Fourier-Series Approximation of a Square Wave

rotate the linkage

select a joint

Black

Orange

Green

Blue

Red

A square wave with amplitude 1 and wavelength

λ

can be approximated by the Fourier expansion

∞

∑

n=1,3,5,…

4

nπ

sin

2πn

λ

,

a sum of sines with amplitudes

4

nπ

and frequencies

n

.

This Demonstration uses an open five-bar linkage to represent Fourier approximations of increasing accuracy. The

th

n

bar of the linkage has length

4

nπ

and rotates at angular speeds

n

around its joint with the preceding bar.

The ordinates of the joints represent progressive approximations to the sum, as you can see by operating the time gauge.

Snapshots 1 through 5 show the progressive approximations of the square wave using one through five bars of the linkage.

Erik Mahieu