How does adding odd sine harmonics build the flat levels and steep transitions of a square wave?

Conditions

• Using the partial sum $S_N(x)=\frac4\pi\sum_{k=0}^{N-1}\frac{\sin((2k+1)x)}{2k+1}$.
• $N$ increases to include more harmonics.

Reasoning, step by step

1. Begin with the fundamental frequency ($k=0$).
2. Add the next odd harmonic ($k=1$, frequency 3).
3. Observe how the superposition modifies the shape: peaks broaden/flattened, valleys deepen.
4. Continue adding terms to refine the approximation towards the rectangular shape.

Example

The script instructs: 'Start with the fundamental sine wave, then add frequencies 3, 5 and so on to build the flat levels and steep transitions.'

Common misconceptions

• Thinking that each added harmonic makes the entire wave taller; amplitudes decay as $1/n$.
• Believing that convergence is uniform across the whole domain immediately.

Watch the explanation

Explore next

• What happens to the approximation error at the jump discontinuities as N increases?
• Why are even harmonics excluded from this construction?