Fourier's Extraordinary Idea — Making Complicated Waves From Simple Ones
What does a violin have in common with a mobile phone signal? Sines, cosines — and one extraordinary mathematical idea.
At A-level, students become very familiar with sine and cosine.
They sketch their graphs. They solve equations involving them. They differentiate and integrate them. In physics, they meet sinusoidal oscillations and alternating currents.
It is therefore quite easy to come away with the impression that sine and cosine are simply two particularly useful functions that happen to describe smooth, repetitive behaviour.
But there is a much bigger idea hiding behind them.
An idea so powerful that it appears in music, acoustics, electronics, radio, telecommunications, medical imaging, astronomy and digital image processing.
The idea is associated with the French mathematician and physicist Joseph Fourier:
A complicated repeating waveform can be constructed from a collection of much simpler sine and cosine waves.
That sounds remarkable.
Even better, we can actually see it happening.
Start With the Simplest Possible Wave
Consider:
sin(x)
There is nothing particularly surprising here. It is the familiar smooth oscillating curve.
Now add another sine wave:
sin(x) + (1/3)sin(3x)
The second wave has three times the frequency but only one-third of the amplitude.
The resulting graph starts looking slightly less like an ordinary sine wave.
Now add another:
sin(x) + (1/3)sin(3x) + (1/5)sin(5x)
Then another:
sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + (1/7)sin(7x)
Something extraordinary begins to happen.
The smooth curves start producing something that increasingly resembles a square wave.
Keep adding the odd harmonics and the approximation becomes increasingly convincing.
In more general terms, an ideal square wave can be represented by an infinite Fourier series containing odd harmonics:
sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + (1/7)sin(7x) + ...
There is a constant scaling factor if we want a particular amplitude, but that is not the important idea for our first investigation.
The important thing is what we have just done.
We have built something containing apparently sharp corners from functions that contain no corners at all.
Try It Yourself
This is an excellent investigation for an A-level Maths or Further Maths student because it needs surprisingly little equipment.
A graphical calculator, Desmos, GeoGebra or a spreadsheet is enough.
Plot:
y = sin(x)
Then:
y = sin(x) + sin(3x)/3
Then:
y = sin(x) + sin(3x)/3 + sin(5x)/5
Continue with:
sin(7x)/7
sin(9x)/9
sin(11x)/11
and watch what happens.
Do not simply look at the final graph.
The interesting part is watching the square wave gradually emerge.
At first, there is no obvious reason why adding curved waves should produce anything remotely square.
Yet it does.
That is precisely the sort of mathematical experience I like students to encounter beyond the examination syllabus.
Rather than being told that mathematics is powerful, they actually see something happen that seems almost impossible.
But Look Carefully at the Corners
There is another interesting feature.
Zoom in near one of the sudden transitions in the square wave.
You may notice that the approximation overshoots and oscillates around the discontinuity.
Adding more terms does not simply make this little feature disappear.
This is connected with the Gibbs phenomenon, another fascinating piece of mathematics that students can investigate.
It is a useful reminder that saying an infinite series "becomes" a square wave needs some mathematical care.
That opens the door to deeper questions about:
convergence;
infinite series;
approximation;
discontinuities;
limits.
Suddenly a visually simple experiment has taken us into some quite sophisticated mathematics.
So What Did Fourier Actually Realise?
Joseph Fourier was studying the flow of heat in the early nineteenth century.
In trying to solve problems involving heat conduction, he developed the idea that complicated functions could be represented using combinations of trigonometric functions.
At the time, this was mathematically controversial.
Today, Fourier analysis has become one of the fundamental tools of mathematical physics and engineering.
A general Fourier series can be written in the form:
f(x) = a0/2 + a1 cos(x) + b1 sin(x) + a2 cos(2x) + b2 sin(2x) + ...
The precise coefficients depend upon the function we are trying to reproduce.
Students do not need to calculate all those coefficients to appreciate the central idea.
Think of it this way:
Fourier analysis gives us a mathematical recipe for taking a complicated signal apart and discovering which simple frequencies are hiding inside it.
And that takes us from pure mathematics straight into music.
What Does a Musical Note Really Look Like?
Play a pure sine wave through a loudspeaker.
It sounds rather plain.
Now play middle C on a piano.
Then play the same note on an organ.
Then a guitar.
Then a violin.
They are all playing approximately the same fundamental frequency, so why don't they sound identical?
Because a musical instrument generally does not produce just one frequency.
It produces a fundamental frequency together with additional harmonics and other spectral components.
The relative strengths and behaviour of these components contribute enormously to the characteristic sound of the instrument.
This means Fourier's mathematical idea gives us a way of looking inside a musical sound.
A Superb Practical Investigation
This is where I would take the mathematics off the page.
Record the same musical note played on several different instruments.
For example:
piano;
organ;
guitar;
violin;
flute;
electronically generated sine wave.
Try to keep the fundamental note the same.
Now examine each recording using audio software capable of displaying a frequency spectrum.
Instead of displaying amplitude against time, display amplitude against frequency.
Suddenly the differences become visible.
The pure sine wave should have a strongly concentrated fundamental frequency.
A real instrument may show the fundamental plus a whole collection of additional frequency components.
A student can now ask:
Which harmonics are strongest?
Are even and odd harmonics equally prominent?
How rapidly do the higher harmonics decrease?
Does the spectrum change as the note develops?
What happens during the attack of the note?
Why does a flute look different from a violin?
We have transformed listening to music into mathematical investigation.
My Organ Becomes a Mathematics Laboratory
This is one reason I particularly like Fourier analysis as a teaching topic.
An electronic organ or synthesiser provides an extraordinary experimental laboratory for investigating sound.
I can select one sound, record a note and examine its spectrum.
Then I can change the registration or instrument sound while keeping the actual musical note unchanged.
What changed?
Not primarily the note being played.
What changed was the mixture of frequencies producing it.
With suitable software, students can both hear and see the difference.
That is a much richer experience than simply being told that musical instruments contain harmonics.
And an organ provides an especially interesting connection because organ stops are explicitly associated with different pitches.
An 8-foot stop sounds at the written pitch.
A 4-foot stop sounds an octave above.
A 2-foot stop sounds another octave higher.
Other stops introduce different harmonic relationships.
We can therefore build complicated sounds by combining components — conceptually remarkably close to the mathematical idea we have just explored.
From Fourier to Synthesisers
Now reverse the problem.
Instead of analysing an existing sound, suppose we want to create one.
Start with a sine wave.
Add another sine wave at twice the frequency.
Perhaps add another at three times the frequency.
Change their amplitudes.
Listen again.
We are performing additive synthesis.
A synthesiser can construct a complicated sound from simpler components in much the same spirit as our mathematical Fourier construction.
This creates a wonderful Maths + Physics + Music crossover experiment:
Construct a square-wave approximation mathematically.
Generate the corresponding frequencies electronically.
Listen to the result.
Examine its spectrum.
Compare mathematical prediction with the actual sound.
A formula has become something we can hear.
Square Waves Are Not Just Mathematical Curiosities
Square waves are extremely important in electronics.
Digital systems often switch between two voltage levels.
An idealised digital signal therefore contains sudden transitions rather than smooth sinusoidal changes.
But Fourier analysis tells us something important.
Producing those sharp transitions requires high-frequency components.
Remove enough of the higher frequencies and the edges become rounded.
This matters when transmitting digital information.
A communications system with limited bandwidth cannot reproduce arbitrarily rapid changes perfectly.
Suddenly our little graph of:
sin(x) + sin(3x)/3 + sin(5x)/5 + ...
has led us towards real questions about data transmission and communications engineering.
Radio — Finding Signals Hidden Inside Signals
Radio gives us another application.
A radio receiver is surrounded by electromagnetic signals.
Different transmitters operate at different frequencies, and information is encoded onto signals in various ways.
Fourier methods allow engineers to examine signals in terms of their frequency components.
Rather than asking:
"What is the signal doing at this particular moment?"
we can ask:
"Which frequencies are present, and how strong are they?"
Those are two different ways of looking at the same information.
This distinction between the time domain and the frequency domain is one of the most important conceptual steps students can take.
Two Ways of Looking at the Same Thing
Imagine recording one second of music.
We could plot:
amplitude against time.
That shows us how the air pressure — or electrical signal from the microphone — changes during that second.
That is the time domain.
Alternatively, we could analyse the same recording and plot:
amplitude against frequency.
Now we can see which frequencies contribute to the sound.
That is the frequency domain.
Neither representation is inherently "the real signal".
They are two mathematical views of the same phenomenon.
That idea extends far beyond music.
Fourier Analysis and Images
An image may seem to have little to do with sound.
But mathematically there is a connection.
A digital photograph contains variations in brightness and colour across space.
Slow changes correspond to low spatial frequencies.
Fine details and sharp edges involve higher spatial frequencies.
Fourier techniques can therefore be applied to images.
They can help with:
filtering;
sharpening;
noise reduction;
compression;
pattern analysis;
astronomical imaging;
medical imaging.
Once students understand the idea with a sound wave, it becomes much easier to appreciate how the same mathematics can be applied elsewhere.
Spectroscopy — A Particularly Interesting Connection
Fourier mathematics also appears in spectroscopy.
Some spectroscopic techniques record signals that are not initially in the form of the familiar spectrum we ultimately want.
A Fourier transform can convert measured information into a frequency spectrum.
Fourier transform infrared spectroscopy — FTIR — is an important example.
Once again, the basic philosophy is similar:
A complicated measured signal can reveal its hidden frequency components through mathematics.
For a student studying both Maths and Chemistry or Physics, this is a wonderful example of subjects meeting each other.
The Fast Fourier Transform
There is, however, a practical problem.
Real digital signals may contain thousands or millions of data points.
Calculating their frequency components directly can require enormous amounts of computation.
This is where the Fast Fourier Transform, usually abbreviated FFT, becomes important.
The FFT is an efficient family of algorithms for computing a discrete Fourier transform.
Its impact on computing, engineering and signal processing has been enormous.
When audio software instantly displays the frequency spectrum of a recording, mathematics and algorithms are working behind the scenes.
The student sees a graph appear almost immediately.
Hidden underneath it is some extraordinarily elegant mathematics.
A Challenge for an A-Level Student
Here is a good investigation.
Create a spreadsheet containing values of x.
Calculate:
y1 = sin(x)
Then:
y2 = sin(x) + sin(3x)/3
Then:
y3 = sin(x) + sin(3x)/3 + sin(5x)/5
Continue until perhaps the first ten odd harmonics have been included.
Plot each approximation.
Then investigate:
How many terms are needed before the graph looks convincingly like a square wave?
But do not stop there.
Try changing the coefficients.
What happens if every component has the same amplitude?
What happens if the amplitudes decrease more quickly?
What happens if even harmonics are introduced?
Can you deliberately create a different waveform?
Now the student is no longer merely following instructions.
They are experimenting with mathematics.
Could We Hear the Mathematics?
This would be my next step.
Generate the individual sine-wave components as sounds.
Listen first to the fundamental.
Then add the third harmonic.
Then the fifth.
Then the seventh.
The graph is gradually becoming more square.
But the sound is changing as well.
That is an extraordinarily powerful teaching moment.
The equation:
sin(x) + sin(3x)/3 + sin(5x)/5 + ...
is no longer simply ink on paper.
It is a graph.
It is an electrical signal.
And it is something we can hear.
Why Go Beyond the A-Level Syllabus?
A student might reasonably ask:
"Will Fourier series be on my A-level Maths examination?"
For most students, no.
But I think that is precisely why topics like this deserve occasional exploration.
The examination syllabus is necessarily selective.
It cannot contain everything interesting about mathematics.
If students only ever encounter mathematics that is immediately required for the next examination, they can develop a rather distorted picture of the subject.
Sine and cosine can become:
"those functions I need for the trig question."
Fourier transforms reveal something much bigger.
The trigonometric functions they have been manipulating are part of a mathematical language capable of describing and analysing the physical world.
Mathematics Is About Connections
Some of the most memorable lessons are those in which the artificial boundaries between school subjects disappear.
Fourier's idea connects:
Mathematics — functions, trigonometry, series and approximation.
Physics — waves, oscillations and electromagnetic signals.
Music — harmonics, timbre and synthesis.
Computing — digital sampling, algorithms and signal processing.
Electronics — waveforms, bandwidth and communications.
Chemistry — spectroscopy and molecular analysis.
It is difficult to think of many mathematical ideas with such an extraordinary reach.
The Bigger Lesson
There is something else I would want a student to take away from this investigation.
Mathematics is not simply about obtaining exact answers.
It is also about finding useful ways of representing complicated things.
A violin note looks complicated.
A radio signal looks complicated.
A square wave looks simple until we ask what frequencies are required to construct it.
Fourier's extraordinary insight gives us another way of looking at all of them.
Break the complicated thing into simpler pieces.
Understand the pieces.
Then understand how the pieces fit together.
That principle extends far beyond Fourier analysis.
It is one of the great strategies of mathematics and science.
Conclusion — Hear a Sine Wave Differently
The next time an A-level student sees:
y = sin(x)
I would like them to see more than a trigonometric graph.
That simple curve can become one component of a violin note.
Add others and it can approximate a square wave.
Analyse a complicated sound and sine waves can help reveal what is hidden inside it.
Extend the idea and we arrive at radio communications, digital electronics, spectroscopy and image processing.
That is why exploring mathematics beyond the syllabus can be so valuable.
Sometimes one familiar equation opens a door into an unexpectedly large part of science and technology.
And Fourier's extraordinary idea is a magnificent example.
A complicated world can sometimes be understood by adding together very simple waves.
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