Fourier Synthesis (Web Site from Lab Manual)
A periodic function can be represented by linear combinations of sines and cosines. This java applet lets you play with different combinations, watch the resulting functions, and hear the sound wave with the same waveform.
Some sample Fourier calculations from the Fourier synthesis site.
Fourier Approximations and Music
This project is from the Duke Connected Curriculum Project and introduces Fourier approximations of periodic functions in the context of musical sounds. Maple V is available on the acpub computers.
Some more biographical information about Joseph Fourier.
He even has a university named after him!
A Pictorial Introduction to Fourier Analysis
Here is one of the many uses of Fourier analysis. In the late 1960's, Blakemore and Campbell suggested that the neurons in the visual cortex might process spatial frequencies instead of particular features of the visual world. This means that instead of piecing the visual world together like a puzzle, the brain performs something akin to the mathematical technique of Fourier analysis to detect the form of objects. While this analogy between the brain and the mathematical procedure is at best a loose one (since the brain doesn't really "do" a Fourier analysis), whatever the brain actually does when we see an object is easier to understand within this context.