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I'm familiar with the sine wave being something that can be used to model many types of oscillation in nature (and the way that multiple sine waves can be seen as sum to produce complex repeating waveshapes, a la Fourier's theorem).

However, I'm struggling to think of any other waveshapes that can be associated with phenomena in nature. Are there any, or does the sinusoid stand alone as the basic 'shape' of most naturally-occurring cyclic phenomena?

(To give another perspective on my question - when it comes to static values, there are various well-known mathematical constants such as π, e, The imaginary unit i, the golden ratio φ - but are there any well-known mathematical or physical cycle shapes, apart from the sinusoid?)

Нет войне
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  • Hardly anything is exactly sinusoidal. Picking nice functions for waveforms tends to be simply wanting an analytic solution, rather than that they are somehow realistic. – jacob1729 May 21 '19 at 21:39
  • @jacob1729 sure - I'm taking that as read. – Нет войне May 21 '19 at 21:52
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    Twisting words a bit: is it that nature always oscillates in sine waves, or is it that the thing you have chosen to define with the word "oscillate" is that which is described by sine waves, thus rejecting all natural motions which are not sine waves? – Cort Ammon May 21 '19 at 22:08
  • @CortAmmon I think the two answers so far are already examples of things that can be 'naturally' seen as non-sinusoidal (though I must admit I had previously been thinking of the pendulum as sinusoidal) – Нет войне May 21 '19 at 22:56
  • Would an impulse count as a "waveshape"? :) – pipe May 22 '19 at 09:45
  • What do you mean "waveform"? A wave is NOT a sine, it is any function satisfying the wave equation (which in turn can be proven to decompose as infinite sum of sine and cosine, in some cases). – gented May 22 '19 at 10:33
  • @gented Here, I meant it in the same sense as 'waveshape', i.e. a repeating pattern. I've edited the question body to remove 'waveform'. (and @ pipe - I didn't mean an impulse, because in general I don't think that implies any repetition?) – Нет войне May 22 '19 at 11:47
  • @gented and - I don't think I said that a wave is (necessarily) a sine - but I would think of sine / sinusoid as an example of a waveshape (or 'waveform', if that word is used in the sense of 'waveshape') – Нет войне May 22 '19 at 11:56
  • @topomorto That is exactly my point, a wave isn't necessarily a repeating pattern. Take for example Maxwell's equations or some solutions of the Einstein's equations approximated to low orders: they represent "waves" in standard terminology and are clearly manifesting in nature. – gented May 22 '19 at 12:39
  • @gented In the question I'm not talking about all waves. I'm talking about waves that do have repeating patterns. Is there any more specific vocabulary that you think I should put in the question to make that clear? Most people who have answered the question so far seem to have understood, but I mostly deal with waveforms in a musical rather than physics context, so I may well need to improve the domain-specific vocabulary. – Нет войне May 22 '19 at 13:11
  • @topomorto "Most people who have answered the question so far seem to have understood" but what is your definition of "repeating patterns"? You want equations whose solutions are not like sine, but that at the same time are periodic, but that at the same time are not a combination of sine and cosine? What exactly do you mean with "repeating pattern"? Moreover all answers seem very different from each other to be honest, so I don't know exactly what question they have in mind. – gented May 22 '19 at 14:41
  • @gented what is your definition of "repeating patterns"? - I'm not interested in defining it too tightly - if that makes the question too broad, so be it, and no problem at all from my POV - I'm very happy with the variety of answers I got! thanks to all who answered. – Нет войне May 22 '19 at 15:10

7 Answers7

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stick-slip friction cycling gives rise to a sawtooth waveform, which is nonsinusoidal- although it can be built up out of a series of sine waves by superposition.

niels nielsen
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  • Thanks! yes, I guess all recurring waveforms can be built up of sines, but this is a great answer as the phenomenon itself isn't really fundamentally like that (unlike, say, a body orbiting around another orbiting body) – Нет войне May 21 '19 at 21:54
  • @topomorto not all recurring waveforms can be built of sines accurately, a Fourier sum will always overshoot at a jump discontinuity (such as in a perfect square wave) and that overshoot never disappears but instead converges toward ~9% error (for standard Fourier series summation) as the number of terms approaches infinity, it's called the Gibbs phenomenon. – zakinster May 22 '19 at 12:08
  • @zakinster True - thinking a bit more deeply about it, I guess in the back of my mind I wasn't imagining any true discontinuities in waveforms relating to observed natural phenomena, but perhaps I should be (a la the Old Faithful answer!) – Нет войне May 22 '19 at 12:27
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However, I'm struggling to think of any other waveforms that can be associated with phenomena in nature.

The motion of an ordinary pendulum of length $L$ is cyclic, but non-sinusoidal (it is only approximately sinusoidal for small angles). The exact non-sinusoidal motion is governed by the non-linear equation: $$ \frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin(\theta) $$

hft
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However, I'm struggling to think of any other waveforms that can be associated with phenomena in nature

This one has been getting some attention recently.

enter image description here

Image credit

From the comments:

That looks like a sinusoidal wave, though? It's just got its amplitude and frequency changing over time.

From the Wikipedia article Sine wave:

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation.

But the inspiral waveform is not periodic - it does not describe a smooth periodic oscillation.

Alfred Centauri
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    That looks like a sinusoidal wave, though? It's just got its amplitude and frequency changing over time. – nick012000 May 22 '19 at 06:04
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    If you magnify up the end of the chirp (black hole coalescence and ring-down), it becomes very non-sinusoidal. Of course, Fourier proved that any periodic function can be decomposed into a sum of sinusoids. – nigel222 May 22 '19 at 10:07
  • @nick012000, the inspiral waveform is not even periodic. How can it "look like a sinusoidal wave"? – Alfred Centauri May 22 '19 at 11:45
  • @nigel222, to be sure, the inspiral waveform is not periodic, and so there is no Fourier series for this waveform. – Alfred Centauri May 22 '19 at 12:23
  • @AlfredCentauri The sinusoidal function has a period of 2 * pi * f, right? In this case, f varies over time t - maybe it’s t^2 or e^t or something, I’m not sure (and I’m not sure if that graph gives you enough information to tell the difference). Similarly it looks like the amplitude varies exponentially over time as well. – nick012000 May 22 '19 at 20:20
  • @nick012000, the inspiral waveform is clearly aperiodic. Think about it: there is just one merger event. A periodic waveform repeats every $T$ seconds where $T$ is the period, i.e., $f(t + nT) = f(t)$. – Alfred Centauri May 22 '19 at 23:52
  • @AlfredCentauri A lot of waveforms run out eventually - the displacement of a spinning wheel is the archetypical periodic event, but it'll stop due to friction eventually. – nick012000 May 23 '19 at 04:43
  • @nick012000, such a waveform that "runs out eventually" isn't periodic. Please see Periodic function. – Alfred Centauri May 23 '19 at 11:30
  • @AlfredCentauri If that’s your definition of periodic, then nothing in nature is periodic - because the Second Law of Thermodynamics is a bitch. – nick012000 May 24 '19 at 01:19
  • @nick012000, (1) it's not my definition and (2), in a comment yesterday to another answer here, I explicitly make the point that there are no genuinely periodic signals in nature. Your comment doesn't make much sense. What point are you trying to make? – Alfred Centauri May 24 '19 at 01:55
  • @AlfredCentauri I was arguing that your definition was wrong outside the idealised world of mathematics. – nick012000 May 24 '19 at 05:23
  • @nick012000, (1) it's not my definition and (2), that's an odd way to think about it. Physics textbooks and papers are chock full of "the idealised world of mathematics". Consider, for example, the plane wave solutions to, e.g., Maxwell's equations. Are there any genuine plane waves in nature? No? Does this make the definition of a plane wave wrong outside of the idealized world of mathematics? This is all I have to say about this. – Alfred Centauri May 24 '19 at 13:11
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Bessel Functions were the first wave-like things to spring to mind. Amongst quite a lot of other things, these are the vibrational modes of a circular membrane, which is why drums like timpani never quite sound in tune.

Taking a liberal view of "waveshapes" would allow in electron wavefunctions in everyday matter (s, p, d, f atomic "orbitals", excitations, covalent bonds, etc.)

nigel222
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Geysers, like for example Old Faithful in Yellowstone national park, erupt in cycles which appear to follow a square waveform of varying frequency and amplitude. Even when we plotted the height of the column of water instead of a binary on/off pattern, it still is squareish, with an impressive slew rate.

http://www.geyserstudy.org/geyser.aspx?pGeyserNo=OLDFAITHFUL

dlatikay
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The premise of the question is faulty. Sine waves never appear in nature. Nor do squares, circles, spheres, logarithms, exponentials, or any other pure mathematical construct. Some things in nature are variously well approximated by these constructs, but you will never find a perfect one anywhere.

Fourier's theorem states that any periodic signal can be constructed with an infinite sum of sine waves. Some quasi-periodic signals in nature will be well approximated with a single frequency component, others will not. In reality there are no sine waves, just occasionally quasi-periodic phenomena. The sine waves are imaginary - a construct to ease our own cognition and calculation.

We could just as well have decided that triangle waves would be the basis of our mathematics - in that case we could represent sinusoidal motion as a fourier series of triangle components. The mathematics would become entirely more complex but the model would nevertheless be perfectly valid. The sine wave is a natural pure component, however, so we use it because it is the most primitive and the simplest and clearest to manipulate.

Roll a lumpy, odd shaped rock down a hill. Is there a sine wave there? Maybe. But there are other things as well. The rise and fall of rivers might be square-ish, but with logarithmic-ish rises and exponential-ish falls. A bolt of lightning you might think of as a dirac delta function on human timescales, but something completely different at 100,000fps. If you look closely enough you can find circles, squares, sine waves, square waves, straight lines, logarithms, exponentials, and all manner of other things in nature. Speakers of Arabic often find "words" written in trees, bushes, vines, etc, much as you might see a rabbit in the clouds.

Are any other things in nature approximated by rabbits? It's a silly question, depending on how you think about it.

J...
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  • Of course you are right on this - FWIW that's why I used the slightly woolly wording "can be associated with phenomena in nature" in the question body. Of course they are only associated as approximations or models of real behaviour. – Нет войне May 22 '19 at 11:58
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    "In reality there are no sine waves, just periodic phenomena" - are there any genuinely periodic signals in nature? Recall that a signal $f(t)$ is periodic with period $T$ if $f(t + nT) = f(t)$ for any integer $n$. – Alfred Centauri May 22 '19 at 13:11
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    @AlfredCentauri quasi-periodic, fair. – J... May 22 '19 at 13:12
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The tonic firing pattern of neurons shows distinctive spikes, as a consequence of the underlying physical and chemical process, a discharge on exceeding a treshold.
On a more macroscopic level, consider the excitation pattern of cardial pathways in any creature with a heart, this beautiful waveform of life, which is anything but sinusoidal (confusingly, the signal is named sinus after the biological structure that creates it, coming from a different meaning of the word).

sinusrythm

image source: Wikipedia, public domain

dlatikay
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