaborate upon it at length, your opinion is not universal. Brews ohare (talk) 19:16, 15 May 2012 (UTC)
You have raised this issue before, multiple times. We have spent far more time discussing it than it was worth. No further discussion of this issue is merited. Please stop trying to disrupt the editing process by repeatedly bringing forward the same issues over and over again with only slight variations. --Srleffler (talk) 03:45, 16 May 2012 (UTC)
I think, Srleffler, that your objections are in fact against an earlier proposal to introduce Fourier series as a definition of wavelength, which is not proposed here. You may have a different opinion about the present proposal. Brews ohare (talk) 19:31, 15 May 2012 (UTC)
The truth is that few sources make that connection, and they don't take it anywhere useful. You have gone back to a formulation that would be just as true and useful if triangle waves were used instead of sine waves; that is, not useful at all, since the sinusoidal components provide no help in analyzing such a situation, where the medium is either nondispersive or nonlinear. The Schaum's Outline book that you cite introduces the Fourier series there only as a step toward getting a Fourier transform, to get a way to represent waves that are NOT period in space, which is useful; and it says it's more common to do it in terms of k than lambda, which is true, so it's not very related to wavelength. And your statement that "The wavelength, say λ, of a general spatially periodic waveform is the spatial interval in which one cycle of the function repeats itself" is contrary to typical usage of the term "wavelength" (that is, for the local wavelength of approximately sinusoidal waves) and is not supported by the source; in fact, your source defines the term "wavelength" only with respect to sinusoidal components, and applies it only fleetingly to a spatially periodic function. The text (if you can call it that) is also flaky in that when it introduces sinusoids in section 1.3 it completely misses their point, again saying something that would be just as true with triangle waves or square waves or a variety of other basis sets. We have been through all this many times. The objections of numerous editors are in the record if you'd like to review them further. Dicklyon (talk) 23:29, 15 May 2012 (UTC)
Dick: You are missing the point here. There is no attempt to propose that Fourier series is the one and only way to expand an arbitrary function in terms of other functions, which might fall under the rubric of generalized Fourier series. The point here is simply to make the connection of a spatially periodic function of general form that satisfies f(x+λ)=f(x) to the Fourier series. A Fourier series, as you must know, inevitably results in a periodic function throughout space. Fourier series is, moreover, a very well known and important aspect of mathematical analysis, and a link to make readers aware of the connection is just an ordinary use of an aside that widens the reader's appreciation of the topic wavelength and its connection to the mathematical analysis of periodicity. The text contains a direct quote from a textbook, and virtually the same language occurs in other sources as well: "Fourier's theorem states that any periodic function f(x) can be expressed as the sum of a series of sinusoidal functions which have wavelengths that are integral fractions of the wavelength λ of f(x)"
There is nothing misleading or inappropriate here, as you well know, and your unsupported assertions to the contrary do not reflect well upon your understanding of the subject, nor indeed, upon your appreciation of one of the major benefits of WP: helping readers widen their awareness of a topic. Brews ohare (talk) 14:11, 16 May 2012 (UTC)
There is no need in Fourier series to define what wave length is. Hence one should not tell about this in the definition. On the other hand, there is nothing wrong to mention Fourier series somewhere in the article. Bringing that kind of dispute to Arbcom seems incredibly strange to me. My very best wishes (talk) 04:35, 17 May 2012 (UTC)
I had already crafted a paragraph to say what could sensibly be said about Fourier series, applied to periodic-in-time waves, but nobody much liked it and it wasn't particularly relevant to wavelength, so I took it out; nobody objected to that. Dicklyon (talk) 05:37, 17 May 2012 (UTC)
So, what is exactly the problem with describing non-sinusoidal waves using Fourier series? I do not see any problems. But probably this belongs to other articles about waves. My very best wishes (talk) 04:14, 18 May 2012 (UTC)

Introduction edit

The sentence "For example, in sinusoidal waves over deep water a particle in the water moves in a circle of the same diameter as the wave height, unrelated to wavelength" should be edited to mention that this is true for a particle on the surface, particles below the surface moving in smaller circles. — Preceding unsigned comment added by 64.134.138.137 (talk) 17:30, 1 March 2013 (UTC)

Fixed.--Srleffler (talk) 00:14, 2 March 2013 (UTC)

Citation to "Subquantum Kinetics" pseudoscience

Why on earth is there a citation to "Paul A. LaViolette (2003). Subquantum Kinetics: A Systems Approach to Physics and Cosmology" (citation for "the notion of a wavelength also may be applied to these wave packets"). That book is utterly pseudoscientific (and the author is known for some way out fringe ideas), and does not belong in a science article in my opinion. Aren't there better references to use for wave packets? Rolf Schmidt (talk) 23:42, 1 January 2015 (UTC)

Fixed. Thanks for pointing that out.--Srleffler (talk) 05:01, 2 January 2015 (UTC)

First use of lambda

When was lambda first used for wavelength? I noticed Fresnel used λ in 1819 in 'Memoire on the diffraction of light' and Herschel used λ = v T in 1828 in 'On the Theory of Light'. Ceinturion (talk) 09:49, 23 October 2015 (UTC)

How to calculate wavelength Hlelokuhle (talk) 18:40, 11 March 2020 (UTC)

Vacuum wavelength

Presently the wikilink Vacuum wavelength goes to this article, but it is not clarified. I would suggest adding something like the following blurb.

When light passes between different materials, the wavelength changes although the frequency stays the same. In the field of optics, it is rare to refer to the invariant frequency of light, but instead to refer to the vacuum wavelength of light, which is c divided by the light frequency. This convention is commonly used even when describing light inside materials where the actual wavelength of the light is not the same as the vacuum wavelength.

(More or less?) --Nanite (talk) 16:40, 21 July 2016 (UTC)

Seems like a good idea.--Srleffler (talk) 03:26, 22 July 2016 (UTC)

Lede doesn't provide a concise definition

The first sentence of the lede implies that wavelength is a characteristic of a sine wave only. Later in the lede it says: oh by the way, it's a characteristic of any periodic wave. The lede should be written so that it is in compliance with MOS:FIRST (...If its subject is definable, then the first sentence should give a concise definition). The first sentence should be something like, "In physics, a wavelength is the distance between any two successive parts of a periodic wave that are in phase, i.e., that are at idential points of its cycle." Sparkie82 (tc) 06:12, 23 March 2018 (UTC)

That's better , thank you. Sparkie82 (tc) 15:54, 7 April 2018 (UTC)
Categories:
Talk:Wavelength: Difference between revisions Add topic