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Fourier Series

対象: Universityカテゴリ: Math

説明

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.

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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]