## Fourier synthesis

A periodic function with period $a$ can be written as a Fourier series of the form,

$$f(x) = A_0 +\sum\limits_n A_n\left(\cos(\theta_n)\cos (2\pi nx/a)+\sin(\theta_n)\sin (2\pi nx/a)\right).$$
 $f_n(x)$ $x/a$
 $S$ $\frac{ka}{2\pi}=n$
 $A_0=$ 0 $A_1=$ 1 $\theta_1=$ 0$\pi$ $A_2=$ 0 $\theta_2=$ 0$\pi$ $A_3=$ 0 $\theta_3=$ 0$\pi$ $A_4=$ 0 $\theta_4=$ 0$\pi$ $A_5=$ 0 $\theta_5=$ 0$\pi$ $A_6=$ 0 $\theta_6=$ 0$\pi$ $A_7=$ 0 $\theta_7=$ 0$\pi$ $A_8=$ 0 $\theta_8=$ 0$\pi$ $A_9=$ 0 $\theta_9=$ 0$\pi$ $A_{10}=$ 0 $\theta_{10}=$ 0$\pi$ $A_{11}=$ 0 $\theta_{11}=$ 0$\pi$
 $f(x)$ $x/a$

Number of periods displayed:

The sliders can be used to compose a periodic function. The plot in the upper right is the power spectrum. The power $A_n^2$ is plotted vs. wave number $k=2\pi /\lambda$ where $\lambda$ is the wavelength of the Fourier component. (If the layout is awkward, try pressing Ctrl- to get all of the components on your screen.)