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.)