Talk:Wavelength/Archive 5

708:, 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, 350:. 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).-- 2702:, 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. 2543:: 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." 3994:

actually many periods of the driver, depending how many driver pulses contribute high speed precursors to the location of the slowly propagated, rounded form of the filtered square wave. So a Fourier series in time could represent this situation at a fixed location, contrary to my earlier intuition, albeit with a period set not solely by the driver, but also depending on the distance from the driver and the dispersion relation determining how speed varies with wavelength.

1073:. 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 3969:

received. At greater separations, the faster components of the square pulse will move (let's say) faster than the rest, so the pulse begins to lose definition. The faster components provide a precursor and a rounded pulse arrives later. Depending upon the size of the separation of the observer from the driver, a precursor may even overtake an earlier rounded pulse, resulting in something quite different as the slow components of pulse leaving at time

4504: 31: 3075: 174: 240:

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.

3831:

clearly set forth in the discussion) is the change in spatial form of a general disturbance as it propagates in space in a dispersive medium. That analysis cannot be based upon a Fourier series, but demands a Fourier integral. If you wish to expand this talk page discussion to explore this mathematical analysis beyond all possibility of slip-shod argument, perhaps progress can be made. Let me know.

2540: 2695: 739:. 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 1564:, 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. 3968:

Srleffler: Thanks again for attempting an explanation that DickLyon apparently could not attempt himself. To try to visualize what you are telling me, I imagine a driver producing a square pulse at regular intervals, so it is a time-periodic driver. At distances close to the driver, a square pulse is

2868:

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

2454:

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 λ? I imagine you could construct such a basis

817:

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

792:

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 λ in this regard, that is an argument for adding to the proposed text, not for eliminating it. For example, one could digress

681:

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

366:

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,

4406:

removed the tag) and the consensus was clearly against you your proposal failed. Stop now trying to re-argue the point as if you are right and every other editor is wrong. That's not how it works. You tried to convince other editors. You failed. Please don't try to start the same argument, now or in

4153:

Dick: It's really silly to blame the extensive talk page discussion here on me, when it is almost all due to your unwillingness to engage in actual dialogue, ignore all comments, refuse to explain yourself, and make silly assertions based upon your own opinion and nothing else, and instead castigate

3993:

So I guess with some work I could figure out how this whole thing evolves. I can imagine that because a steady state is envisioned, whatever the form of the disturbance at a distance, it will have no obvious resemblance to the initiating pulse, and it will repeat at some interval in time that may be

3846:

Brews, you should re-read the paragraph of the article, about which you are complaining. It explicitly specifies that it is talking about waves that are periodic in time. It is you who have failed to grasp the context. Your points below are irrelevant because you are thinking about the wrong special

2952:

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

2796:

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

1330:

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

519:

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

4475:

In his desire to appear as always right, DickLyon has introduced actual errors of fact into the article, and has reduced the article's value to WP by deleting useful and sourced material of general interest to readers, such as the mathematical definition of wavelength described by Dick as "trivial"

4439:

Let me close what I have to say: Brews I have nothing against you, and I'm not "taking sides" with Dicklyon and Blackburne. The earlier response was a perfectly neutral observation of the RfC (not a "ping pong match"). The RfC was what it was. Don't get me wrong - much appreciated you would like to

4137:

Brews, we've got 120 KB of new talk this month (April 2012) due to your attempts to add trivial and inappropriate content to this article, with zero support from any other editor, while you ignore everything we try to tell you and keep returning to your idiosyncratic ways. Are you trying to repeat

3637:

I've moved a modification of the statements about integral wavelengths passing an observer in one period to the section on spatially periodic waves, where it does apply. If Dick Lyon believes some version of this wording will work in the dispersive section, I suggest he put the appropriately worded

3254:

I have restored Dicklyon's version as a far better version of the section, as explained in detail above, and per the consensus in the RfC on the Fourier digression. I would note that the problem with the changes was much amplified by the lack of reason given for them, with edit summaries consisting

3212:

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

2926:

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

2713:

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

2550:

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.

2331:

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

2072:

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

1431:

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

1199:

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

1120:

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

504:

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

485:

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

3756:

Now, this definition goes all the way back to Joseph Fourier, and has been a part of all books on Fourier series for centuries. It states the subject that Fourier series describes. Its exploration by Fourier has huge historical and mathematical significance. To claim this basic statement is to be

3660:

The problem is that the periodic-in-space stuff is an odd corner case, in which Fourier series decomposition does nothing useful. But the periodic-in-time case, in more general linear media, leads to something worth saying. It seems that Brews hasn't yet been able to see that difference. As for

3562:

have a well-defined wavelength, note that the assignment of the term "wavelength" to the spatial period of non-sinusoidal waves is uncommon, and that there are other possible definitions of the term "wavelength" for such waves. The example that comes to mind is a modulated carrier wave. Such waves

2608:

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

689:

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

665:

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

1814:

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

440:

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.

129:

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

4471:

F=q(E+v^B): I was not under the impression that you were siding with these two. The RfC figures large in Blackburne's mind because it involved some outside participants, making it look less like just a quarrel. However, most of the ping-pong activity on this page is just DickLyon and myself with

3830:

However that may be, the incorrect usage of "Fourier series" where "Fourier integral" clearly applies is simply wrong, and can't be justified by this confused usage. A "temporally periodic wave" can be expressed as a Fourier series if regarded only as a function of time, but the context (as very

3567:

exactly periodic, but it is certainly possible to construct a perfectly periodic modulated signal. The wavelength of such a signal is, by conventional definition, the wavelength of the carrier wave, not the spatial period of the modulated signal. The statement about general periodic waves should

3039:

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

2920:

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

2864:

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

750:

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

439:

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

2813:

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

1822:

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

1480:

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

661:

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

4424:

Blackburne: You mistake my purpose in my remarks. I wish to register dissatisfaction with the rejection of material that is sourced and relevant on the supposed basis that it is trivial and discursive, an imposition of subjective matters of taste that cannot be supported objectively. Dick Lyon

4325:

an aside that was condemned by many, apparently on the grounds that if Fourier series did not illuminate wavelength, then wavelength as the fundamental parameter in Fourier series should not be mentioned. Such a stance is hardly WP policy, and on most pages an aside like this wouldn't raise an

4168:

Brews, it is you that started an RfC after you refused to accept Dicklyon's arguments, then have continued to argue after the consensus in the RfC was clearly against you. So yes, the extensive talk page discussions were all initiated by you. You have had all the arguments above. That you keep

3866:

function of time, and therefore cannot be analyzed using a Fourier series in time because a Fourier series is necessarily periodic. The dispersing wave does not repeat in space, so at a fixed location one does not see a repeating pattern as the waveform travels past a fixed point. So the point

3208:

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

1839:

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

239:

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

3802:

The matter is that belaboring a spatially periodic wave is completely trivial, and adds undue weight to the rare application of the notion of wavelength to arbitrary spatially periodic functions. And the temporally periodic wave is what is typically decomposed by a Fourier series so that its

2946:

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

2844:

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

1377:

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

979:

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

3011:

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

3492:"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." 3661:

wavelength of periodic waves being well defined, that's highly suspect on a number of levels; trying to introduce broad mathematical generalities like a least period naturally drags in all the weird excpetions, too, like functions that are periodic but have no minimum period.

1389:, 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. 559:

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

1166:

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.

599:

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

3539:

Dick: I have made very obvious points and you have not responded to them. IMO, you are in error, and with no specific indication from you why you think otherwise, I am inclined to think you have no sensible response to my points and, in fact, have not read them at all.

3717: 2068:

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

297:

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?

1244:. 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 3255:

often single words or entirely missing, and never giving a proper rationale for the changes, making it impossible to tell what the edits were intended to do. It was only possible to judge them by the actual result which was a far worse rendering of the section.--

2781:

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.

2469:

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:

1976:(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 3272:

That seems sensible. John, if you or anyone wants clarification of any of the issues raised in the various numbered points above, I'll be happy to discuss. I know from experience here and elsewhere that discussing them just with Brews will go nowhere useful.

2064:

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

3329:

I've undertaken to repeat the edits that are non-controversial and correct some of the more obvious problems here. I have done this with individual edits so they can be examined one by one by Blackburne, instead of repeating his wholesale mindless reversion.

1481:

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?

1116:: 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 4425:

refuses to accept even the most minor changes in this article based on his personal views, many mistaken or arbitrary matters of taste, and you facilitate his actions with your "me too" approach that contributes nothing substantive to the discussion.

3582:

Srleffler: You discuss ambiguity of the wavelength assigned to a modulated wave, where "wavelength" could refer to the carrier wave rather than to the true periodicity of the waveform. A more extended discussion like that appears in the subsection on

3764:

upon introducing Fourier series as the basis for expansion of non-periodic waveforms, when it is entirely trivial to show that is not the case, quite apart from its basis in the periodicity condition you deleted. You can also read the WP article

1627:. 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: 4092:

the stuff on envelope waves is too odd to rate coverage here; especially with that source that says the envelope wave propagates at a velocity that depends on the envelope wavelength, which just shows that author to be clueless or careless.

2869:

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.

4138:

your mess of June/July 2009, where we had over 200 KB of talking past each other before you went away? If you can find anyone at all who agrees with you, maybe there will be something to talk about. For now, there's not. Please stop.

4169:

rewriting your arguments only makes things worse. It does not mean that every time you do so editors have to anew refute them point by point. You have had your chance to convince people, you've failed, repeatedly. Now please stop.--

751:

theory of separating a signal from background, where one might fit some experimental data with λ as a variable parameter and adjust it for a best fit. However, that or other practical instances where one would use Fourier series to

2707:

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

3505:

governed by the possibly (in general at least) arbitrary dispersion relation for the medium. There is no reason why each component should move an integer number of wavelengths in a given fixed time interval, including the period

2531:

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

3592:

The discussion of this particular paragraph is accompanied by a figure that shows the context intended for it. Also, the following math defining the periodicity condition in one dimension also makes clear what the subject is

3179:, 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 2814:

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.

927: 3020:) 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 2616:. 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. 2693:

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

2436:

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?

1098:

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.

3086:. 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 1152:: 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? 1432:

Fourier series, if you want to take it that far. That said, this might not be the most appropriate place in Knowledge for that information. And it might already exist someplace else that can simply be Wikilinked. --

3803:

sinusoidal components of different wavelengths and frequencies can be separately analyzed. I have not written anything about waves that are not periodic (in time). It seems that others understood; why haven't you?

590:

This is a general issue that comes up now and again in discussions of appropriate content for Knowledge articles. The fact that topic A is of great importance to field B justifies mentioning A in the article on B. It

2839:

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:

3826:

Dick, you make confused explanations that fail to distinguish between time and spatial periodicity, and simply use "periodicity" to mean time periodicity even in paragraphs where spatial periodicity is the topic.

2455:

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.

2638:. 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 2848:

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

2927:

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.

572:. 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 1635:

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.

3622:

I've replaced the word "may" by a footnote referring to some exceptions, but not to different usages like carrier wavelengths, which I feel is covered in other subsections, notably the discussion of "local

206:

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.

1077:

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.

3496:

out of the paragraph describing periodic waveforms to the paragraph describing general waveforms in dispersive media. Evidently in dispersive media all components of the waveform move at different speeds

785:), 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 537:

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. __

2896:

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

1293:

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

2947:

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.

1775:

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

3287:

If you want to improve the section, I'd pull out everything about nonlinear into a final paragraph, as it does confuse when inappropriately mixed up with the linear and sinusoidal decomposition bits.

2642:

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.

1586:"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.)" 3950:), which then propagate at different velocities. At any given location there will be a period for the wave as a whole, which is an integer multiple of the periods of each component wave (but not the 1964:(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 860:. 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. 3602:

To use the word "may" in this paragraph to refer to a possible usage of "wavelength" markedly different from the topic of this paragraph places a perplexing obligation upon the single word "may".

2852:

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

1200:

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.

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intended to replace a few paragraphs written by myself with DickLyon's version, two unrelated subsections not subject to dispute were removed with their accompanying figures and sources. I

2727:

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

758:

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

3348:

two unrelated subsections Blackburne removed that were not involved in the discussion with DickLyon, possibly the unfortunate victims of an enthusiastic response too hastily implemented.

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I added the source (also removed), which is only one of hundreds that could be cited. This little piece of math says formally what is described in the figure accompanying this paragraph.

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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

1529:

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

726:

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:

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F=q(E+v^B): I am discouraged to hear you think things have been repeatedly explained to me. Perhaps your observation is based upon what looks like a ping-pong match on this Talk page.

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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.

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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.

2080:

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.

638:). 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. 1664:. 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 2013:

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.

765:

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

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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.

2675:

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.

3194:

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.

3187:. 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. 603:

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. --

505:

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.

3311:

removed the baby with the bathwater, deleting updates of references and some useful rewording of some completely uncontroversial matters. In addition, of course, and as

622:: 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 2936:. 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? 2901:

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.

2144:. 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." 977: 957: 2369:

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...

4000:(i) Is it helpful to be told that at a fixed point in space the behavior can be represented as a Fourier series in time with a hard-to-find period? Shouldn't a 3867:

remains that one needs a Fourier integral to capture the behavior, whether one seeks to capture it in space at a certain time or in time at a certain location.

2656:

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.

555:: 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 2906:

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.

4381:

which has attracted about 1100 hits so far. Perhaps an article on the analysis of propagating waveforms in a dispersive medium would be a helpful resource.

4115:

was placed here long, long ago, after some discussion, and is not something recently introduced by myself. It serves a purpose, although you don't like it.

1343:

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. --

2855:

4. "Just which wavelengths can contribute to a waveform is decided by boundary conditions. (See the section on standing waves.)" is an unnecessary tangent

1378:

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.

1183:(@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. 4480:

just to avoid DickLyon, which may have turned out to be a plus for WP, but the circumstances leading to its birth are a testimony to his intransigence.

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has a wavelength, by the definition of spatial periodicity, so the word "may" should be removed, returning this text to its form before your revision.

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This text describes propagation in nonlinear media. While this is an interesting subject, it is not germane to this discussion which concerns simply

3954:

integer for each). As the article states, "During every period, an integer number of each component wavelength of the wave passes the observer..."--

3769:

that says the same thing. The Fourier integral, on the other hand is not so restricted, as I am sure you are aware. So why not use the correct term?

874: 2634:

This text discusses Fourier series as one example of eigenfunction methods. Again, this matter has been dealt with in the above recapitulation of

1956:

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

1262:

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.

865:

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:

3886:

special cases, since many waves are periodic in time at their source. One needs a Fourier integral for waves that are not periodic in time, but

1535:

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?

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variable. However, when applied to spatial periodicity, the Fourier series directly incorporates wavelength as pointed out in the quotation in

1869:

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.--

793:

to suggest that different musical instruments playing the "same" note have different voices because their Fourier series have different terms.

294:). So I don't accept your stance that "wavelength" is being awkwardly squeezed into the discussion by using an uncommon usage of terminology. 258:

This author's use of "wavelength" in the quotation is entirely appropriate and accurate and often used. Ordinarily (although not invariably)

211:

Some discussion appears on this Talk page regarding a previous suggestion involving a more elaborate discussion of Fourier series presented

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I've rearranged Dick's revision, changed some wording, and inserted a version of the proposed text. I think the result is less disjointed.

2718:, the historical and mathematical foundation of Fourier series. The removal of the general definition eliminates this simple segue to the 2254:

section though, since this is more of a sidenote with no interruption of continuity, but will not do so. Not fussed about the reference...

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of Dick's revision. Blackburne has contributed nothing beyond "I agree with Dick" and Dick has contributed little more than snide remarks.

4455: 4352:(I'd say obviously a spatial concept) is really about analyzing time dependence of a dispersing waveform as a "temporally periodic wave". 4265: 4220: 3227:

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

3004:

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:

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DickLyon's remark: "you need to know the wavelength before you can do a Fourier series". This comment relates to using Fourier series to

4402:

On "Such a stance is hardly WP policy" One core policy is consensus, and a formal mechanism for achieving it is a RfC. As that is over (

3702:. If this periodicity requirement is satisfied by λ, it also is satisfied by 2λ, 3λ, and so on. The least such λ is the wavelength of 4578: 1287:: Why is it "incongruous" or as DIck says "odd content" to point out the mathematical apparatus known as Fourier series that is the 3409:

periodic. Consequently, to cover the general case under consideration that includes non-periodic waveforms, one must resort to the

2536:, and those that require a considerable expansion of this proposal to go into nuances not so far entertained as a possible insert. 1581:: Your claim that "no sensible connection" has been made between Fourier series and wavelength seems to ignore the quote provided: 1517:, a statement I agree with, and (ii) Fourier series is a "strange", "incongruous" and "odd" addition to an article on wavelength. 1385:

that wavelength is a fundamental concept in the powerful, widely used, and seminal apparatus of Fourier series. The insertion is a

3882:

There are certainly special cases where the waveform can be a periodic function of time at a fixed location. Moreover, these are

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No-one seems brave enough to tackle the point here: the formal apparatus for dealing with arbitrary waveforms periodic in space

3862:

Upon further reflection, even at a fixed location where a dispersing waveform is strictly a function of time, it will not be a

96:. Some editors appear to find connecting Fourier series to wavelength is a digression. The text in question is provided below. 1935:? Why not include the Fourier series there also for the hell of it, because waves are also periodic in time and phase angle??? 278:

where this text referring to Fourier series is proposed to be placed. The mathematical definition is stated in the article as

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engaging in dialogue, and responding what you say, but you ignore them: its always about your own points. In other words see

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Note also that a wave that starts out periodic in time at one location will be periodic in time at other points as well, but

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in the opening paragraphs where it was pointed out that sine waves propagate with fixed form enabling the application of the

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to the propagation of arbitrary waveforms. These words were cribbed from your first draft, except the more accurate Fourier

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with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium. An example is the

2767:

I believe you added several kilobytes and destroyed the point of it. I'll wait and see if anyone else cares to comment.

1215:"Other decompositions" are not able to represent a periodic function throughput its domain, so Fourier series are special. 4296:

of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral submultiples of λ (

2646: 2219:{{Hatnote|For further descriptions of periodic functions in a more general context to wavelength and phase, see also ].}} 2180:

perspective (or words to that effect). That way the series is mentioned in passing with no obfuscation. Perhaps the best

147:

of spatial period λ, can be synthesized as a sum of harmonic functions whose wavelengths are integral submultiples of λ (

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and "mathematical pretense". These actions also appeared some time ago on the topic of "envelope", and led me to write

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link or template will indicate "for further descriptions of (spatial) periodicity and wavelength see the main articles

215:. That discussion is not pertinent in the present case because what we have here is a much more curtailed description. 4014:(iii) A question no-one can answer: why wouldn't Dick Lyon try to explain this matter, rather than sneering about it? 2122:"summation of sinusoidally varying waves with wavelengths related to the wavelength of the periodic waveform itself:" 690:

not clear to me that there is a non-negligible set of readers for whom this treatment would be beneficial, however.--

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Every time I explained it, you pronounced it incorrect, and changed it to something wrong. Eventually I gave up.

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is alluded to because the context is general waveforms and not spatially periodic ones. Are you clear that Fourier

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have a mention in this article so that someone learning about wavelength here can go on and learn about Fourier.

38: 3928:

Again, you missed the crucial phrase in the first sentence of the paragraph: "a wave that is periodic in time". --

844:

Just because wavelength and Fourier series relate to waves, doesn't imply there is any important connection, even

4545: 4540: 4532: 671: 543: 72: 67: 59: 2185: 1218: 1184: 1079: 3909:"To an observer at a fixed location, the amplitude varies in time and repeats itself with a certain period, T." 3049: 2932:

6. A periodic wave, as opposed to some approximation of a periodic wave, always has a wavelength as defined by

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John, no, the point is you "agree" with these editors but offer no reason for doing so, as I have pointed out.

266:

invariably refers to periodicity in space. Of course, periodicity can be expressed in many different ways, but

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Brews, I don't know if this is what Dick has in mind, but regarding the statement that general periodic waves

1466:

No meaningful connection between wavelength and sinusoidal Fourier series components has yet been mentioned.

666:

reader. Sourced, you say... it is our job to be selective about which sources are relevant to the subject. __

4448: 4258: 4213: 3384:, so may not have a well-defined wavelength. Such waves are typically analyzed into sinusoidal waves via the 2377: 2262: 2021: 1045: 3481: 3033: 2133: 1740:

in an interval , so these expansions do not represent the periodic function outside this selected interval.

989: 115:

This text is to appear following the equation defining the wavelength of a periodic function in the section

1248:)" But it would still seem incongruous in the proposed location. A footnote would be a little better. -- 762:, so it is a digression to ask whether Fourier series are more easily generalized using different concepts. 3922: 3876: 3840: 3519: 3437: 3315:, no attempt has been made to address substantive issues carefully laid out above and never responded to. 2911:

3. The definition of an eigenfunction is that it satisfies an operator equation with an eigenvalue, as in

2128:

is quite a mouthfull and not much help, a typical reader could get lost. Perhaps change the statement to:

825: 4284:

Some of that was about including a brief quotation about the connection of wavelength to Fourier series,

2612:

This general discussion is what was already alluded to in the above recapitulation of material regarding

1889:, as that is where wavelength has a role. It happens that Fourier series apply to a function periodic in 4485: 4430: 4386: 4360: 4159: 4120: 4078: 4024: 3918: 3872: 3836: 3777: 3643: 3607: 3545: 3515: 3477: 3433: 3353: 3335: 3320: 3240: 3218: 2991: 2819: 2787: 2750: 2736: 2661: 2460: 2290: 2085: 1902: 1845: 1797: 1756: 1657: 1641: 1598: 1540: 1486: 1394: 1307: 1205: 1157: 1130: 802: 643: 581: 510: 491: 445: 410: 372: 331: 303: 220: 101: 2892:

1. The dispersion relation definitely is related to the physics of the medium. A common example is the

2558:

Quote: "any periodic function can be represented as the superposition of harmonic terms of frequencies

255:

Dick, the source is cited as the origin of the quotation. I've moved the footnote to make this clearer.

1427:

I just don't find that approach interesting enough to get my vote. A more interesting approach is to

712:: "Sinusoidal waves are important because they occur in many physical situations..." -- seriously?). 235:"It is noteworthy that the wavelength of a general periodic waveform is related mathematically to its 1665: 1593:

Doesn't this statement make the connection between Fourier series and wavelength? Of course it does.

667: 539: 262:

is taken to refer to periodicity in time or with respect to some general variable, say ξ or θ, while

4594:

Roger Grimshaw (2007). "Solitary waves propagating over variable topography". In Anjan Kundu (ed.).

4409: 4171: 3959: 3933: 3895: 3852: 3573: 3257: 3025: 2845:

wavelength is functionally related to frequency by the physics of the medium" meaningless or false.

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on wavelength. You call it a "fluttery" statement, which I can only assume means something similar.

1871: 1825: 1780:, or spatial periodicity of the arbitrary but periodic waveform. The real point, however, is that 1733: 1037:?? Obviously not used in practice, its only for theoretical interest and definitely not essential. 695: 608: 528: 472: 425: 391: 352: 4442: 4252: 4207: 3638:

sentence or two there, and not feel obligated to remove the discussion from the periodic section.

3213:

formulation without addressing comments upon it is a waste of what little time you have to spend.

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it doesn't belabour the issue and focuses most directly on the connection between the two topics.

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I'll wait until we get other comments. This is supposed to be a RFC, not us arguing some more.

245: 4011:(ii) Is the discussion presented likely to be clear to a representative reader as it now stands? 1550:

The problem is that you haven't yet made a sensible connection. If your assertion is true that

4340:

are discarded as "trivial" and "mathematical pretense". Even a simple matter like substituting

523:

Regarding the second paragraph: It is not clear to me that this observation is "nontrivial". --

92:

Comment is sought as to whether a reference to Fourier series is appropriate under the heading

4631: 4605: 4575: 3723: 3417: 3410: 2954: 1711: 1339:? Personally, I would rather see the continuous transform of one cycle of the waveform. The 486:

function? Either claim would directly contradict the cited sources (or in fact, any sources).

367:

and as is the topic here, it is irrelevant to point out its use to analyze spatial variation.

180: 2605: 2555: 2487: 2478: 2475: 2471: 1033:. More intuitive than the Fourier series, becuase this formula actually says what wavelength 755:

wavelength, seems beyond the intended scope of simply pointing out that Fourier series use λ.

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be periodic. As you know, and as you have provided in your one-line edit rationale, Fourier

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that describe linear physical media, and the frequencies and wavelengths are related by the

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Dick has identified the need to rewrite this section, and has made a first attempt to do so.

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is any need for the quotations but you may as well keep the references for inline citations.

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It's hard to use Fourier series to define wavelength, just as it's hard to use it to define

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You seem to be very confused and wrong on each of these points. Please read them again.

3380:"In more general linear media (that is, dispersive media), a wave that is periodic in time 2481: 1701: 467:

It has nothing in particular to add to the topic of the article and thus strays off topic.

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the following text, which has been in the article in one form or another for a long time:

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In any event, sourced and factual matters like the mathematical definition of wavelength

2665: 2653:, which is again a wonderful subject that has nothing to do with the present discussion. 1706:(reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. ix 1381:

For the proposed text, however, a much more limited purpose is entertained. It is simply

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Consequently, the present statement in the article about a wave in a dispersive medium:

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probably be more strongly qualified, to indicate the rarity of this usage of the term.--

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The one-line edit justification given is "period waves are analyzed by Fourier series".

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the topic of dispersion and the special advantages of sine waves might go better in the

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It is evident from the figure referred to in the paragraph containing this remark that

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at all. This conversation on including something non-essential to an article is also

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do not depend on wavelength at all" is a flat contradiction of the quotation in the

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they will switch on to the Fourier series when they read "a superposition of waves

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Please notice the first part of the section states the waves under discussion may

2714:

by eliminating the generalized definition of wavelength for an arbitrary waveform

1302:

of including such a pointer? I see none at all: it is all upside and no downside.

1672:, or spatial periodicity, is not fundamental to these generalized Fourier series. 4519:

If you wish to start a new discussion or revive an old one, please do so on the

3017: 1668:. The length of the chosen interval appears in the analysis, but the concept of 576:

entitled to know about this connection to a truly gigantic area of mathematics?

46:

If you wish to start a new discussion or revive an old one, please do so on the

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only for ξ sitting inside some selected interval ? In particular, we can use ξ=

2881: 2823: 2808: 2791: 2776: 2754: 2740: 2684: 2498: 2464: 2446: 2395: 2294: 2280: 2089: 2039: 1906: 1880: 1849: 1834: 1801: 1760: 1645: 1602: 1573: 1544: 1490: 1475: 1441: 1398: 1352: 1311: 1271: 1257: 1228: 1209: 1194: 1176: 1161: 1134: 1108: 1089: 1063: 922:{\displaystyle {\frac {x}{\lambda }}={\frac {\phi }{2\pi }}={\frac {\tau }{T}}} 835: 806: 721: 699: 675: 647: 612: 585: 547: 532: 514: 495: 476: 449: 434: 414: 400: 376: 361: 335: 321: 307: 249: 105: 4349: 4001: 3232: 3228: 3172: 3041: 3013: 2251: 1785: 1433: 1344: 1249: 573: 556: 275: 120: 116: 93: 3040:

sharper crests and flatter troughs than those of a sinusoid. Large-amplitude

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the formal apparatus for dealing with arbitrary waveforms periodic in space

1552:

the formal apparatus for dealing with arbitrary waveforms periodic in space

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is the number of wave cycles passing through two fixed points of seperation

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periodicity is meant, not time periodicity. Now, a spatially periodic wave

595:, however, that topic B deserves a mention in the article on A. Usually, B 2137: 134:, and is simply a summation of the sinusoidally varying component waves: 3757:

removed because it is only mathematical pretense strikes me as bizarre.

2941:

7. Fine, we'll make it the least value of λ for which this is the case.

1885:

Blackburne, what you still fail to recognize is that the topic here is

1513:

take on two forms: (i) Fourier series do not illuminate the concept of

381:

You can reinterpret my statement as you like; my point is I agree with

2893: 2887:

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

2472:

see "nothing arbitrary" in this Britannica article by Lord Rayleigh

2285:

F = q(E+v×B): I have taken a whack at rewording along these lines.

520:

to an article on wavelength. Relevance is not always bidirectional.

4330: 3073: 4250:, not much help reducing the byte count but better than nothing. 3946:. One can decompose the initial wave into sinusoids (via Fourier 3048:, a periodic traveling wave named because it is described by the 2118:

Perhaps we can compromise on this: part of your current proposal

3888:

that is not what is being discussed in the paragraph in question

3029: 3976:

are overtaken by the fast components of pulses leaving at time

4498: 1736:

defined over a finite interval, for example, solutions to the

25: 2005:

How many editors have said the same thing? Isn't it obvious?

3082:

If a traveling wave has a fixed shape that repeats, it is a

2973:

values of the argument ξ and not restricted to representing

4348:

is refused based on a specious argument that an article on

4624:

Alexander McPherson (2009). "Waves and their properties".

3694:"Mathematically, a periodic wave in one spatial dimension 2597:

The contribution of this source to the discussion of the

2546:

The contribution of this source to the discussion of the

1840:

wording right has proved so distracting to your thought.

274:

is used to define a periodic function in the subsection

4440:

improve the article but this isn't getting anywhere...

4374: 4370: 4154:

my efforts, as this little addition of yours examples.

4112: 4070: 4066: 3761: 3688: 3428:

would be better, but that is a redirect at the moment.

3368: 3312: 3308: 420: 2609:

already emerges when one calculates the eigenvalues."

769:

does not come up because its underlying definition is

4355:

I'll leave this for a few weeks and try again later.

3487:

Finally, in this reversion you moved these sentences:

3078:

Wavelength of a periodic but non-sinusoidal waveform.

3032:

solutions, more complex solutions can be built up by

992: 965: 945: 877: 3028:

of the equations or those media. From these simple

1784:is a huge topic, a powerful apparatus, and it uses 3235:where dispersion is described. What do you think? 1011: 971: 951: 921: 3090:of an unchanging waveform moving with a velocity 2627:Fourier series and its connection to wavelength. 735:wavelength, but do not illuminate the concept of 176:Schaum's Outline of Theory and Problems of Optics 3997:So I guess that leaves me with a few questions: 3510:. Consequently this change introduces an error. 2451:Dick: I rewrote the contents of the "green box". 731:Fourier series for spatially periodic waveforms 2965:represent periodic functions with the property 1897:. Please try to adopt the appropriate context. 125: 4627:Introduction to Macromolecular Crystallography 4019:In any event, thank you for your explanation. 2601:is only to repeat what is said there already. 2188:'s version above. I would simply prefer to add 2184:perspective (if preferred by others) would be 1613:Dick, apparently you doubt the assertion that 3634:, which I believe is the correct terminology. 2140:, which can be mathematically described by a 8: 3448:"If a traveling wave has a fixed shape that 2915:. It is not defined in terms of propagation. 4369:The battles long ago with Dick Lyon over a 1509:To this point, objections to including the 423:as you seem to have trouble finding them.-- 4197:repeatedly explained things to you, which 1865:Perhaps you could point out which part of 999: 991: 964: 944: 909: 891: 878: 876: 4090:Talk:Wavelength#phase_and_group_velocity 3362: 1854:BTW, Blackburne, your observation that " 4560: 4472:Blackburne as the "me too" cheerleader. 4371:reference to the envelope of a waveform 3760:By the same token, I cannot fathom why 3413:, which treats waveforms of all kinds. 3171:In more general linear media (that is, 1692: 1012:{\displaystyle \lambda ={\frac {L}{N}}} 326:I removed "noteworthy" as unnecessary. 165: 4517:Do not edit the contents of this page. 3363:Dick Lyon's recent incorrect changes 2731:might fit into the ultimate revision. 1732:expressions most often are based upon 564:might be referred to in an article on 44:Do not edit the contents of this page. 3698:satisfies the periodicity condition: 3443:In a second change, you have written: 3416:Consequently, Dick, your change from 3375:They include the following statement: 3022:linear partial differential equations 2986:I await your further comments, Dick. 1703:Fourier Analysis and Its Applications 1700:Gerald B Folland (2009). "Contents". 7: 4246:I plan to archive all up to section 3382:will not necessarily repeat in space 179:. McGraw-Hill Professional. p. 205. 4407:a few weeks. Drop it and move on.-- 4088:As I and others told you before at 1819:do not depend on wavelength at all. 662:obliterating the previous one(s). 3424:has introduced an error. Probably 3117:= time. The amplitude at location 24: 4571:Global environment remote sensing 1823:minutes after they were posted.-- 1815:mathematical topic, not least as 1654:Generalizations of Fourier series 685:I do see the appeal in trying to 4567: 4502: 3137:is the same as that at location 2136:and create a spatially periodic 1931:of wave oscillations? And about 1650:To repeat what was said before: 1125:to including the proposed text. 111:Text referring to Fourier series 29: 3722:. John Wiley & Sons. p. 1. 3460:have a well-defined wavelength. 2698:. This fact is made a segue to 4061:Restoration of two subsections 3913:is an egregious misstatement. 3056:-th order, usually denoted as 1331:opens another issue, which is 1: 3371:that I think are incorrect. 2009:of "failing to understand" - 4377:) led to my construction of 1337:continuous Fourier transform 4597:Tsunami and Nonlinear Waves 4441: 4251: 4248:Mathematical representation 4206: 2370: 2255: 2252:Mathematical representation 2014: 2007:Stop accusing other editors 1927:What about time period and 1562:the Fourier series, no less 1038: 88:Reference to Fourier series 4657: 4490:17:30, 27 April 2012 (UTC) 4467:17:08, 27 April 2012 (UTC) 4435:16:46, 27 April 2012 (UTC) 4419:16:26, 27 April 2012 (UTC) 4391:15:43, 27 April 2012 (UTC) 4365:14:08, 27 April 2012 (UTC) 4329:More recently, I raised a 4277:12:35, 27 April 2012 (UTC) 4232:11:09, 27 April 2012 (UTC) 4181:10:42, 27 April 2012 (UTC) 4164:06:10, 27 April 2012 (UTC) 4148:05:19, 27 April 2012 (UTC) 4125:06:37, 27 April 2012 (UTC) 4111:Dick: This material about 4103:03:53, 27 April 2012 (UTC) 4083:17:23, 26 April 2012 (UTC) 4051:05:18, 28 April 2012 (UTC) 4029:21:53, 27 April 2012 (UTC) 3964:17:23, 27 April 2012 (UTC) 3938:17:10, 27 April 2012 (UTC) 3923:16:22, 27 April 2012 (UTC) 3900:17:10, 27 April 2012 (UTC) 3877:15:26, 27 April 2012 (UTC) 3857:17:10, 27 April 2012 (UTC) 3841:06:24, 27 April 2012 (UTC) 3813:03:42, 27 April 2012 (UTC) 3782:22:27, 26 April 2012 (UTC) 3772:Dick, what is the matter? 3671:20:20, 26 April 2012 (UTC) 3648:19:10, 26 April 2012 (UTC) 3612:17:54, 26 April 2012 (UTC) 3578:17:31, 26 April 2012 (UTC) 3550:15:19, 26 April 2012 (UTC) 3535:14:54, 26 April 2012 (UTC) 3520:05:57, 26 April 2012 (UTC) 3482:05:48, 26 April 2012 (UTC) 3438:05:48, 26 April 2012 (UTC) 3358:17:30, 26 April 2012 (UTC) 3340:19:30, 25 April 2012 (UTC) 3325:15:10, 25 April 2012 (UTC) 3297:15:03, 25 April 2012 (UTC) 3283:15:01, 25 April 2012 (UTC) 3267:14:43, 25 April 2012 (UTC) 3245:14:26, 25 April 2012 (UTC) 3223:05:40, 25 April 2012 (UTC) 3204:05:24, 25 April 2012 (UTC) 2996:23:32, 24 April 2012 (UTC) 2882:21:41, 24 April 2012 (UTC) 2824:17:31, 24 April 2012 (UTC) 2809:17:20, 24 April 2012 (UTC) 2792:16:40, 24 April 2012 (UTC) 2777:15:38, 24 April 2012 (UTC) 2755:13:58, 24 April 2012 (UTC) 2741:12:22, 24 April 2012 (UTC) 2685:01:44, 24 April 2012 (UTC) 2666:03:13, 24 April 2012 (UTC) 2651:discrete Fourier transform 2636:generalized Fourier series 2614:generalized Fourier series 2499:01:23, 24 April 2012 (UTC) 2465:00:35, 24 April 2012 (UTC) 2447:23:22, 23 April 2012 (UTC) 2396:20:31, 23 April 2012 (UTC) 2295:19:28, 23 April 2012 (UTC) 2281:17:55, 23 April 2012 (UTC) 2090:17:11, 23 April 2012 (UTC) 2040:16:42, 23 April 2012 (UTC) 1907:15:54, 23 April 2012 (UTC) 1881:15:47, 23 April 2012 (UTC) 1850:15:26, 23 April 2012 (UTC) 1835:14:55, 23 April 2012 (UTC) 1802:14:22, 23 April 2012 (UTC) 1761:12:44, 23 April 2012 (UTC) 1730:Generalized Fourier series 1646:12:33, 23 April 2012 (UTC) 1633:generalized Fourier series 1603:12:55, 23 April 2012 (UTC) 1574:06:16, 23 April 2012 (UTC) 1545:04:35, 23 April 2012 (UTC) 1491:05:15, 23 April 2012 (UTC) 1476:04:14, 23 April 2012 (UTC) 1442:00:56, 24 April 2012 (UTC) 1399:13:33, 23 April 2012 (UTC) 1353:12:29, 23 April 2012 (UTC) 1312:04:08, 23 April 2012 (UTC) 1272:02:39, 23 April 2012 (UTC) 1258:01:52, 23 April 2012 (UTC) 1229:16:43, 23 April 2012 (UTC) 1210:13:08, 23 April 2012 (UTC) 1195:11:53, 23 April 2012 (UTC) 1177:06:18, 23 April 2012 (UTC) 1162:04:08, 23 April 2012 (UTC) 1135:04:08, 23 April 2012 (UTC) 1109:02:39, 23 April 2012 (UTC) 1090:00:22, 23 April 2012 (UTC) 1064:23:38, 22 April 2012 (UTC) 836:18:20, 22 April 2012 (UTC) 807:21:47, 22 April 2012 (UTC) 722:18:37, 22 April 2012 (UTC) 700:18:13, 22 April 2012 (UTC) 676:17:29, 22 April 2012 (UTC) 648:16:25, 22 April 2012 (UTC) 613:18:13, 22 April 2012 (UTC) 586:16:25, 22 April 2012 (UTC) 548:16:17, 22 April 2012 (UTC) 533:16:00, 22 April 2012 (UTC) 515:01:35, 22 April 2012 (UTC) 496:20:44, 21 April 2012 (UTC) 477:20:24, 21 April 2012 (UTC) 450:13:39, 22 April 2012 (UTC) 435:11:04, 22 April 2012 (UTC) 415:00:57, 22 April 2012 (UTC) 401:21:08, 21 April 2012 (UTC) 377:20:44, 21 April 2012 (UTC) 362:19:56, 21 April 2012 (UTC) 336:16:07, 21 April 2012 (UTC) 322:15:46, 21 April 2012 (UTC) 308:15:25, 21 April 2012 (UTC) 250:01:57, 21 April 2012 (UTC) 225:13:47, 22 April 2012 (UTC) 106:20:09, 20 April 2012 (UTC) 94:general periodic waveforms 4568:Ken'ichi Okamoto (2001). 3229:general waveforms section 1867:Fourier series#Definition 389:that it doesn't belong.-- 270:is the subject here, and 119:, just before the header 3944:not with the same period 3050:Jacobi elliptic function 2797:terse edit summaries).-- 2649:This text describes the 1738:Sturm-Liouville equation 4630:(2 ed.). Wiley. p. 77. 4292:states that a function 4073:these two subsections. 4067:reversion by Blackburne 3313:accoladed by Dick Lyon 3309:reversion by Blackburne 2526: 1984:" (or words to effect). 1662:Fourier-Legendre series 1290:entire formal treatment 1217:← That's just wrong. 200: 143:states that a function 4375:continues on this page 3079: 2671:New section in article 2132:"Sinusiodal waves can 1656:include, for example, 1013: 973: 953: 923: 850:really isn't essential 276:More general waveforms 117:More general waveforms 4574:. IOS Press. p. 263. 4515:of past discussions. 4004:article focus on the 3307:The recent wholesale 3233:general media section 3077: 1658:Fourier-Bessel series 1014: 974: 972:{\displaystyle \tau } 954: 952:{\displaystyle \phi } 939:is a spatial length, 924: 173:Eugene Hecht (1975). 42:of past discussions. 4331:three-point critique 3405:apply to waves that 3094:can be expressed as 3026:dispersion relations 1962:each with wavelength 1887:periodicity in space 1734:orthogonal functions 990: 963: 943: 875: 4600:. Springer. pp. 52 4373:(which it seems he 4338:f(x−vt+λ) = f(x−vt) 3716:Eric Stade (2011). 3700:f(x−vt+λ) = f(x−vt) 3303:High-handed actions 848:the Fourier series 4133:Brews, please stop 3563:are of course not 3080: 2527:DickLyon's sources 2250:at the top of the 1625:the Fourier series 1009: 969: 959:is a phase angle, 949: 919: 500:The point is that 4558: 4557: 4527: 4526: 4521:current talk page 4412: 4290:Fourier's theorem 4174: 3418:Fourier transform 3411:Fourier transform 3260: 2955:Fourier transform 2934:f(x-vt+λ)=f(x-vt) 2802: 2716:f(x−vt+λ)=f(x−vt) 1874: 1828: 1813: 1007: 917: 904: 886: 834: 428: 394: 355: 198: 197: 141:Fourier's theorem 85: 84: 54: 53: 48:current talk page 4648: 4641: 4640: 4621: 4615: 4614: 4591: 4585: 4584: 4565: 4554: 4529: 4528: 4506: 4505: 4499: 4478:Envelope (waves) 4465: 4408: 4379:Envelope (waves) 4342:Fourier integral 4275: 4230: 4170: 3732: 3719:Fourier Analysis 3632:Fourier integral 3426:Fourier integral 3369:these reversions 3256: 3167: 3136: 3126: 3108: 3070: 2798: 2394: 2279: 2038: 1974:why its measured 1870: 1824: 1807: 1741: 1727: 1721: 1720: 1697: 1532:no-cost addition 1221: 1187: 1082: 1075:Fourier analysis 1062: 1018: 1016: 1015: 1010: 1008: 1000: 978: 976: 975: 970: 958: 956: 955: 950: 928: 926: 925: 920: 918: 910: 905: 903: 892: 887: 879: 830: 823: 819: 424: 390: 351: 201:Author's remarks 190: 189: 170: 126: 81: 56: 55: 33: 32: 26: 4656: 4655: 4651: 4650: 4649: 4647: 4646: 4645: 4644: 4637: 4623: 4622: 4618: 4611: 4593: 4592: 4588: 4581: 4566: 4562: 4550: 4503: 4463: 4459: 4417: 4273: 4269: 4228: 4224: 4179: 4135: 4063: 3988: 3981: 3974: 3729: 3715: 3367:Dick, you made 3365: 3305: 3265: 3154: 3153:are related by 3128: 3118: 3113:= position and 3095: 3057: 2921:eigenfunctions. 2900: 2807: 2673: 2592: 2585: 2581: 2574: 2570: 2563: 2529: 2392: 2388: 2277: 2273: 2036: 2032: 1879: 1833: 1773: 1745: 1744: 1728: 1724: 1717: 1699: 1698: 1694: 1507: 1219: 1185: 1080: 1060: 1056: 988: 987: 961: 960: 941: 940: 896: 873: 872: 828: 821: 668:Just plain Bill 620:Just plain Bill 593:does not follow 540:Just plain Bill 433: 419:My reasons are 399: 360: 233: 203: 194: 193: 186: 172: 171: 167: 130:described as a 113: 90: 77: 30: 22: 21: 20: 18:Talk:Wavelength 12: 11: 5: 4654: 4652: 4643: 4642: 4635: 4616: 4609: 4586: 4579: 4559: 4556: 4555: 4548: 4543: 4538: 4535: 4525: 4524: 4507: 4497: 4496: 4495: 4494: 4493: 4492: 4473: 4461: 4457: 4413: 4410:JohnBlackburne 4400: 4399: 4398: 4397: 4396: 4395: 4394: 4393: 4367: 4353: 4346:Fourier series 4334: 4327: 4316: 4315: 4314: 4313: 4312: 4311: 4310: 4309: 4308: 4307: 4306: 4305: 4282: 4271: 4267: 4239: 4238: 4237: 4236: 4235: 4234: 4226: 4222: 4186: 4185: 4184: 4183: 4175: 4172:JohnBlackburne 4134: 4131: 4130: 4129: 4128: 4127: 4106: 4105: 4062: 4059: 4058: 4057: 4056: 4055: 4054: 4053: 4034: 4033: 4032: 4031: 4017: 4016: 4015: 4012: 4009: 3995: 3991: 3986: 3979: 3972: 3940: 3911: 3910: 3903: 3902: 3860: 3859: 3824: 3823: 3822: 3821: 3820: 3819: 3818: 3817: 3816: 3815: 3791: 3790: 3789: 3788: 3787: 3786: 3785: 3784: 3770: 3767:Fourier series 3758: 3754: 3744: 3743: 3742: 3741: 3740: 3739: 3738: 3737: 3736: 3735: 3734: 3733: 3727: 3712: 3678: 3677: 3676: 3675: 3674: 3673: 3653: 3652: 3651: 3650: 3635: 3628:Fourier series 3624: 3617: 3616: 3615: 3614: 3597: 3596: 3595: 3594: 3587: 3586: 3585: 3584: 3555: 3554: 3553: 3552: 3494: 3493: 3489: 3488: 3462: 3461: 3445: 3444: 3422:Fourier series 3390: 3389: 3386:Fourier series 3377: 3376: 3364: 3361: 3304: 3301: 3300: 3299: 3285: 3261: 3258:JohnBlackburne 3252: 3251: 3250: 3249: 3248: 3247: 3225: 3210: 3189: 3177:Fourier series 3030:traveling wave 3009: 3008: 3007: 3006: 3005: 2999: 2998: 2983: 2982: 2949: 2948: 2943: 2942: 2938: 2937: 2929: 2928: 2923: 2922: 2917: 2916: 2908: 2907: 2903: 2902: 2898: 2889: 2888: 2871: 2870: 2866: 2862: 2859: 2856: 2853: 2850: 2846: 2837: 2836: 2835: 2834: 2833: 2832: 2831: 2830: 2829: 2828: 2827: 2826: 2803: 2800:JohnBlackburne 2760: 2759: 2758: 2757: 2724: 2723: 2710: 2709: 2704: 2703: 2700:Fourier series 2691: 2672: 2669: 2590: 2583: 2579: 2572: 2568: 2561: 2528: 2525: 2524: 2523: 2522: 2521: 2520: 2519: 2518: 2517: 2516: 2515: 2514: 2513: 2512: 2511: 2510: 2509: 2508: 2507: 2506: 2505: 2504: 2503: 2502: 2501: 2452: 2415: 2414: 2413: 2412: 2411: 2410: 2409: 2408: 2407: 2406: 2405: 2404: 2403: 2402: 2401: 2400: 2399: 2398: 2390: 2386: 2350: 2349: 2348: 2347: 2346: 2345: 2344: 2343: 2342: 2341: 2340: 2339: 2338: 2337: 2336: 2335: 2334: 2333: 2312: 2311: 2310: 2309: 2308: 2307: 2306: 2305: 2304: 2303: 2302: 2301: 2300: 2299: 2298: 2297: 2275: 2271: 2235: 2234: 2233: 2232: 2231: 2230: 2229: 2228: 2227: 2226: 2225: 2224: 2223: 2222: 2221: 2220: 2202: 2201: 2200: 2199: 2198: 2197: 2196: 2195: 2194: 2193: 2192: 2191: 2190: 2189: 2186:Sławomir Biały 2161: 2160: 2159: 2158: 2157: 2156: 2155: 2154: 2153: 2152: 2151: 2150: 2149: 2148: 2147: 2146: 2142:Fourier series 2126: 2125: 2124: 2103: 2102: 2101: 2100: 2099: 2098: 2097: 2096: 2095: 2094: 2093: 2092: 2074: 2070: 2066: 2051: 2050: 2049: 2048: 2047: 2046: 2045: 2044: 2043: 2042: 2034: 2030: 1994: 1993: 1992: 1991: 1990: 1989: 1988: 1987: 1986: 1985: 1982:Fourier series 1945: 1944: 1943: 1942: 1941: 1940: 1939: 1938: 1937: 1936: 1916: 1915: 1914: 1913: 1912: 1911: 1910: 1909: 1875: 1872:JohnBlackburne 1856:Fourier series 1852: 1829: 1826:JohnBlackburne 1820: 1817:Fourier series 1782:Fourier series 1772: 1769: 1768: 1767: 1766: 1765: 1764: 1763: 1743: 1742: 1722: 1715: 1691: 1690: 1689: 1688: 1687: 1686: 1678: 1677: 1676: 1675: 1674: 1673: 1648: 1608: 1607: 1606: 1605: 1591: 1590: 1589: 1588: 1587: 1506: 1503: 1502: 1501: 1500: 1499: 1498: 1497: 1496: 1495: 1494: 1493: 1457: 1456: 1455: 1454: 1453: 1452: 1451: 1450: 1449: 1448: 1447: 1446: 1445: 1444: 1412: 1411: 1410: 1409: 1408: 1407: 1406: 1405: 1404: 1403: 1402: 1401: 1379: 1364: 1363: 1362: 1361: 1360: 1359: 1358: 1357: 1356: 1355: 1341:Fourier series 1333:Fourier series 1319: 1318: 1317: 1316: 1315: 1314: 1298:? What is the 1277: 1276: 1275: 1274: 1246:Fourier series 1239: 1238: 1237: 1236: 1235: 1234: 1233: 1232: 1231: 1220:Sławomir Biały 1186:Sławomir Biały 1181: 1180: 1179: 1142: 1141: 1140: 1139: 1138: 1137: 1114:Sławomir Biały 1093: 1092: 1081:Sławomir Biały 1067: 1066: 1058: 1054: 1022: 1021: 1020: 1019: 1006: 1003: 998: 995: 982: 981: 968: 948: 932: 931: 930: 929: 916: 913: 908: 902: 899: 895: 890: 885: 882: 867: 866: 862: 861: 812: 811: 810: 809: 796: 795: 794: 790: 763: 756: 744: 728: 727: 683: 659: 658: 657: 656: 655: 654: 653: 652: 651: 650: 617: 616: 615: 601: 521: 498: 480: 479: 461: 460: 459: 458: 457: 456: 455: 454: 453: 452: 429: 426:JohnBlackburne 417: 395: 392:JohnBlackburne 356: 353:JohnBlackburne 343: 342: 341: 340: 339: 338: 295: 256: 237:Fourier series 232: 229: 228: 227: 208: 207: 202: 199: 196: 195: 192: 191: 184: 164: 163: 157: 156: 136: 135: 132:Fourier series 121:Envelope waves 112: 109: 89: 86: 83: 82: 75: 70: 65: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 4653: 4638: 4633: 4629: 4628: 4620: 4617: 4612: 4607: 4603: 4599: 4598: 4590: 4587: 4582: 4580:9781586031015 4577: 4573: 4572: 4564: 4561: 4553: 4549: 4547: 4544: 4542: 4539: 4536: 4534: 4531: 4530: 4522: 4518: 4514: 4513: 4508: 4501: 4500: 4491: 4487: 4483: 4479: 4474: 4470: 4469: 4468: 4464: 4454: 4452: 4447: 4445: 4438: 4437: 4436: 4432: 4428: 4423: 4422: 4421: 4420: 4416: 4411: 4405: 4392: 4388: 4384: 4380: 4376: 4372: 4368: 4366: 4362: 4358: 4354: 4351: 4347: 4343: 4339: 4335: 4332: 4328: 4324: 4323: 4322: 4321: 4320: 4319: 4318: 4317: 4303: 4300:λ, λ/2, λ/3, 4299: 4295: 4291: 4288: 4287: 4286: 4285: 4283: 4280: 4279: 4278: 4274: 4264: 4262: 4257: 4255: 4249: 4245: 4244: 4243: 4242: 4241: 4240: 4233: 4229: 4219: 4217: 4212: 4210: 4204: 4200: 4196: 4192: 4191: 4190: 4189: 4188: 4187: 4182: 4178: 4173: 4167: 4166: 4165: 4161: 4157: 4152: 4151: 4150: 4149: 4145: 4141: 4132: 4126: 4122: 4118: 4114: 4110: 4109: 4108: 4107: 4104: 4100: 4096: 4091: 4087: 4086: 4085: 4084: 4080: 4076: 4072: 4071:have restored 4068: 4060: 4052: 4048: 4044: 4040: 4039: 4038: 4037: 4036: 4035: 4030: 4026: 4022: 4018: 4013: 4010: 4007: 4003: 3999: 3998: 3996: 3992: 3989: 3982: 3975: 3967: 3966: 3965: 3961: 3957: 3953: 3949: 3945: 3941: 3939: 3935: 3931: 3927: 3926: 3925: 3924: 3920: 3916: 3908: 3907: 3906: 3901: 3897: 3893: 3889: 3885: 3881: 3880: 3879: 3878: 3874: 3870: 3865: 3858: 3854: 3850: 3845: 3844: 3843: 3842: 3838: 3834: 3828: 3814: 3810: 3806: 3801: 3800: 3799: 3798: 3797: 3796: 3795: 3794: 3793: 3792: 3783: 3779: 3775: 3771: 3768: 3763: 3759: 3755: 3752: 3751: 3750: 3749: 3748: 3747: 3746: 3745: 3730: 3725: 3721: 3720: 3713: 3711: 3708: 3707: 3705: 3701: 3697: 3693: 3692: 3690: 3687:Hi Dick: You 3686: 3685: 3684: 3683: 3682: 3681: 3680: 3679: 3672: 3668: 3664: 3659: 3658: 3657: 3656: 3655: 3654: 3649: 3645: 3641: 3636: 3633: 3629: 3626:I've changed 3625: 3621: 3620: 3619: 3618: 3613: 3609: 3605: 3601: 3600: 3599: 3598: 3591: 3590: 3589: 3588: 3581: 3580: 3579: 3575: 3571: 3566: 3561: 3557: 3556: 3551: 3547: 3543: 3538: 3537: 3536: 3532: 3528: 3524: 3523: 3522: 3521: 3517: 3513: 3509: 3504: 3500: 3491: 3490: 3486: 3485: 3484: 3483: 3479: 3475: 3471: 3467: 3459: 3456:. Such waves 3455: 3454:periodic wave 3451: 3447: 3446: 3442: 3441: 3440: 3439: 3435: 3431: 3427: 3423: 3419: 3414: 3412: 3408: 3404: 3403: 3398: 3393: 3387: 3383: 3379: 3378: 3374: 3373: 3372: 3370: 3360: 3359: 3355: 3351: 3347: 3342: 3341: 3337: 3333: 3327: 3326: 3322: 3318: 3314: 3310: 3302: 3298: 3294: 3290: 3286: 3284: 3280: 3276: 3271: 3270: 3269: 3268: 3264: 3259: 3246: 3242: 3238: 3234: 3230: 3226: 3224: 3220: 3216: 3211: 3207: 3206: 3205: 3201: 3197: 3193: 3192: 3191: 3190: 3188: 3186: 3182: 3178: 3174: 3169: 3166: 3162: 3158: 3152: 3148: 3144: 3140: 3135: 3131: 3125: 3121: 3116: 3112: 3106: 3102: 3098: 3093: 3089: 3085: 3084:periodic wave 3076: 3072: 3068: 3064: 3060: 3055: 3051: 3047: 3043: 3037: 3035: 3034:superposition 3031: 3027: 3023: 3019: 3015: 3003: 3002: 3001: 3000: 2997: 2993: 2989: 2985: 2984: 2980: 2976: 2972: 2968: 2967:f(ξ) = f(ξ+λ) 2964: 2960: 2956: 2951: 2950: 2945: 2944: 2940: 2939: 2935: 2931: 2930: 2925: 2924: 2919: 2918: 2914: 2910: 2909: 2905: 2904: 2895: 2891: 2890: 2886: 2885: 2884: 2883: 2879: 2875: 2867: 2863: 2860: 2857: 2854: 2851: 2847: 2843: 2842: 2841: 2825: 2821: 2817: 2812: 2811: 2810: 2806: 2801: 2795: 2794: 2793: 2789: 2785: 2780: 2779: 2778: 2774: 2770: 2766: 2765: 2764: 2763: 2762: 2761: 2756: 2752: 2748: 2744: 2743: 2742: 2738: 2734: 2730: 2729:proposed text 2726: 2725: 2721: 2720:proposed text 2717: 2712: 2711: 2706: 2705: 2701: 2697: 2696:Lord Rayleigh 2692: 2689: 2688: 2687: 2686: 2682: 2678: 2670: 2668: 2667: 2663: 2659: 2654: 2652: 2648: 2643: 2641: 2637: 2633: 2628: 2626: 2622: 2617: 2615: 2610: 2607: 2602: 2600: 2599:proposed text 2595: 2593: 2586: 2575: 2564: 2557: 2552: 2549: 2548:proposed text 2544: 2542: 2537: 2535: 2534:proposed text 2500: 2496: 2492: 2488: 2485: 2482: 2479: 2476: 2473: 2468: 2467: 2466: 2462: 2458: 2453: 2450: 2449: 2448: 2444: 2440: 2435: 2434: 2433: 2432: 2431: 2430: 2429: 2428: 2427: 2426: 2425: 2424: 2423: 2422: 2421: 2420: 2419: 2418: 2417: 2416: 2397: 2393: 2383: 2381: 2376: 2374: 2368: 2367: 2366: 2365: 2364: 2363: 2362: 2361: 2360: 2359: 2358: 2357: 2356: 2355: 2354: 2353: 2352: 2351: 2330: 2329: 2328: 2327: 2326: 2325: 2324: 2323: 2322: 2321: 2320: 2319: 2318: 2317: 2316: 2315: 2314: 2313: 2296: 2292: 2288: 2284: 2283: 2282: 2278: 2268: 2266: 2261: 2259: 2253: 2249: 2248: 2247: 2246: 2245: 2244: 2243: 2242: 2241: 2240: 2239: 2238: 2237: 2236: 2218: 2217: 2216: 2215: 2214: 2213: 2212: 2211: 2210: 2209: 2208: 2207: 2206: 2205: 2204: 2203: 2187: 2183: 2179: 2175: 2174: 2173: 2172: 2171: 2170: 2169: 2168: 2167: 2166: 2165: 2164: 2163: 2162: 2145: 2143: 2139: 2135: 2130: 2129: 2127: 2123: 2120: 2119: 2117: 2116: 2115: 2114: 2113: 2112: 2111: 2110: 2109: 2108: 2107: 2106: 2105: 2104: 2091: 2087: 2083: 2079: 2078:proposed text 2075: 2071: 2067: 2063: 2062: 2061: 2060: 2059: 2058: 2057: 2056: 2055: 2054: 2053: 2052: 2041: 2037: 2027: 2025: 2020: 2018: 2012: 2008: 2004: 2003: 2002: 2001: 2000: 1999: 1998: 1997: 1996: 1995: 1983: 1979: 1975: 1971: 1967: 1963: 1959: 1955: 1954: 1953: 1952: 1951: 1950: 1949: 1948: 1947: 1946: 1934: 1930: 1926: 1925: 1924: 1923: 1922: 1921: 1920: 1919: 1918: 1917: 1908: 1904: 1900: 1896: 1895:proposed text 1892: 1888: 1884: 1883: 1882: 1878: 1873: 1868: 1864: 1863: 1861: 1860:proposed text 1857: 1853: 1851: 1847: 1843: 1838: 1837: 1836: 1832: 1827: 1821: 1818: 1811: 1810:edit conflict 1806: 1805: 1804: 1803: 1799: 1795: 1791: 1790:proposed text 1787: 1783: 1779: 1770: 1762: 1758: 1754: 1751: 1750: 1749: 1748: 1747: 1746: 1739: 1735: 1731: 1726: 1723: 1718: 1713: 1709: 1705: 1704: 1696: 1693: 1685: 1682: 1681: 1680: 1679: 1671: 1667: 1663: 1659: 1655: 1652: 1651: 1649: 1647: 1643: 1639: 1634: 1631:of the other 1630: 1626: 1623: 1622: 1618: 1617: 1612: 1611: 1610: 1609: 1604: 1600: 1596: 1592: 1585: 1584: 1583: 1582: 1580: 1577: 1576: 1575: 1571: 1567: 1563: 1560: 1559: 1555: 1554: 1549: 1548: 1547: 1546: 1542: 1538: 1534: 1533: 1527: 1525: 1524: 1518: 1516: 1512: 1511:proposed text 1504: 1492: 1488: 1484: 1479: 1478: 1477: 1473: 1469: 1465: 1464: 1463: 1462: 1461: 1460: 1459: 1458: 1443: 1439: 1435: 1430: 1426: 1425: 1424: 1423: 1422: 1421: 1420: 1419: 1418: 1417: 1416: 1415: 1414: 1413: 1400: 1396: 1392: 1388: 1384: 1380: 1376: 1375: 1374: 1373: 1372: 1371: 1370: 1369: 1368: 1367: 1366: 1365: 1354: 1350: 1346: 1342: 1338: 1334: 1329: 1328: 1327: 1326: 1325: 1324: 1323: 1322: 1321: 1320: 1313: 1309: 1305: 1301: 1297: 1292: 1291: 1286: 1283: 1282: 1281: 1280: 1279: 1278: 1273: 1269: 1265: 1261: 1260: 1259: 1255: 1251: 1247: 1243: 1240: 1230: 1226: 1222: 1216: 1213: 1212: 1211: 1207: 1203: 1198: 1197: 1196: 1192: 1188: 1182: 1178: 1174: 1170: 1165: 1164: 1163: 1159: 1155: 1151: 1148: 1147: 1146: 1145: 1144: 1143: 1136: 1132: 1128: 1124: 1119: 1115: 1112: 1111: 1110: 1106: 1102: 1097: 1096: 1095: 1094: 1091: 1087: 1083: 1076: 1072: 1069: 1068: 1065: 1061: 1051: 1049: 1044: 1042: 1036: 1032: 1028: 1024: 1023: 1004: 1001: 996: 993: 986: 985: 984: 983: 966: 946: 938: 934: 933: 914: 911: 906: 900: 897: 893: 888: 883: 880: 871: 870: 869: 868: 864: 863: 859: 856:, for such a 855: 851: 847: 843: 840: 839: 838: 837: 833: 832: 824: 816: 808: 804: 800: 797: 791: 788: 784: 780: 776: 772: 768: 764: 761: 757: 754: 749: 745: 742: 738: 734: 730: 729: 725: 724: 723: 719: 715: 711: 707: 706:pitch (music) 703: 702: 701: 697: 693: 688: 684: 680: 679: 678: 677: 673: 669: 663: 649: 645: 641: 637: 633: 629: 625: 621: 618: 614: 610: 606: 602: 598: 594: 589: 588: 587: 583: 579: 575: 571: 567: 563: 558: 554: 551: 550: 549: 545: 541: 536: 535: 534: 530: 526: 522: 518: 517: 516: 512: 508: 503: 499: 497: 493: 489: 484: 483: 482: 481: 478: 474: 470: 466: 463: 462: 451: 447: 443: 438: 437: 436: 432: 427: 422: 418: 416: 412: 408: 404: 403: 402: 398: 393: 388: 384: 380: 379: 378: 374: 370: 365: 364: 363: 359: 354: 349: 346:I agree with 345: 344: 337: 333: 329: 325: 324: 323: 319: 315: 311: 310: 309: 305: 301: 296: 293: 289: 285: 281: 277: 273: 269: 265: 261: 257: 254: 253: 252: 251: 247: 243: 238: 230: 226: 222: 218: 214: 210: 209: 205: 204: 187: 182: 178: 177: 169: 166: 162: 159: 158: 154: 151:λ, λ/2, λ/3, 150: 146: 142: 138: 137: 133: 128: 127: 124: 122: 118: 110: 108: 107: 103: 99: 95: 87: 80: 76: 74: 71: 69: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 4626: 4619: 4601: 4596: 4589: 4570: 4563: 4551: 4516: 4510: 4450: 4443: 4403: 4401: 4337: 4301: 4297: 4293: 4289: 4260: 4253: 4247: 4215: 4208: 4198: 4194: 4136: 4064: 4005: 3984: 3977: 3970: 3951: 3947: 3943: 3912: 3904: 3887: 3883: 3863: 3861: 3829: 3825: 3718: 3709: 3703: 3699: 3695: 3631: 3627: 3623:wavelength". 3564: 3559: 3507: 3502: 3498: 3495: 3469: 3465: 3463: 3457: 3453: 3449: 3415: 3406: 3401: 3400: 3396: 3394: 3391: 3385: 3381: 3366: 3343: 3328: 3306: 3253: 3184: 3180: 3170: 3164: 3160: 3156: 3150: 3146: 3142: 3138: 3133: 3129: 3123: 3119: 3114: 3110: 3104: 3100: 3096: 3091: 3087: 3083: 3081: 3066: 3062: 3058: 3053: 3046:cnoidal wave 3038: 3010: 2978: 2974: 2970: 2966: 2962: 2958: 2933: 2912: 2872: 2838: 2715: 2674: 2655: 2644: 2639: 2629: 2624: 2618: 2611: 2603: 2596: 2588: 2577: 2566: 2559: 2553: 2545: 2538: 2530: 2379: 2372: 2264: 2257: 2182:mathematical 2181: 2177: 2131: 2121: 2023: 2016: 2010: 2006: 1977: 1973: 1969: 1965: 1961: 1957: 1890: 1886: 1777: 1774: 1725: 1707: 1702: 1695: 1683: 1669: 1628: 1624: 1621: 1619: 1616: 1614: 1578: 1561: 1558: 1556: 1553: 1551: 1531: 1530: 1528: 1522: 1521: 1519: 1514: 1508: 1428: 1386: 1383:to point out 1382: 1299: 1295: 1289: 1288: 1284: 1241: 1214: 1149: 1122: 1117: 1113: 1070: 1047: 1040: 1034: 1030: 1026: 936: 857: 854:fairly silly 853: 849: 845: 841: 826: 822:Waleswatcher 820: 815:Weak Support 814: 813: 786: 782: 778: 774: 770: 766: 759: 752: 747: 740: 736: 732: 686: 664: 660: 635: 631: 627: 623: 619: 596: 592: 569: 565: 561: 553:Hi Srleffler 552: 501: 464: 291: 287: 283: 279: 271: 267: 263: 259: 234: 175: 168: 160: 152: 148: 144: 140: 114: 91: 78: 43: 37: 4509:This is an 4482:Brews ohare 4427:Brews ohare 4383:Brews ohare 4357:Brews ohare 4193:Brews they 4156:Brews ohare 4117:Brews ohare 4075:Brews ohare 4021:Brews ohare 3915:Brews ohare 3869:Brews ohare 3833:Brews ohare 3774:Brews ohare 3640:Brews ohare 3604:Brews ohare 3542:Brews ohare 3512:Brews ohare 3474:Brews ohare 3430:Brews ohare 3350:Brews ohare 3332:Brews ohare 3317:Brews ohare 3237:Brews ohare 3215:Brews ohare 3042:ocean waves 3018:wave vector 2988:Brews ohare 2865:wavelength. 2849:distractor. 2816:Brews ohare 2784:Brews ohare 2747:Brews ohare 2733:Brews ohare 2708:discussion. 2658:Brews ohare 2457:Brews ohare 2287:Brews ohare 2134:superimpose 2082:Brews ohare 1899:Brews ohare 1842:Brews ohare 1794:Brews ohare 1753:Brews ohare 1638:Brews ohare 1595:Brews ohare 1537:Brews ohare 1483:Brews ohare 1391:Brews ohare 1304:Brews ohare 1202:Brews ohare 1154:Brews ohare 1127:Brews ohare 858:petty issue 799:Brews ohare 640:Brews ohare 578:Brews ohare 507:Brews ohare 488:Brews ohare 442:Brews ohare 407:Brews ohare 369:Brews ohare 328:Brews ohare 300:Brews ohare 217:Brews ohare 98:Brews ohare 36:This is an 4636:0470185902 4610:364209032X 4350:Wavelength 4002:Wavelength 3762:you insist 3728:1118165519 3583:envelopes. 3452:, it is a 3344:I've also 3209:agreement. 3173:dispersive 3014:wavenumber 2632:Gockenbach 2625:mentioning 2587:... where 1933:wave phase 1786:wavelength 1778:wavelength 1716:0821847902 1670:wavelength 1515:wavelength 1296:wavelength 1118:wavelength 846:mentioning 789:waveforms. 767:wavelength 760:wavelength 741:vice versa 737:wavelength 574:Wavelength 570:vice versa 566:vegetables 557:Wavelength 502:wavelength 272:wavelength 268:wavelength 264:wavelength 185:0070277303 161:References 4552:Archive 5 4546:Archive 4 4541:Archive 3 4533:Archive 1 4113:envelopes 3956:Srleffler 3930:Srleffler 3892:Srleffler 3884:important 3849:Srleffler 3570:Srleffler 3565:generally 2959:transform 2913:O f = λ f 1968:and what 1929:Frequency 1771:The point 753:determine 748:determine 692:Srleffler 605:Srleffler 525:Srleffler 469:IRWolfie- 387:IRWolfie- 79:Archive 5 73:Archive 4 68:Archive 3 60:Archive 1 4326:eyebrow. 4203:WP:STICK 4140:Dicklyon 4095:Dicklyon 4043:Dicklyon 4008:aspects? 3864:periodic 3847:case. -- 3805:Dicklyon 3663:Dicklyon 3527:Dicklyon 3420:back to 3346:restored 3289:Dicklyon 3275:Dicklyon 3196:Dicklyon 3141:at time 3127:at time 2874:Dicklyon 2769:Dicklyon 2677:Dicklyon 2606:Beerends 2556:Rogalski 2541:Rayleigh 2491:Dicklyon 2439:Dicklyon 2178:physical 2138:waveform 2076:All the 1978:see also 1958:assuming 1579:DickLyon 1566:Dicklyon 1468:Dicklyon 1387:heads up 1264:Dicklyon 1169:Dicklyon 1150:DickLyon 1101:Dicklyon 787:periodic 714:Dicklyon 710:this one 385:and now 383:Dicklyon 348:Dicklyon 314:Dicklyon 242:Dicklyon 231:Comments 4512:archive 4006:spatial 3689:removed 3466:spatial 3450:repeats 3109:, with 2621:Osborne 2176:from a 2065:merits. 1970:its for 1505:Summary 1123:no cost 1071:Comment 842:Oppose: 687:replace 600:having. 562:pumpkin 39:archive 3948:series 3710:Source 3503:f=f(λ) 3470:always 3402:series 3388:, ..." 3181:period 3145:, if Δ 2963:series 2894:phonon 2640:merely 2594:=1/λ" 1666:domain 1242:Oppose 1025:where 935:where 775:x−vt+λ 733:employ 628:x−vt+λ 568:, and 465:Oppose 284:x−vt+λ 260:period 4451:E+v×B 4415:deeds 4261:E+v×B 4216:E+v×B 4177:deeds 4065:In a 3593:here. 3501:with 3263:deeds 3149:and Δ 2897:BaTiO 2805:deeds 2647:Smith 2571:= 2 f 2380:E+v×B 2265:E+v×B 2069:like. 2024:E+v×B 1877:deeds 1831:deeds 1684:Items 1434:Bob K 1429:start 1345:Bob K 1285:Bob K 1250:Bob K 1048:E+v×B 431:deeds 397:deeds 358:deeds 139:.. 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