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# Manim 傅里叶级数逼近方波演示代码

设计思路

• 方波函数：f(x) = 1（0<x<π），-1（π<x<2π），周期 2π
• 傅里叶级数：f(x) ≈ (4/π)·Σ sin((2k-1)x)/（2k-1)，只含奇次谐波
• 逐项累加：每帧增加一项谐波，同时显示公式和当前项数

完整代码

```python

from manim import *

import numpy as np

class FourierSquareWave(Scene):

def construct(self):

axes = Axes(

x_range=[0, 2 * TAU, PI / 2],

y_range=[-1.6, 1.6, 1],

x_length=11, y_length=5,

axis_config={"include_numbers": False},

).shift(DOWN * 0.5)

labels = axes.get_axis_labels("x", "f(x)")

self.play(Create(axes), Write(labels))

# 目标方波

square_wave = axes.plot(

lambda x: 1 if (x % TAU) < PI else -1,

color=GRAY,

)

sq_label = Text("方波", font_size=24, color=GRAY).to_edge(UP, buff=0.2)

self.play(Create(square_wave), Write(sq_label))

title = Text("逐项叠加谐波", font_size=28).next_to(sq_label, DOWN, buff=0.1)

counter = Integer(0, font_size=36).to_corner(UR)

self.play(Write(title), Write(counter))

accumulated = VMobject()

for n in range(1, 11): # 前10个奇次谐波项

k = 2 * n - 1

term = axes.plot(

lambda x, k=k: 4 / PI * np.sin(k * x) / k,

color=interpolate_color(BLUE, RED, n / 10),

)

counter.set_value(n)

# 累加曲线

if n == 1:

accumulated = axes.plot(

lambda x, k=k: 4 / PI * np.sin(k * x) / k, color=YELLOW

)

else:

prev_k_terms = list(range(1, 2 * (n - 1), 2)) + [k]