How do adding odd sine harmonics build the flat levels and steep transitions of a square wave?
By starting with the fundamental sine wave and sequentially adding higher odd frequencies (3, 5, etc.), the partial sums interfere constructively in the middle of the half-periods to flatten the peaks, and destructively near the boundaries to steepen the transitions toward vertical jumps.
Conditions
- The series consists of the fundamental and odd harmonics.
- Terms are added sequentially to form partial sums.
Reasoning, step by step
- Begin with the fundamental sine wave.
- Add the third harmonic.
- Add the fifth harmonic.
- Observe the flattening of the central regions.
- Observe the steepening of the transition regions.
- Conclude that higher odd harmonics refine the square wave shape.
Example
The script states: 'Start with the fundamental sine wave, then add frequencies 3, 5 and so on to build the flat levels and steep transitions.'
Common misconceptions
- Believing that adding more harmonics makes the transitions smoother.
- Thinking that even harmonics are required to create the flat tops.
Watch the explanation
BilibiliFourier series
0:08 – 0:19Watch this moment ↗
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