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?)

  • $\begingroup$ 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. $\endgroup$ – jacob1729 May 21 '19 at 21:39
  • $\begingroup$ @jacob1729 sure - I'm taking that as read. $\endgroup$ – topo Reinstate Monica May 21 '19 at 21:52
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    $\begingroup$ 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? $\endgroup$ – Cort Ammon May 21 '19 at 22:08
  • $\begingroup$ @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) $\endgroup$ – topo Reinstate Monica May 21 '19 at 22:56
  • $\begingroup$ Would an impulse count as a "waveshape"? :) $\endgroup$ – pipe May 22 '19 at 9:45

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.

  • $\begingroup$ 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) $\endgroup$ – topo Reinstate Monica May 21 '19 at 21:54
  • $\begingroup$ @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. $\endgroup$ – zakinster May 22 '19 at 12:08
  • $\begingroup$ @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!) $\endgroup$ – topo Reinstate Monica May 22 '19 at 12:27

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: $$ \frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin(\theta) $$


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.

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    $\begingroup$ That looks like a sinusoidal wave, though? It's just got its amplitude and frequency changing over time. $\endgroup$ – nick012000 May 22 '19 at 6:04
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    $\begingroup$ 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. $\endgroup$ – nigel222 May 22 '19 at 10:07
  • $\begingroup$ @nick012000, the inspiral waveform is not even periodic. How can it "look like a sinusoidal wave"? $\endgroup$ – Alfred Centauri May 22 '19 at 11:45
  • $\begingroup$ @nigel222, to be sure, the inspiral waveform is not periodic, and so there is no Fourier series for this waveform. $\endgroup$ – Alfred Centauri May 22 '19 at 12:23
  • $\begingroup$ @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. $\endgroup$ – nick012000 May 22 '19 at 20:20

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.)


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.



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).


image source: Wikipedia, public domain


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.

  • $\begingroup$ 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. $\endgroup$ – topo Reinstate Monica May 22 '19 at 11:58
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    $\begingroup$ "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 $n$. $\endgroup$ – Alfred Centauri May 22 '19 at 13:11
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    $\begingroup$ @AlfredCentauri quasi-periodic, fair. $\endgroup$ – J... May 22 '19 at 13:12

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