Talk:Sine wave

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One of the 500 most frequently viewed mathematics articles.

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.

Merged material[edit]

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

What is the in [edit]

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

Great article! <a href="">Hauberk</a>

What are the sources for these assertions?[edit]

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)

Wave equation and Helmholtz equation[edit]

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)

Beat (acoustics)[edit]

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)

initial phase[edit]

In the first section, it says: initial phase (t=0) = , referencing the general form of the sinusoid which was stated: . shouldn't it say: initial phase (t=0) = (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   ?"   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:   (ignoring another annoying detail that one article uses sine and the other uses cosine).  So:  
--Bob K 19:53, 29 September 2006 (UTC)

length of a sine[edit]

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)

 ? (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: , 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 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)

Parsing Errors Report[edit]

The math formulas yield the following errors on my browser (Firefox version 3.04; Microsoft Windows XP Version 5.1.2600).

"Failed to parse (Cannot write to or create math output directory): y (t) = A \cdot \sin(\omega t + \theta)"

"Failed to parse (Cannot write to or create math output directory): \omega,"

"Failed to parse (Cannot write to or create math output directory): y(t) = A\cdot \sin(kx - \omega t+ \theta ) + D.\,"

"Failed to parse (Cannot write to or create math output directory): k = { \omega \over c } = { 2 \pi f \over c } = { 2 \pi \over \lambda }"

"Failed to parse (Cannot write to or create math output directory): \cos(x) = \sin(x + \pi/2),"

Sorry I can't be of any more help than mechanically reporting; I don't have enough background to understand these errors. -- (talk) 16:13, 6 December 2008 (UTC)

I tried Firefox and saw the same thing. Then to my surprise, I went back to IE and saw it again, but only on that page (Sine wave). While there I noticed and removed some whitespace within the formulas. Then both browsers displayed the formulas without errors. I think it was just coincidence, because when I now go back and look at old versions, I no longer see the problem. Something else was changed somewhere while I was doing my edits.
--Bob K (talk) 17:54, 6 December 2008 (UTC)
I think the parsing happens when the edit is submitted, and if you get such an error it will often fix itself if you do anohter edit, changing nothing. Dicklyon (talk) 19:42, 25 December 2008 (UTC)

Sampling Sine Wave[edit]

Hello! I hope that the paper below will help to evaluate the sampling process of a sine wave:

ET 4 CO 198.pmd

Best regards Petre Petrov —Preceding unsigned comment added by (talk) 18:57, 25 December 2008 (UTC)

General form?[edit]


I would like to clarify the topic about the “general form”:

  • This is not the general form of the sine wave. (The general form has a DC component equal to zero).
  • This represents SUM OF TWO FUNCTIONS:

1. Linear function parallel to the axis of the time (It (DC or vertical displacement) does not depend on the time t);

2. Sine function which depends on time t. (The "phase" is displacement into time).


The given “general form” is the equation of the simplest band limited signal (SBLS) and that should be mentioned in the front page.

I hope that is useful.


Petre Petrov

References:, page 18, Table 1., page 55, Figure 2. —Preceding unsigned comment added by (talk) 12:27, 26 December 2008 (UTC)

No, it's not helpful. You keep spamming talk pages with references to your own nonsense papers. Whether one includes the DC term in the general form is a matter of convention, of how general you want to be. See the first entry in your Table 1, page 18. The is no reliable source for your new notion of a "simplest band limited signal", so, no, we shouldn't mention that. Dicklyon (talk) 19:06, 26 December 2008 (UTC)

PP: How you will make a difference between a "sinusoidal signal" and the "simplest band limited signal" or "the simplest signal with two lines into its spectrum?

Is there a difference or not? —Preceding unsigned comment added by PetrePetrov (talkcontribs) 13:06, 27 December 2008 (UTC)

Sure, there's a difference between a signal with one frequency and one with two. So what is the relevant point here? Dicklyon (talk) 00:26, 28 December 2008 (UTC)

PP: The points is that may be there should be clear definitions of what are:

a SS,

a CS.

a SBLS with SS component,

a SBLS with CS component

and not to make a mixture of them on the from page. (talk) 03:56, 28 December 2008 (UTC)

It's not appropriate for wikipedia to use your new terminology or taxonomy. If there's a reliable source for this distinction, cite it here. Dicklyon (talk) 05:10, 28 December 2008 (UTC)

PP: The definitions of the circle, triangle, SS, CS, XY coordinate systems and the evolution in time are quite old. You could find them even in school books. Nothing new. OK?

Is DC offset a convention? Revisiting one of the above points, can we really call it DC offset in the general form? That's certainly a common way to refer to it, especially among electrical engineers, but I think to be general it should just be called the "offset". Mattski (talk) 06:52, 15 April 2009 (UTC)

“DC offset” is a bad and unclear replacement or the “horizontal function” in the formula above written by me.

I apologize for the inaccurate language.

May be it is better to say “constant signal”, “constant voltage signal”, “constant current signal” according to the case.

In my opinion the SS has two possible equations:

A = Am*sin(w*t+i) + 0


A = Am*sin(w*t+0) + 0

Zeros are important from electrical point of view and should be mention explicitly in the definition.

Also in my opinion the SBLS with SS component has two possible equations:

A = Am*sin(w*t+i) + B


A = Am*sin(w*t+0) + B

Zeros are important from electrical point of view and should be mention explicitly in the definition.

I think that only accurate definitions (widely used or not at the moment) must be included on the front page articles in Wikipedia.

May be I am wrong again?

BR P Petrov (talk) 19:13, 5 August 2009 (UTC)

Why the minus sign?[edit]

We specify the "general form" as:

Why not:

And it should also be explained in the article.

--Bob K (talk) 21:27, 20 February 2009 (UTC)

With this convention, positive wavenumber corresponds to propagation in the positive x direction. That's why. Dicklyon (talk) 05:03, 21 February 2009 (UTC)
nice answer. that is, distance represents a phase delay. It is only a convention, the software I use regards positive x as a backward motion, for example, so +t and +x are aligned, for +w. Greglocock (talk) 10:08, 21 February 2009 (UTC)

Oscillation or wave?[edit]

I dislike using the term 'wave' for

In my terminology (I am a Dane) the above is named an 'oscillation', while the term 'wave' is used for a spatial variation. (talk) 02:38, 14 June 2009 (UTC)

The term "sine wave" often means a waveform as opposed to a travelling wave. Dicklyon (talk) 21:43, 14 June 2009 (UTC)
In my opinion a (sine)waveform denotes a sine wave in space and it has the form:
where x is a spatial coordinate, and k denotes the wave number
oposed to a sine oscilation, which has the form:
However, the first equation of the article reads;
which is named a harmonic oscillation!
A traveling (sine) wave reads:
—Preceding unsigned comment added by (talk) 02:49, 23 June 2009 (UTC)
You can check what others mean by "sine waveform", and then maybe adjust your opinion. Dicklyon (talk) 06:32, 6 August 2009 (UTC)

Who discovered the sine wave and how did it originate from geometry?[edit]

How did someone decide that it would be interesting to perform repeated measures on a leg of a pair of superimposed right triangles where the radius of the circle was a constant hypotenuse?Bmarmie (talk) 16:26, 11 August 2009 (UTC)

Maybe you can read this book and summarize the answer for us. Dicklyon (talk) 21:31, 11 August 2009 (UTC)

Trigonometric function, mechanical motion, physics of fluids, electrical engineering…?[edit]

The fields above need different definition of the “sinus” applied to them.

There are differences in the mathematical description and the properties between:

“Sinus” as trigonometric function.

“Wave” as material object, e.g. in fluids.

“Sinusoidal signal” as electrical signal.

“Mechanical oscillation” as motion.

“Electrical oscillation” as process in electronic circuit, etc.

Each of these cases and even some more should be treated separately.

Also “natural waves”, e.g. in water and air are very different from the “sinusoidal shape”.

I have never seen natural “sine wave” in water (“ocean waves”).

In conclusion the term “wave” in sense of “sinusoidal wave” or “sine wave” seems misleading to me.

Also to say that “cosine wave is said to be sinusoidal” is like to say that the axis Y is the axis X because both of them are lines or axis.

The text about Fourier seems different from historical and engineering facts.

In my opinion the material should be rewritten in more precise way.

I hope that is useful.


Petre Petrov —Preceding unsigned comment added by (talk) 19:04, 12 August 2009 (UTC)

Complex exponential function[edit]

I think that we should mention something about the complex exponential function at the bottom of the article. Akilaa (talk) 06:03, 21 September 2009 (UTC)

Sine sample[edit]

The supposed 220hz sound sample sounds a little too low to be 220hz, or perhaps my software is a little too high for 220 hz. In using NCH's Tone Generator, I played the 220 hz sine frequency at the same time as the sample on the page, and no resonance was present. I tuned mine down to around 215 hz and I could feel the resonance fine, so I think the tone is either a little off, or maybe a lossy codec was used in producing the tone used for the sample 220 hz noise on the page. Just pointing this out; not much of a big deal, but it's still noticeable when comparing the sample to the actual 220hz sine tone. —Preceding unsigned comment added by (talk) 04:34, 3 February 2010 (UTC)

A lossy codec would not shift frequencies consistently. As you should know. Greglocock (talk) 11:00, 3 February 2010 (UTC)

Hey tough guy, I never said I was an expert on the subject. I'm just a kid with a tone generator who knows what resonance is. Maybe that sound clip isn't 220 hz, have you considered that? IOA94 (talk) 03:18, 21 November 2010 (UTC)

I recorded the lossy sound into my computer and analyzed it. The analyzer says 220.2 Hz. I counted samples making up one cycle and there were 200 at 44,100 samples per second. 200 goes into 44,100 220.5 times. It's not a super clean sine wave (there are distortion harmonics present) but it is good enough for this article. Binksternet (talk) 04:55, 21 November 2010 (UTC)

I want to day thank you and I'm lost on what to do[edit]

I want to day thank you and I'm lost on what to do Shannonrice788 (talk) 06:40, 30 March 2016 (UTC)