# Why is the sine wave considered the most "fundamental" wave? **URL:** \ **Category:** Factual Questions\ **Created:** [October 21, 2005, 10:55pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350 "2005-10-21T22:55:57Z")\ **Posts on this page:** 13\ **Page:** 1
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [October 21, 2005, 10:55pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/1 "2005-10-21T22:55:57Z")
As an engineer, I understand the Fourier Series and all that stuff. And amongst my peers, it’s “common knowledge” that a sine wave is the most “pure” or “fundamental” wave. Musical instruments are a perfect example; the sound of each instrument is nothing more than the fundamental frequency + harmonics of various amplitudes and phases. Each component is a sine wave. But I have to wonder… can _ **any** _ wave be chosen as the “fundamental” wave? Could you base all Fourier Analysis on any arbitrary wave shape, or is there something inherently special or natural about the sine wave? ---
**Author:** ![Patty\_O\_Furniture](https://avatars.discourse-cdn.com/v4/letter/p/96bed5/32.png) [@Patty\_O\_Furniture](https://boards.straightdope.com/u/Patty_O_Furniture)\ **Post date:** [October 21, 2005, 10:59pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/2 "2005-10-21T22:59:17Z")
It’s been years since I studied this, but isn’t the fundamental supposed to be the one with the greatest amplitude and the harmonics are of lesser and lesser amplitude as they get higher in pitch? ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [October 21, 2005, 11:05pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/3 "2005-10-21T23:05:54Z")
> [@Patty O'Furniture](#): > > It’s been years since I studied this, but isn’t the fundamental supposed to be the one with the greatest amplitude and the harmonics are of lesser and lesser amplitude as they get higher in pitch? Amplitude of _what_? A _sine_ wave? If you say “yes,” then aren’t you implying that a sine wave must be the “fundamental” wave? My point is this: we naturally think of complex waveforms as being made of pure _sine_ waves. Why is that? Couldn’t I invent my own waveform (something that kind of looks like a sine wave, but isn’t really a sine wave), and use **it** as a “fundamental” wave? ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [October 21, 2005, 11:20pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/4 "2005-10-21T23:20:22Z")
> [@](#): > > But I have to wonder… can any wave be chosen as the “fundamental” wave? Could you base all Fourier Analysis on any arbitrary wave shape, or is there something inherently special or natural about the sine wave? Didn’t you do this in your studies? I remember that pretty much any periodic signal can be used in much the same wave as a sine wave for this sort of analysis. They are useful in signal processing because they move through linear filters undistorted. A triangle wave going through a generic linear filter is no longer a triangle wave. ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [October 21, 2005, 11:21pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/5 "2005-10-21T23:21:46Z")
Yes, you may expand periodic functions in terms of other sets of functions, but not just any old set you dream up. You want a complete orthogonal set. Look up the “generalized” Fourier series. Rather than try to describe it, I’ll just refer you to this article: [http://mathworld.wolfram.com/GeneralizedFourierSeries.html](http://mathworld.wolfram.com/GeneralizedFourierSeries.html) It mentions the alternate example of the Laplace series, and one link away, you will find it mentions the Fourier-Legendre series (using Legendre polynomials) and the Fourier-Bessel series (using Bessel functions). ---
**Author:** ![Kimstu](https://avatars.discourse-cdn.com/v4/letter/k/ecd19e/32.png) [@Kimstu](https://boards.straightdope.com/u/Kimstu)\ **Post date:** [October 21, 2005, 11:58pm UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/6 "2005-10-21T23:58:29Z")
Also, the sine wave is what’s naturally produced by simple harmonic motion, which is pretty fundamental in physical terms. ---
**Author:** ![Polerius](https://avatars.discourse-cdn.com/v4/letter/p/d78d45/32.png) [@Polerius](https://boards.straightdope.com/u/Polerius)\ **Post date:** [October 22, 2005, 12:52am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/7 "2005-10-22T00:52:53Z")
> [@gazpacho](#): > > They are useful in signal processing because they move through linear filters undistorted. Exactly, they are the eigenfunctions of any linear system, and so useful in signal processing as well as communications, among other fields. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 22, 2005, 1:40am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/8 "2005-10-22T01:40:46Z")
Don’t forget that sines and cosines are very easy to calculate, especially compared to the special functions already mentioned. ---
**Author:** ![GargoyleWB](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gargoylewb/32/199_2.png) [@GargoyleWB](https://boards.straightdope.com/u/GargoyleWB)\ **Post date:** [October 22, 2005, 2:01am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/9 "2005-10-22T02:01:59Z")
Also, a sine or cosine wave is simply a relationship of the angle and radius of a point on a unit circle. If the angle changes at a constant rate, this is what we define as a sine wave. These two qualities, constant angular velocity, and constant radius, are common to many basic harmonic functions, and the math is very simple. Thus, sine is a natural choice to use as a fundamental waveform. You could base a system of math or wave analysis on some other non-trivial waveform, but you’d only be needlessly complicating things. Similarly, you could write financial software that computes everything in base 7 instead of base 10 arithmetic, but it doesnt mean you should. 😉 ---
**Author:** ![TJdude825](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@TJdude825](https://boards.straightdope.com/u/TJdude825)\ **Post date:** [October 22, 2005, 2:03am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/10 "2005-10-22T02:03:38Z")
> [@Kimstu](#): > > Also, the sine wave is what’s naturally produced by simple harmonic motion, which is pretty fundamental in physical terms. Yes. Basically, if an object is moving in a circle with constant speed, its x-coordinate and its y-coordinate can both be described by a sine wave, or a variation on it (such as y=3_sin(t) or y = 0.75_sin(t-pi/2)) ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [October 22, 2005, 2:09am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/11 "2005-10-22T02:09:32Z")
> [@ultrafilter](#): > > Don’t forget that sines and cosines are very easy to calculate, especially compared to the special functions already mentioned. But many other functions are _faster_ to calculate. Computers don’t care about the weirdness of the formula, just how long does it take to get a certain number of bits. My long time favs are [Chebyshev Polynomials](http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html) which do a much better job than Sine/Cosine in interpolation for a given computational effort. (And periodicity is overrated, IMHO.) ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 22, 2005, 2:11am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/12 "2005-10-22T02:11:49Z")
> [@ftg](#): > > But many other functions are _faster_ to calculate. Computers don’t care about the weirdness of the formula, just how long does it take to get a certain number of bits. Fourier analysis predates computers by quite some time. ---
**Author:** ![dotchan](https://avatars.discourse-cdn.com/v4/letter/d/9d8465/32.png) [@dotchan](https://boards.straightdope.com/u/dotchan)\ **Post date:** [October 22, 2005, 2:44am UTC](https://boards.straightdope.com/t/why-is-the-sine-wave-considered-the-most-fundamental-wave/327350/13 "2005-10-22T02:44:17Z")
There’s also wavelet analysis, where (iirc, it’s been a while since I’ve read up on it) you can define your own base functions (within a few rules). It’s more useful for noisy data like the crash of a cymbal.