Corners and Smoothness

How fast do the harmonics fade?

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Smoother functions have Fourier coefficients that fade faster.
• Jumps: about 1/n. Corners: about 1/n². Smooth: faster than any power.
• Parseval's theorem turns the square wave into $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$.

Steps

1. Which is easier?
2. Three waves
3. Smoothness and decay
4. The triangle’s third harmonic
5. Triangle versus square
6. Energy in the harmonics