# Are there pure sine waves in nature or are they a mathematical construct that helps us understand more complex phenomena?

I've studied a bit of frequency analysis with FFT and optimal phase binning and was taught that we can represent any composite waveform as the sum of its component frequencies.

I understand the maths works and gives meaningful results that we can use for design or to solve problems, but does this mean that sine waves are a natural 'element', like particles are for matter but in the time domain (4th dimension) - something that occurs in nature? Or are they a mathematical construct that helps us interpret nature?

Do pure, single frequencies occur through natural phenomena or processes?

I was taught about tuning forks but (without having tested it) I assume they will produce some harmonics as the straight bars have more than one mode of vibration.

Then I thought about the rotation of the planets but they are not pure sinusoids either since the gravity of other planets affects their rotation.

Finally I thought about light, but only lasers have a single frequency and as far as I know they don't occur naturally.

I assume I'm not the first human asking this question. Are you aware of any academic work on this matter?

• What makes you think a particle is anything other than a mathematical construct that helps us interpret nature? – WillO Aug 28 '17 at 13:45
• Good question. I'll have to think about it a bit but I guess the fact that I can interact with groups of them (matter). I have a difficulty understanding how we interact with things that occur in time. – Schizomorph Aug 28 '17 at 13:49
• Not sure how to phrase this but don't we interact with "groups of sine waves" in the sense that we can hear some frequencies for example? – Tycho's Nose Aug 28 '17 at 15:22
• " but only lasers have a single frequency" - real lasers don't actually produce a single frequency of light. Also, the fact that a real laser must 'turn on' at some time in the past and 'turn off' at some time in future, the light produced during that time, no matter how stable in frequency while the laser is on, necessarily consists of a continuum of frequencies (as can be shown analytically by finding the spectrum of the product of a pure sine and a rectangular pulse) – Hal Hollis Aug 28 '17 at 15:36
• You may be surprised at the answer to "Are there naturally occurring lasers in nature?" . – David Cary Aug 28 '17 at 15:42

## 12 Answers

Since no phenomenon is completely periodic (nothing keeps repeating from minus infinity to infinity), you could say that sine waves never occur in nature. Still, they are a good approximation in many cases and that is usually enough to consider something physical.

Or are they a mathematical construct that helps us interpret nature?

I would even go further and say that it is reasonable that everything in physics is a mathematical construct that helps us interpret nature, but that would lead to the philosophical debate of what is nature an so on. After all, almost everything in physics breaks down or at least becomes problematic at some regime: the notion of particles in strongly-interacting theories, energy in general relativity, the notion of a sound wave at the atomic scale...

• The phrase "interpret nature" doesn't even compile. Scientists try to predict some parts of reality, that's all. – kubanczyk Aug 28 '17 at 17:44
• Your statement that something needs to repeat over all time||space to be truly periodic isn't true - very common geometries limit that repetition. – user121330 Aug 28 '17 at 18:11
• "everything in physics is a mathematical construct that helps us interpret nature..." - xkcd is relevant here – Robotnik Sep 1 '17 at 0:47
• @kubanczyk I am not sure I am putting this correctly, but don't you think to "interpret nature" might just be to predict parts of reality (or all of it)? I mean in what else way could we ever truly "interpret" reality. To know the future may be to understand the present. – PhyEnthusiast Sep 1 '17 at 9:20
• Aren't quantum effects potentially infinite (or at least they do not change between changing states)? Are there not sine waves involved somewhere? I think of e.g. a hydrogen atom in vacuum, with the electron in a stable state. The actual math is way beyond me but it is some oscillation which should have a (perfect) sine in there somewhere. – Peter A. Schneider Sep 2 '17 at 20:48

This is really more of a supplement to jinawee's answer, but you might want to consider what, if anything, makes your question different from the following analogous questions:

• Are there lines in nature, or are they a mathematical construct that helps us understand more complex phenomena?
• Are there points in nature, or are they a mathematical construct that helps us understand more complex phenomena?
• Are there spheres in nature, or are they a mathematical construct that helps us understand more complex phenomena?

At a fundamental level, physics is about building mathematical models of the observable world. These models are "real" only to the extent that they make testable predictions that can be checked against that observable world. Since any experimental observation is only accurate up to a certain precision, it's never possible to say that one of these mathematical models is exactly the same as the thing that it describes. But without the language of mathematical idealizations, physics would be unable to do much of anything.

As jinawee said they cannot be physical due to their temporal extent. Nevertheless they are extremely useful because they (sine, cosine and combination of them) are the eigen-functions of the operator $\partial_t^2$ that shows up in a lot of differential equations:

$\partial_t^2(A\sin(\omega t+\phi))=-\omega^2A\sin(\omega t+\phi)$.

You can easily check that this is also true for any linear combination of sine and cosine of the same frequency with arbitrary constant phase. On the other hand, this property (eigen-function) is not satisfied by other functions, even periodic ones like $\sin(\omega t)+\sin(2\omega t)$. This why these functions are special and omnipresent.

Now since wave-like equations are often linear we are naturally lead to use Fourier analysis: we can decompose ~any signals* as a linear combination of harmonic function sine and cosine, transform the derivative operators into algebraic multiplication $(\partial_t^2 \rightarrow -\omega^2)$ easily solve the now algebraic equation, and add together the solutions to recreate a physical signal (i.e. bounded in time). Check Wave- vs Helmholtz- equation

*there are some restrictions but this is not bothering for usual physical signals

• "this property (eigen-function) is not satisfied by other functions..." Not cosh and sinh? I guess you could clarify that it is not satisfied by any other bounded functions. – Ryan Sep 3 '17 at 4:56

To my knowledge, it seems that sine waves occur in nature.

For example, light is, in a sense, an oscillation of the electromagnetic field, which has no harmonics if we consider a single photon. I'd like to add that lasers do occur in "nature", more precisely they are real, simply because it is possible to make them. I feel like whether or not they are actually created by some non man-made process is irrelevant, as we humans still need to obey laws of nature, which means that lasers do follows laws of nature.

In conclusion, I think that it can argued that everything is just a mathematical construct int the end, because we cannot what happens with absolute precision in phenomena and are forced to come up with models to describe our observations as best as we can, so we can never know for sure if our model are 100% exact or if they are just exact to a certain degree, but with an error that is too small for us to observe.

• The first paragraph is inaccurate: it is perfectly possible to have a single photon that's in a superposition of multiple frequencies, which can include the harmonics of some fundamental. Such a state can in no way be described as a pure sine wave. – Emilio Pisanty Aug 28 '17 at 16:33
• To continue @EmilioPisanty's comment, a photon having a well-defined frequency would have a position with infinite uncertainty, and I don't think you can say those really occur in nature. – JiK Aug 28 '17 at 17:49
• @JiK Could you expand on this? Isn't the frequency of the photon equivalent to its energy, therefore wouldn't the uncertainty principle affect the time? Also how does relativity from the PoV of the photon come into this discussion? – Lenzuola Aug 28 '17 at 20:15
• Also, real lasers do not actually emit a single frequency, but (simplified version) a kind of "hashed gaussian" spectrum (frequency between hashes corresponds to the cavity length, gaussian width to the temperature) – Nicolas Aug 30 '17 at 8:52
• So where is the photon moving towards? Passing the universe with (slightly) different gravitational potentials? Then then frequency is changing anyway. – lalala Sep 1 '17 at 19:14

Every signal we try to measure will have background noise from other phenomena. With so many phenomena in the universe, it might be reasonable to say that we could never measure a pure "natural" sinusoid, but that's a different question than whether phenomena exist that have purely sinusoidal behavior.

Many, but not all phenomena are composed of multiple frequencies. For your statement to be true, all (there are about $10^{80}$ atoms in the known universe - to count all the 2 atom interactions, that'd be $10^{160}$, etc..., and this ignores photons which are far more numerous) phenomena must have multiple frequency components. Just for some perspective, if we ignore everything but hydrogen, your statement is orders of magnitude less likely than winning the lottery while drowning and getting struck by lightning in the same second.

Every second order linear differential equation has solutions of $e^{p v}$ where $p$ is a complex parameter and $v$ is the variable. For sound and light, we expect a purely imaginary $p$ in most cases simply because energy is conserved. For example, every electron around a free hydrogen atom (hydrogen represents more than 73% of the (non-dark) matter in the universe, much of which is free) has a radial solution (page 6) that's a pure sinusoid. If you quibble that there are other charges that perturb the orbit of the electron, I completely agree that there are often perturbations to that orbit, but they are so small you couldn't measure them (the uncertainty in the energy levels of the hydrogen atom may be derived from the theory ignoring the presence of other atoms), and if you did, you'd find that the most likely behavior was exactly sinusoidal anyway.

In other words, it's absurd to say that there are no pure sinusoids in nature (ignoring the philosophical quagmire of what's unnatural). Also, yes, sine waves also help us understand more complex phenomena.

• TL;DR. Basically, it's safe to assume that there are ONLY sine waves in nature, and everything else is composed of sine waves. However, we need infinite time to measure these sine waves - which most (if not all) of us don't have. Also, most of us (if not all) cannot measure stuff from before the big bang. – Klaws Aug 29 '17 at 15:23
• More interesting philosophical result of this: when someone fires a gun, can you observe pre-ringing (and therefore maybe predict this event)? Would that mean that this event was pre-determined - or that events affect the past? If you could measure the amount of pre-ringing, you might be able to establish whether there was time before the big bang - and the complete absence of pre-ringing could establish that time is infinite in both directions? – Klaws Aug 29 '17 at 15:33
• @Klaws I get that if you do a Fourier transform, you have to integrate over all space||time to nail down a frequency, but that says nothing of about whether the signal was, in fact, sinusoidal. In any event, this argument revolves around the geometry of a sphere which doesn't require all space at all. Also, if "there are ONLY sine waves in nature, and everything else is sine waves", what are you trying to say?!? – user121330 Aug 29 '17 at 16:23

A slightly ontological answer which ends up as "pick your poison" without needing quantum stuff.

I see two aspects to your question:

• If you have an arbitrary wave in any medium, is its FFT decomposition into individual sine waves "real"?
• Can you find anything that, when measured, plots a, for all intents and purposes, perfect sine curve?

The answer to the first question is a very definite "maybe". You will find plenty of processes which absolutely do not have a "real" FFT analysis. Take an earthquake; it creates a solid matter wave traveling through Earth. It is incredibly unlikely that this wave is a perfect sine wave; and there is no part of this process (of rocks gliding past each other) that would invite any suspicion that if you do a FFT for the random jumble we see on our meters, the constituent sine wave would have any "real" couterparts in the rocky bottoms of our Earth.

On the other hand, you can imagine processes which we could, indeed, treat as if they were naturally occuring FFTs. Find a magic lake made out of liquid unobtanium which, when you drop stones in it, somehow produces perfect sine waves. Now, drop three stones right next to each other. Yes, you will get a seemingly random wave; yes, you can FFT transform it to get 3 cleanly separated parts, and yes, there is a physical equivalent to this analysis (i.e., the 3 stone drops). So, yes, with enough handwaving of irrelevant details, you could use a FFT on a seemingly random wave to reconstruct physically "real" events.

The answer to part two would depend a bit on your assumptions. What would you accept as "perfect"? Measuring stuff is annoyingly difficult at small resolutions (darn you, Heisenberg). Where would you place the "cut off point"? Would you accept a measurement that is perfect up to the 10-nanometer scale? Within the 1mm-scale? If so, sure, take a very big pendulum in thin or no air and very well oiled parts, and measure its angle. Voilá, within your arbitrary measure accuracy, you have a perfect sine wave, c/f the pertinent Physics.SE question.

At least for a little while, until friction slows the pendulum down enough to notice it even in the arbitrary resolution you picked for your measurement. And yes, per our current understanding, certainly if we have a contracting universe, everything will slow down in the end. Or, worse, if we find out that the universe is ever expanding, every process still started with the Big Bang, so it is not eternal in that direction. So if you need an eternal process, you're right out of luck.

In a comment you said "in my mind frequencies are for sound what chemical elements are for matter".

So, sin waves are not extremely special. The process of breaking down a wave phenomina into linear combination some set of base component and being able to reconstruct it from the coefficients is not at all special to sin (or cos) waves.

Any sufficiently dense set of funtions (in a formal sense) that separates things sufficiently and contain the constant functions will do.

As it happens, sin waves have nice easy to work with mathematical properties, and the fourier transform being its own inverse has a certain elegance.

You can see practical applications of that fact, such as wavelets used in jpeg compression. These wavelets aren't periodic like sin waves, yet a linear combination of such wavelets is dense in amplitude space.

You can step back and look at the fourier transform. You start with some wave. You multiply it against the original signal (using convolution), and from the result work out how much they overlap and what is the best scale of the wave to approximate the original signal.

Then you subtract that scaled wave from the original signal. This "removes the frequency component" from the original signal (in that if you convolved it with the wave again, you'd get zero).

We then repeat this with different frequencies, each time "removing a frequency component". So long as the frequency components we remove are orthogonal to each other (a generalization of being "at right angles"), removing new frequency components doesn't "bring back" the old ones.

As it happens, "removing the frequency component" corresponds to an operation.

If you set of a resonance cavity of length L down which pressure waves travel at speed S, and have some repeating set of pressure waves, the cavity will amplify the part of the pressure wave of frequency L/S that roughly corresponds to a the convolution of the sin wave with the amplitude of the pressure wave over time.

That seems pretty academic, but have you ever looked at your ears?

They are resonance cavities. Pressure waves go in, and bounce back and forth.

Along the side of it, there are hairs that pick up changes in pressure. Waves of various frequencies are amplified and damped by the resonance chamber and excite and ignore a predictable set of hairs.

In short, our ears split pressure waves into something a lot like what fourier analysis does. We have physical fourier transformers on our head attached to our brain.

So when we do a fourier analysis and say that there is a strong signal at 550 Hz, this corresponds to what our ears hear because our ears are doing something that the math approximates and mapping the pressure waves into a spectrum of frequencies for us to hear.

Our eyes don't do that.

When you do fourier analysis on images, you do get useful results, but there often are nasty singularities and artifacts.

For a given frequency of photon, the universe of light is very similar to the universe of sound at human scales (one moves faster). But instead of a resonance cavity for photons, we have a pinhole camera and lens. This gives us great directional resolution on light. Meanwhile, the ear gives us great temporal resolution on sound. With our ears, we can hear if something is vibrating at 500 Hz or at 600 Hz really obviously; with our eyes, if you took a light and flashed it on and off at 500 or 600 Hz you wouldn't even see it.

Our eyes instead have pigments that absorb certain frequencies of photon, split the infinite dimensional photon frequency space into a 1 to 4 dimensional cube, and give us high resolution positional information about where photons come from.

The step of mapping photon frequencies to the 1-4 pigments corresponds to a convolution, which you can approximate with a fourier transform, but the spacial positioning doesn't really correspond to a resonance cavity like frequency tool. Thus when you use fourier analysis on positions of lights,it doesn't map that well to our perceptual experience.

In short, no, the pure sin curve as a fundamental component of sound is an artifact of how we hear. Given something vibrating in a specific way, you'll get a pure sin curve, but vibrating in that specific way isn't fundamental to the universe either.

First, from a systemic point of view, if you can model a physical system as outputs that linearly depend on (potentially unknown) inputs, and that the system characteristics are stable over time, you end up with a so called Linear-Time-Invariant system. For such a system, complex sines are the most natural functions, even if you cannot really observe them. They are "natural", because a complex sine input is converted into a complex sine output of the very same frequency. It is called "an eigenfunction" of the said system.

And the good news is: any other solution for such a system, as complicated it can be, can be decomposed into a weighted sum of complex sine eigenfunctions, with makes the analysis of LTI systems much simpler in the Fourier domain. Fourier diagonalizes LTI systems, hence the efficiency of FFT for faster computations.

Second, as this as not be mentioned directly yet, the heat equation is derived from Fourier's law or law of heat conduction:

the flow rate of heat energy per unit area through a surface is proportional to the negative temperature gradient across the surface.

To solve the resulting heat equation, Fourier "invented" the so-called Fourier series, that turned out, on its fast version (FFT), to be one of the most important algorithms.

Whether true sines exist could be of philosophical nature (Platonism). However, for less linear, or less time-invariant systems, physicists developed more localized versions of the complex sines, called wavelets, that are akin to solitons, and that can be used to analyze non-linear differential equations, turbulence phenomena,quantum field theory, etc.

Nota: on "rotation of the planets but they are not pure sinusoids either since the gravity of other planets affects their rotation". Gauss is sometimes first credited for the fast Fourier transform, used for the prediction of the position of celestial bodies.

FWIW, the movement of a pendulum mass on the end of a string is very close to a perfect single frequency.

But in answer to your question, the sine waves in a Fourier transform do NOT represent a physical reality. Only the waveform itself is a physical reality. The sine wave components in a Fourier transform are merely a mathematical construct which allows us to analyse the waveform in a particular way.

This should be clear when you look at the maths behind it. For a square wave (and for various other waveforms), you can only get a fully accurate Fourier transform if you have an infinite number of harmonics - anything less leaves you with a mere approximation. This should tell you straight off that it is merely a mathematical tool.

I've studied a bit of frequency analysis with FFT and optimal phase binning and was taught that we can represent any composite waveform as the sum of its component frequencies.

Whoever first tried Fourier transform, why would he even try it? What would make someone to think that it's a good idea to break a signal into sine waves?

It turns out that the idea's sensible if you look at how the sound could be generated. For instance, consider a guitar string. Its ends are fixed, they can't move. The only move that can happen with a string would have the ends fixed at zero. Next, what would be the simplest wave on the string that has ends at zero? It's a half sine wave with the length equal to the string length. The second possible wave is the full sine wave, etc. These were all so called standing waves. Now it only sounds sensible that one might represent the dynamics of the shape of the string that produces sound as a combination of standing waves.

Are the standing waves real? You can observe the waves that look like standing waves, sure. Whether they are real is a different question. I don't think that sine or cosine are real unless you believe in God, who used these function to design the world. However, you see the shapes that are best described as sine waves all the time. In that regard they are mathematical constructs that correspond to reality, or are represented in reality.

In physics class we took a long square tube, coated the bottom with sawdust, mounted a tuning fork next to one end, beat on it with a rubber mallet and watched the sine wave graph appear in the sawdust.

That's a real enough sine wave for me and I hope it's real enough for you.

Physics explanation: sound wave oscillation in a tube open on both ends and length a multiple of the tuning fork frequency sustains a sine wave shaped sound oscillation in the tube.

• Of course that's not real enough, and you did not see a sine wave at all. What you saw was a "something-like-a-sine-wave-for-children-who-didn't-know-the-difference-and-couldn't-have-measured-it-anyway". As your physics teacher should have covered the following week, the fundamental is only one of an infinite range of resonant frequencies - and ALL that infinite range of harmonics were produced when you hit the fork with the mallet. – Graham Oct 24 '17 at 13:02
• @Graham: We had high quality tuning forks. Almost all of the energy was in the fundamental and the higher-order harmonics couldn't move the sawdust. Oh, and we did measure it. – Joshua Oct 24 '17 at 13:20
• The fact that you say "almost all" is the key here that you're aware it is not a pure sine wave. It's a reasonable approximation, sure, and close enough to measure to a resolution of a few mm either way with a school ruler. It's certainly not a "pure single frequency" per the OP's question though, and a microphone attached to a spectrum analyser would easily show the harmonics. A pendulum is much closer to a single frequency, but again it's still not entirely pure. – Graham Oct 27 '17 at 8:09
• @Graham: Try again. The higher order harmonics didn't have enough energy to overcome static friction. We would later measure the same tuning fork on an oscilloscope and they were barely perceptible on the scope. – Joshua Oct 27 '17 at 13:37
• "Barely perceptible" - and yet they were perceptible. A mass-friction system makes a nice low-pass filter, but the maths of a low-pass filter says that you're still only reducing those harmonics, even if you can't measure the output with a school ruler. I don't understand your objection to someone saying (as the OP does) that a tuning fork has harmonics and the result is not a pure sine wave, when it's simple physical fact. It's real enough for your school ruler, sure. Is it real enough to be "a natural element like a particle"? Of course not. – Graham Oct 30 '17 at 13:45

I believe you are confusing the inability of humans to measure something "perfectly," with the non occurrence in nature of "pure" sine waves. There should be no doubt that pure sine waves do occur in nature, it is only our inability to measure accurately and without disturbing that which is being measured that causes the problem.

Are mathematical sine waves human "constructs"? Absolutely yes! So what? They are in deed, "constructs" that help us understand nature.

Your examples will "work" if you remove "peripheral effects."
The tunning fork can be made to oscillate at only its fundamental frequency.
The orbit of a planet can be declared circular within a certain tolerance (%), thereby it oscillates at a single frequency (within the given tolerance).
Light can be "cleaned" with filters so that only one frequency (within a give tolerance) passes through.
These phenomena, viewed through a proper apparatus, will show the sinuous nature of the frequency.

• "There should be no doubt that pure sine waves do occur in nature" - I doubt it and I doubt that I shouldn't doubt it. – Alfred Centauri Sep 1 '17 at 1:15

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