All the effects of nature are only mathematical
results of a small number of immutable laws.
~Pierre-Simon Laplace

In my discussion last time (corrections here), I discussed how there is a physical limit to how good a recording can sound, whether vinyl or digital. There is a more fundamental limit, however, that I glossed over—a limit that depends not on atoms or compression techniques, but on pure mathematics. This limit was partially discovered by Jean Baptiste Joseph Fourier, and the method we will discuss bears his name.

The Superposition Principle

Before we discuss Fourier’s discovery, let’s take a brief moment to talk about how waves interact with each other. I’ve previously discussed wave interference
and the superposition principle, so if you remember my previous discussions, feel free to skip to the next section.

Imagine waves as wiggles on a very stretchy string. If I try and push up on the string (make a wiggle that goes up) and you try and push down on the string (make a wiggle that goes down) at the same time, neither of us ends up moving the string as much as we intended. This is called destructive interference. Similarly, if I push up on the string at the same time that you push up on the string, we’ll probably stretch it quite a lot. This is called constructive interference. The process of overlaying one wave over another is called superposition.

Sound waves work the same way: they can both constructively and destructively interfere.

The Sound of an Extremely Short Pulse

Last time, I explained that sound is a wave of air pressure. I also described how the tone of the sound is directly related to the frequency of the wave. Usually, when we play music, the frequency of the wave that generates the notes is much higher than, say, the frequency of the beat of the music. This is the beating phenomenon I described before. It looks like a little wave contained inside a big wave, as shown below.

But what happens when we want to make the music so fast (or the pitch so low) that the music changes at approximately the same frequency as the wiggles that control the pitch? For instance, what if we want to make a single loud, very short noise–so short that it’s about as fast as one of the little wiggles in my figure above. What’s the pitch?

To figure this out, we’ll take advantage of interference. By superposing waves of different frequencies, we will attempt to recreate our pulse. Hopefully, the waves will all destructively interfere before and after the pulse, but constructively interfere during the pulse. Then we will know that the pulse contains all the tones required to construct it.

Let’s start with a wave with a frequency of one wiggle per second. This doesn’t bear much resemblance to our pulse yet, but you have to start somewhere.

Now comes the fun part. Let’s add, via superposition, a wave with twice the frequency. The figure below shows the two waves to be added on the left and their sum on the right. It’s important that the two waves line up so that their peaks overlap at the place we want our pulse to be–in this case, the center of the plot.

Already we can see some of the wiggles dying down. Let’s add another wave, this time with three times the original frequency.

After ten or so waves superposed, we get something like following:

Although it’s not perfect, a single pulse does appear in the center. The big pulses on either side of the graph are the pulse repeating itself. To get rid of the repeats and all of the small wiggles, we’d need infinitely many frequencies in different proportions (i.e., we’d need to add the same frequency several times). In fact, we usually need uncountably many frequencies. When we need countably infinite frequencies, we call their sum a Fourier series. When we need uncountably many frequencies, we call their sum a Fourier transform. In general, the shorter the pulse we want in time, the broader the range of frequencies we need to construct it.

Importantly, this means that an extremely short pulse of sound has infinitely many tones. This is a pretty surprising result! It makes some intuitive sense, though. A single pulse of pressure isn’t periodic, so it can’t have a frequency. On the other hand, there was a single pulse, so it has at least one instance. The frequency can’t be zero. Fourier series and Fourier transforms offer an in-between: A pulse as many frequencies.

Because a shorter pulse requires a broader frequency range, an infinitely short pulse requires an infinitely broad frequency range (a sine wave of frequencies actually). It’s impossible to construct a short pulse made of too few frequencies. This is a physical limit on how short of a pulse we can construct.

Resolution

The shortness of a pulse roughly translates into how well we can control sound we make, or resolve sound we hear, as a function of time. Similarly, the number of frequencies roughly translates into how well we can resolve the pitch of a given sound we record. In other words, it’s physically impossible to record a sound with both perfect time resolution and perfect pitch. If you want to get a better time resolution (analogously, a higher sample rate), you need worse frequency resolution, and vice versa. This limit comes not from the number of atoms in a recording tip or the amount of memory on a computer–it comes directly from the properties of waves.

The Quantum Connection

This strange property of waves also makes itself known in quantum mechanics. You may have heard of the Heisenberg Uncertainty Principle. Roughly, the uncertainty principle states that you can’t perfectly know a particle’s position and momentum (mass times velocity) at the same time. In a vacuum, this sounds incredibly weird. However, if we recall that particles are waves, it makes a bit more sense.

A particle is a probability wave. The amplitude of the wave roughly represents the probability of finding a particle in a given place, while the frequency of the wave roughly represents the particle’s momentum. If we imagine a localized particle as a short pulse, rather than a wave, the reason we can’t know a particle’s position and momentum perfectly is because we can’t make an infinitely short pulse out of a single frequency! This is a beautiful example of physical law emerging from a mathematical idea. Waves are mathematical constructs, but they appear everywhere.

Further Reading

• You can find a Fourier series calculator here.
• Wolfram Mathworld has an article on Fourier series here and an article on Fourier transforms here.
• CU’s physics education group has a simulation of Fourier series.
• A nice Fourier series tutorial can be found here.
• Better Explained has a page on the Fourier transform.
• Our understanding of optical frequency combs and mode-locked lasers rely heavily on the Fourier transform.
• Modern physics relies heavily on Fourier analysis. Indeed, it’s built into our very notation.

Now you can understand this comic!

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Questions? Comments? Insults?

Fourier analysis is a very difficult and unintuitive topic, so I may not have explained it well. (Indeed, my own understanding may be incorrect.) If you have any questions, comments, corrections, or insults, please don’t hesitate to say so in the comments!

‘EDIT: Stackexchange has an excellent discussion on Fourier analysis: http://math.stackexchange.com/questions/1002/fourier-transform-for-dummies