Fourier: Making Waves: free math simulation
Add sine harmonics to build square, sawtooth and triangle waves, and hear them.
▶ Open the simulationFree, no sign-up. Works on phones, tablets and laptops.
The science
Any repeating wave can be built by adding sine waves of different frequencies and amplitudes. A square wave is made of odd harmonics with amplitudes 1, 1/3, 1/5 and so on. Adding more harmonics makes the shape sharper.
Square wave ≈ sin x + ⅓ sin 3x + ⅕ sin 5x + …
Things to try
- Build a square wave from odd harmonics.
- Make a sawtooth wave.
- Listen to how the sound changes as harmonics are added.
Topics covered
harmonics series sine square sawtooth sound waveform
How to use it
Open the simulation, then change the experiment live with the sliders, checkboxes and buttons in the side panel. Drag objects on screen where it makes sense, and use Pause, Step, Speed and Reset in the bottom bar to study every moment.
Frequently asked questions
What does the Fourier: Making Waves simulation show?
Any repeating wave can be built by adding sine waves of different frequencies and amplitudes. A square wave is made of odd harmonics with amplitudes 1, 1/3, 1/5 and so on. Adding more harmonics makes the shape sharper.
What is the key equation?
Square wave ≈ sin x + ⅓ sin 3x + ⅕ sin 5x + …
Is the Fourier: Making Waves simulation free?
Yes. It runs free in any web browser on phones, tablets and laptops, with no sign-up or download.
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