Let's look at the sine wave again. Plotting it this way makes the relationship to the trigonometric functions clearer.



Waves can be added together. Adding waves according to certain rules can create surprising geometric patterns. This is called additive synthesis.


These are special cases of what is called a Fourier series, expressed in the form of the following equation. If you look at the left part of the demos, you can see that there are multiple rotational motions with different radii and velocities adding up to create the waveform.


${\displaystyle {\frac {a_{0}}{2}}+\sum {k=1}^{\infty }(a{k}\cos kx+b_{k}\sin kx)}$

Square Wave 矩形波

${\displaystyle {\begin{aligned}x_{\text{square}}(t)&={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin \left(2\pi (2k-1)ft\right)}{2k-1}}\\&={\frac {4}{\pi }}\left(\sin(2\pi ft)+{\frac {1}{3}}\sin(6\pi ft)+{\frac {1}{5}}\sin(10\pi ft)+\dots \right)\end{aligned}}}$