Fourier Series
受众: University类别: Math
简介
Demonstrates how a square wave can be decomposed into and reconstructed from sine waves. Individual harmonic components animate in one by one, and the running partial sum curve updates to show convergence toward the square wave.
受此动画启发？
Fourier Series
Description
Demonstrates how a square wave can be decomposed into and reconstructed from sine waves. Individual harmonic components animate in one by one, and the running partial sum curve updates to show convergence toward the square wave.
Phases
| # | Phase Name | Duration | Description |
|---|---|---|---|
| 1 | Setup | 5s | Title, axes, show target square wave in white |
| 2 | 1st Harmonic | 7s | Show sin(x) term, animate addition to sum, display formula term |
| 3 | 3rd Harmonic | 7s | Add (4/3π)sin(3x), update sum curve, compare to square wave |
| 4 | 5th Harmonic | 7s | Add (4/5π)sin(5x), convergence clearly improving |
| 5 | 7th Harmonic | 7s | Add (4/7π)sin(7x), Gibbs phenomenon visible at jump |
| 6 | Full Formula | 8s | Show Σ (4/nπ)sin(nx) formula, mention n=1,3,5,... |
| 7 | Summary | 5s | Overlay of square wave and partial sum, convergence annotation |
Layout
+--------------------------------------------------+
| Title: "Fourier Series" |
+--------------------------------------------------+
| |
| Axes with square wave | Formula panel |
| (white, dashed) | (right side) |
| | |
| Individual harmonics: | f(x) = Σ 4/(nπ) |
| 1st (blue), 3rd (teal), | · sin(nx) |
| 5th (yellow), 7th (gold) | |
| | Current sum: |
| Running sum (orange) | highlighted |
| | terms |
+--------------------------------------------------+
Area Descriptions
- Left 65%: Main axes showing individual sine harmonics and their cumulative sum
- Right 35%: Fourier series formula and term-by-term display
- Bottom: Gibbs phenomenon note, convergence description
Assets & Dependencies
- Fonts: LaTeX
- Manim version: ManimCE 0.19.1
Notes
- Square wave with period 2π, amplitude 1
- Fourier coefficients: 4/(nπ) for odd n
- Show individual harmonics as thin curves and cumulative sum as thicker curve
- Axes x_range = [-2π, 2π], y_range = [-1.5, 1.5]