Talk:Sine wave

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This article was originally based on material from FOLDOC, used with permission. Update as needed.

It contains material merged from sine curve; see the history of that article for details.


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[edit] What is the x in kx

In the first formula, k (wave number) is multiplied by an undefined quantity. --Bob K 20:57, 18 February 2006 (UTC)

Great article! <a href="http://en.wikipedia.org/wiki/Hauberk">Hauberk</a>

[edit] What are the sources for these assertions?

The human ear can recognize single sine waves because sounds with such a waveform sound "clean" or "clear" to humans; some sounds that approximate a pure sine wave are whistling, a crystal glass set to vibrate by running a wet finger around its rim, and the sound made by a tuning fork. To the human ear, a sound that is made up of more than one sine wave will either sound "noisy" or will have detectable harmonics.

The sound of a violin is full of harmonics and they don't sound noisy (if played well) and I don't hear the harmonics but I know they are there. Vibrating guitar strings have lots of harmonics. Why wouldn't a vibrating tuning fork have them?-Crunchy Numbers 20:30, 25 August 2006 (UTC)

If you can tell the note is coming from a violin, then you are hearing the harmonics. They give an instrument its unique character. You might have a good point about the tuning fork, and I look forward to Omegatron's answer. I can offer you my own source for that same belief... my high school teachers. But back in those days they also taught nonsense like:
  • One should not drink water during a workout.
  • Our solar system has 9 planets.
--Bob K 21:37, 25 August 2006 (UTC)

Of course I am hearing the sum of the harmonics and of course the harmonics give instruments distinctive sounds; but that wasn't the point.

The human ear can recognize single sine waves because sounds with such a waveform sound "clean" or "clear" to humans.

This article talks about discernable harmonics as if someone could pick them out individually. I listened to the example ogg file with the sine wave and as always a sine wave sounds really bad and harsh to me. Whistling doesn't sound the same to me as wine glass tones and I don't believe either is a pure harmonic.-Crunchy Numbers 03:08, 26 August 2006 (UTC)

I think you've got a good point, regardless of whether or not a tuning fork has significant harmonics. I wouldn't object if someone rewrites that section. What would be really cool is to add a couple of more audible links: one with a mixture of 1000 Hz and 2000 Hz, and another with a mixture of 1000 Hz and (say) 1600 Hz. As I recall, in the latter case you also hear the beat frequency (600 Hz). Come to think of it, beat frequencies are usually associated with the mixing (multiplying) process. Tuning forks are presumably additive. So how does that happen, if in fact it does? --Bob K 14:18, 1 September 2006 (UTC)
I guess it was the first sentence about the human ear that annoyed me. I found webpages with waveform pictures and frequency spectrums of tuning forks and flutes. Several webpages mention that human pucker whistling is very, very close to pure. The tuning fork seems to have small overtones at 3 and 6 times the principal but looks like a sine wave to my eye on the chart. If you wait a few seconds after striking it the overtones die out for the most part. The flute didn't look so much like a sine wave. One webpage mentioned that a small percentage of the population are sensitive to pure sinewave sounds and can't stand hearing them.
As much as I like Wikipedia I find myself trusting less and less of what I see, even things that turn out to be true. I wonder if this is a named phenomenom?-Crunchy Numbers 05:04, 1 September 2006 (UTC)
"Healthy skepticism" perhaps? Yes, it is risky to trust Wikipedia. Fertile ground grows both beans and weeds. Pretty cool that you found waveform pictures of tuning forks on the internet. It never ceases to amaze me how people choose to spend their time. --Bob K 05:51, 1 September 2006 (UTC)
Consider adding the webpages you found to the See Also section. --Bob K 14:18, 1 September 2006 (UTC)

See some of the arguments regarding perception at just intonation. Hyacinth 23:16, 25 August 2006 (UTC) I didn't see the connection to harmonics in a single timbre. It is an interesting article though.-Crunchy Numbers 03:08, 26 August 2006 (UTC)

[edit] Wave equation and Helmholtz equation

Why are these sections here? Neither explains why it is included in this article. Also they both have their own articles.-Crunchy Numbers 01:35, 2 September 2006 (UTC)

Why do you explicitly state that D has to be nonzero? D being zero just means that you picked the right spot to start measuring your sine wave, and it falls off as soon as you start taking derivatives (velocity, acceleration, jerk).

I assume this is the statement you refer to:
In general, the function may also have:
  • a spatial dimension, x (aka position), with frequency k (also called wave number)
  • a non-zero center amplitude, D (also called DC offset)
If D is zero, then obviously one can include it in the "generalized" form, because it has no effect. The writer's point is that we can also include offsets that are not zero. I believe it is correct as written, but that doesn't mean it can't be improved.
--Bob K 13:41, 20 July 2007 (UTC)

[edit] Beat (acoustics)

Since Bob K mentioned beat frequencies I looked up this article. It is pretty interesting. Bob, if you don't add a link to this article I might.-Crunchy Numbers 01:48, 2 September 2006 (UTC)

I love the audio links you are finding. There is nothing like hearing it for yourself. In the jargon I am familiar with, the beat frequency is the walk-thru rate of two similar frequencies. In the example I gave (1000 and 1600), the rate would probably be too fast to hear a 600 Hz modulation. I.e., the states of constructive and destructive interference would be too brief. I'd like to hear it anyhow. I wonder how those links are made.
Please add the link yourself, if you feel like it. At the moment I am a little weary of doing things and subsequently watching them get undone by others.
--Bob K 02:56, 2 September 2006 (UTC)

[edit] initial phase

In the first section, it says: initial phase (t=0) = -\varphi, referencing the general form of the sinusoid which was stated: y = A\cdot \sin(\omega t - \varphi). shouldn't it say: initial phase (t=0) = \varphi (the same without the minus sign)?Rgrizza 14:51, 29 September 2006 (UTC)

I think a better question is:  "Why isn't the general form chosen as   A\cdot \sin(\omega t + \varphi)?"   IMO, it should be.
Anyhow, the initial phase is the value of the instantaneous phase at t=0.
The instantaneous phase in this case is:   \phi(t) = \omega t - \varphi\, (ignoring another annoying detail that one article uses sine and the other uses cosine).  So:   \phi(0) = - \varphi\,
--Bob K 19:53, 29 September 2006 (UTC)

[edit] length of a sine

I tried to figure out the length of one period of a sine curve, thinking that it would be a fairly simple task, or at least a simple and elegant answer. Turns out it is not analytically solvable. I was quite surprised. Anybody think that is interesting enough to mention in the article? maxsch 22:28, 18 October 2007 (UTC)

2π / ω ? (Or is this a trick question?)
--Bob K 02:10, 19 October 2007 (UTC)
I think he means the arc length, not the length of the period, but the length of one period's worth of sine curve. And yeah, I would expect that the answer for that would not have a closed form. - Rainwarrior 02:27, 19 October 2007 (UTC)
Of course it has a closed form; well, probably it does, to one who is good at integration. For each dx the arc length is sqrt(dx^2 + dy^2), where dy is the cosine of x times dx; is that impossible to integrate over a period? Or just hard to see the answer? The average factor by which dy exceeds dx is going to be the average of sqrt(1+cos^2) over a period, which is probably something not hard to work out. Oops, I take it back; Wolfram can integrate it here, but the indefinite integral involves an AppellF1 function. The definite integral over one cycle might be simpler. Here's a page on the AppellF1. Dicklyon 04:18, 19 October 2007 (UTC)
Well starting with the arc length integral: \int_{0}^{2\pi} \Big( \sqrt{1+\cos^2{x}} \Big) dx, that Wolfram integrator will say that it's an elliptic integral of the second kind. I tried constructing a Taylor series for this by hand, but it got cumbersome very quickly. After this I looked around, and there was a numerical approximate at Ask Dr. Math of 7.640395578. For the 2π case you can probably express the solution concisely with summation notation, but I don't know if a closed form could be wrestled out of it. - Rainwarrior 06:02, 19 October 2007 (UTC)