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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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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).[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]