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More general waveforms wording

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

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

I added the mention of Fourier integral because it was part of what the source that I cited talk about, and to appease Brews a bit, but I don't mind it being gone. As for the period, I'm not sure I understand. How can the period anywhere be other than the period of the source, that is, the least common period of all the sinusoids that are propagating by the location? Dicklyon (talk) 06:10, 29 April 2012 (UTC)

I may have been mistaken about the period.--Srleffler (talk) 06:57, 29 April 2012 (UTC)

It seems to me that the period is related in only a complicated fashion to the period of the source, involving the separation of the observation point from the source and also the dispersion relation. Maybe we need a source to tie this down? Brews ohare (talk) 16:42, 30 April 2012 (UTC)

With the Fourier series decomposition, it's easy to see that the wave contains only harmonics of the source period. No new frequencies are added by propagation, even if there are reflecting ends, dispersion, or whatever. So you have harmonics of the period everywhere, and therefore the same period everywhere, no? Dicklyon (talk) 22:59, 30 April 2012 (UTC)

Yes. I had in mind that as the components got out of phase with one another the superposition would have a period that was longer than one cycle of the fundamental, but I see I was mistaken.--Srleffler (talk) 03:17, 1 May 2012 (UTC)

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

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

Yes the period might be very long in the two-beating-frequencies case. But the source associates "wavelength" with the components, not with a long pattern. There's actually no "driver" or "source" that's relevant here, just periodic-in-time wave motion, which can be analyzed into harmonic frequency components. I don't see any speculation, but then again I don't see a source that says precisely what our text says, obvious though it is. Dicklyon (talk) 00:48, 2 May 2012 (UTC)

There is no "presumption", only definition. The paragraph we are discussing is specifically about the important special case of waves that are periodic in time. There is no need to presume or speculate; periodicity in time is the specified initial condition. The only question is how the system evolves over time, and how it behaves at other spatial locations.

Since the topic of discussion is waves that are periodic in time, the relevant period in the two-beating-waves case is the long period required for the full waveform to repeat. The Fourier series in this case is particularly simple, and the fact that the period of the wave as a whole is much longer than those of the nonzero Fourier components is not a problem. In my initial comments on this topic, I had presumed that there might be locations where the period of the combined wave might be short (like the periods of the components), but I was mistaken. The component waves maintain their frequencies as they propagate and there will be nowhere along their common path where they do not beat against one another, producing a waveform with the same, long, period.--Srleffler (talk) 03:19, 2 May 2012 (UTC)

Actually, it might be slightly more complicated than that in wave media with reflections that can make nulls for certain frequencies at certain locations. If one of the two components in the two-component beating pattern has a null, then the response at the location of the null will be just the other component, so it will have a short period there. However, what the article says is still true there: the wave is still periodic with the longer period T, even if also with some shorter period. Just to be sure, I have edited the text to try to make sure that the period T is stated as the period of the wave and can't be misunderstood as one of these possible shorter periods at a null in a corner case. Dicklyon (talk) 03:39, 2 May 2012 (UTC)

A periodic disturbance in time at a particular location will result in a disturbance at all distances from the source when steady-state is reached. So a disturbance in time repeated with period T but of duration less than T will involve many frequencies, submultiples of T. Nothing much changes as T changes, if the disturbance maintains the same form in time, but is simply spaced with larger "blank" periods in between. T is not a very useful parameter in describing matters, therefore, and it should not be framed as the key to analysis here. Brews ohare (talk) 22:06, 3 May 2012 (UTC)

True, the periodicity of T is just there to make the Fourier series applicable. Dicklyon (talk) 03:50, 4 May 2012 (UTC)

More general waveforms references

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

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

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

I agree the section remains rather unsatisfying. I did my best to find a sensible way to incorporate the Fourier series into something to do with wavelength, cobbling what was there; but it's still a bit of a misfit. Most sources that talk about dispersion and Fourier analysis don't do in the context of periodic waves, and usually do it in terms of wavenumber, not wavelength. And their analysis doesn't usually conclude anything related to wavelength or to repetition in space. Probably we should just simplify the section, since there are better articles for covering these other concepts of waves in linear dispersive media. As for the concept of wavelength being applied to other than approximately sinusoidal waves, it's unusual at best; discussing it can easily be misleading, or spiral into contradictions, like when you get into claims that it's well-defined for arbitrary periodic functions. Dicklyon (talk) 05:58, 3 May 2012 (UTC)

Hi Dick: Quite possibly the easiest approach is to use wavevector, or maybe to use a simple example instead of trying the general case. As you know, however, I do not agree in the slightest that application of wavelength to a periodic wave in space of general form is in any way misleading, although the occurrence of such waves in nature is not general, but restricted to particular media. From a conceptual point of view, wavelength is what Fourier series is about in space, with the simple interpretation of the general argument ξ as x instead of angle, or time. From this stance, as I have pointed out by direct quotations from at least three sources, many authors do exactly that. Brews ohare (talk) 21:09, 3 May 2012 (UTC)

Not that many authors do that. And none of them seem to reveal any reason for doing a sinusoidal decomposition of the spatial pattern. At least in the case of the dispersive linear system there's a reason to decompose into sinusoids. Dicklyon (talk) 03:52, 4 May 2012 (UTC)

Hi Dick: Perhaps it is just argumentative, but here's a question: why do authors use Fourier series when argument ξ is interpreted as time, or angle, or whatever? Why is it automatically a wasted effort only when ξ is interpreted as a spatial variable? Why is period T more significant as a time period than λ is a a spatial period? Could it be that in fact there is no difference at all? Brews ohare (talk) 20:40, 5 May 2012 (UTC)

It's not unusual to use wavenumber (or reciprocal wavelength) in a Fourier transform, as a way to get a sinusoidal decomposition, especially for wave packets. But periodic-in-space waves are a relatively rare corner, seldom encountered where a sinusoidal decomposition would be helpful. If they're also periodic in time at the same time, in a linear system, the system is nondispersive, and has a trivial wave equation, for which a sinusoidal decomposition is not needed; it adds nothing to the understanding of the system. If they're in a nonlinear system, the sinusoidal decomposition is not particularly illuminating either. Dicklyon (talk) 22:27, 5 May 2012 (UTC)

Here's another question: when an oboe plays a note, it sounds different than when a violin plays the same note. Could it be that the difference can be expressed as a difference in the Fourier series expressing the wavelengths of vibration supported by the general waveform in the oboe's air column for that note compared to the wavelengths present in the general waveform on the violin string when the same note is present? Would that be an interesting enough example of Fourier expansion of the general waveform in space to warrant mentioning the use of Fourier series for spatial analysis of waveforms? Or, perhaps, an article Wavelength (music) is needed? Brews ohare (talk) 20:52, 5 May 2012 (UTC)

The sound difference has much more to do with the waveform in air; this is what propagates and carries the pattern. The physics within the instrument gives rise to different modes, and to a temporal near-periodicity from how the signal interacts with the reed or the bow, but I haven't seen an analysis like you're describing, where the composite signal is analyzed into space-domain sinusoids. Of course it could be done. Dicklyon (talk) 22:27, 5 May 2012 (UTC)

Once it's left the instrument the waveform is not uniform, i.e. there is no 'general waveform', even allowing for distance attenuation. The sound you hear an inch from a source is very different from the sound you hear three feet or thirty feet from it: the best example is the human voice as we're most familiar with it: often you can tell how far away someone is quite accurately by the sound of their voice. This is less noticeable with an instrument because its pure note dominates and so it depends on the particular frequency of the instrument. And of course in most cases performers don't want you to hear different sounds depending on how far away you are, and will go to great lengths to minimise effects of distance (modifying the building to compensate for example).--JohnBlackburne_deeds 22:57, 5 May 2012 (UTC)

To pursue this matter further, harmonics are produced on the guitar by deliberately introducing a node on the guitar string, a direct application of wavelength considerations. Brews ohare (talk) 21:05, 5 May 2012 (UTC)

That doesn't produce harmonics, but kills the fundamental and certain other harmonics. Yes, it is described in terms of wavelength on the string, or the modes of the instrument. But not in terms of a sinusoidal decomposition of a periodic-in-space pattern. Dicklyon (talk) 22:27, 5 May 2012 (UTC)

Dick & Blackburne: You might find some interest in reading a source or two on this subject instead of relying on your recollections. Something interesting can be done here. Brews ohare (talk) 20:51, 10 May 2012 (UTC)

I have plenty of good books on the physics and psychophysics of music and musical instruments; it's not clear what you see as relevant in the page you've linked in that college physics text. Dicklyon (talk) 21:24, 10 May 2012 (UTC)

Dick: Figures 14.26 and 14.27 of this source compare waveforms for a pure note on a tuning fork with the same note on a clarinet and a flute. The point, of course, is that the characteristic voice of the instrument is expressed in its peculiar waveform, which is in each case periodic with the same wavelength but of different shape. Accordingly, the differences between voices is sought in the different harmonics of the fundamental found in each. This difference can be expressed in time or in space, although the latter requires some expression of the characteristics of the medium, which cannot be unduly dispersive. Obviously, instruments usually operate in air, and the slight dispersion of sound in air is no impediment to applying a spatial analysis. The design of an instrument is perhaps even more clearly related to wavelength, as the dimensions of the instrument determine how an excitation of a particular spatial mode will be related to its various harmonics; for example, how the standing wave on a violin string is connected to the various resonances of the instrument. Brews ohare (talk) 13:04, 15 May 2012 (UTC)

Removal of Fourier series in time section

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

This removal was a good step: it tried to introduce Fourier series using the topic of Fourier series in time, applied to waveforms that have no identifiable wavelength in space. The door is now open to introduce Fourier series in a context appropriate to the subject of wavelength, that is, the context of spatially periodic general waveforms which, of course, always have an identifiable wavelength. That such waveforms can be and are analyzed using Fourier series is well documented, and the objection that such waveforms are not necessarily found in general media restricts its applicability in general, but doesn't mean it deserves no mention here. Brews ohare (talk) 12:34, 15 May 2012 (UTC)

I'd suggest a reconsideration of the text below:

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

Brews, drop it. You've proposed this once, you've had your RfC on this, neither time did you convince other editors. Proposing yet another variation on it after failing to convince other editors multiple times is simply disruptive.--JohnBlackburne_deeds 16:18, 15 May 2012 (UTC)

I agree. This is a dead issue. Brews, drop the stick and move away from the horse.--Srleffler (talk) 17:03, 15 May 2012 (UTC)

Srleffler: As Blackburne has not advanced any actual argument against inclusion of this text, your "agreement" with him is only as to his cheer-leading and not about any "agreement" upon substance. Your own comments regarding the RfC on the above text were as follows:

"The introduction of Fourier series appears neither to make the concept of wavelength clearer nor to provide a better fundamental definition of wavelength of a general periodic wave. The current proposed text is admittedly better than the many attempts prior to this RFC, in that it doesn't belabour the issue and focuses most directly on the connection between the two topics.

I do see the appeal in trying to replace the definition of wavelength of a non-sinusoidal wave in terms of the wave's period of repetition with a definition that is tied directly to sinusoidal waves. I'm partial to this for the same reason that I was originally opposed to applying the term "wavelength" to non-sinuoidal waves at all. It's not clear to me that there is a non-negligible set of readers for whom this treatment would be beneficial, however."--Srleffler (talk) 18:13, 22 April 2012 (UTC)

As I understand these points you raise, your objection to including this text is that it does not clarify the concept of wavelength. However, that is not the purpose of this text. What this text aims to do is to alert readers that there is a connection of spatially periodic waveforms (waveforms with a wavelength) to Fourier series. That connection is undeniable.

Your further objection is that nobody cares anyway. Inasmuch as several sources mention this connection, and indeed elaborate upon it at length, your opinion is not universal. Brews ohare (talk) 19:16, 15 May 2012 (UTC)

You have raised this issue before, multiple times. We have spent far more time discussing it than it was worth. No further discussion of this issue is merited. Please stop trying to disrupt the editing process by repeatedly bringing forward the same issues over and over again with only slight variations. --Srleffler (talk) 03:45, 16 May 2012 (UTC)

I think, Srleffler, that your objections are in fact against an earlier proposal to introduce Fourier series as a definition of wavelength, which is not proposed here. You may have a different opinion about the present proposal. Brews ohare (talk) 19:31, 15 May 2012 (UTC)

The truth is that few sources make that connection, and they don't take it anywhere useful. You have gone back to a formulation that would be just as true and useful if triangle waves were used instead of sine waves; that is, not useful at all, since the sinusoidal components provide no help in analyzing such a situation, where the medium is either nondispersive or nonlinear. The Schaum's Outline book that you cite introduces the Fourier series there only as a step toward getting a Fourier transform, to get a way to represent waves that are NOT period in space, which is useful; and it says it's more common to do it in terms of k than lambda, which is true, so it's not very related to wavelength. And your statement that "The wavelength, say λ, of a general spatially periodic waveform is the spatial interval in which one cycle of the function repeats itself" is contrary to typical usage of the term "wavelength" (that is, for the local wavelength of approximately sinusoidal waves) and is not supported by the source; in fact, your source defines the term "wavelength" only with respect to sinusoidal components, and applies it only fleetingly to a spatially periodic function. The text (if you can call it that) is also flaky in that when it introduces sinusoids in section 1.3 it completely misses their point, again saying something that would be just as true with triangle waves or square waves or a variety of other basis sets. We have been through all this many times. The objections of numerous editors are in the record if you'd like to review them further. Dicklyon (talk) 23:29, 15 May 2012 (UTC)

Dick: You are missing the point here. There is no attempt to propose that Fourier series is the one and only way to expand an arbitrary function in terms of other functions, which might fall under the rubric of generalized Fourier series. The point here is simply to make the connection of a spatially periodic function of general form that satisfies f(x+λ)=f(x) to the Fourier series. A Fourier series, as you must know, inevitably results in a periodic function throughout space. Fourier series is, moreover, a very well known and important aspect of mathematical analysis, and a link to make readers aware of the connection is just an ordinary use of an aside that widens the reader's appreciation of the topic wavelength and its connection to the mathematical analysis of periodicity. The text contains a direct quote from a textbook, and virtually the same language occurs in other sources as well: "Fourier's theorem states that any periodic function f(x) can be expressed as the sum of a series of sinusoidal functions which have wavelengths that are integral fractions of the wavelength λ of f(x)"

There is nothing misleading or inappropriate here, as you well know, and your unsupported assertions to the contrary do not reflect well upon your understanding of the subject, nor indeed, upon your appreciation of one of the major benefits of WP: helping readers widen their awareness of a topic. Brews ohare (talk) 14:11, 16 May 2012 (UTC)

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