I just came from a class on Fourier Transformations as applied to signal processing and sound.  It all seems pretty abstract to me, so I was wondering if there were any physical systems that would behave like a Fourier transformation.

That is, if given a wave, a purely physical process that would "return" the Fourier transform in some meaningful way.  Like, you gave it a sound wave and you would see, "Oh, there are a lot of components of frequency 1kHz...a few of frequency 10kHz...some of 500Hz..."

I've seen things happening where, if you put sand on a speaker, the sand would start to form patterns on the speakers that are related to the dominant wavelengths/fundamental frequencies of the sound.  Is this some sort of natural, physical fourier transform?

Your ear is an effective Fourier transformer.

An ear contains many small hair cells.  The hair cells differ in length, tension, and thickness, and therefore respond to different frequencies.  Different hair cells are mechanically linked to ion channels in different neurons, so different neurons in the brain get activated depending on the Fourier transform of the sound you're hearing.

A piano is a Fourier analyzer for a similar reason.

A prism or diffraction grating would be a Fourier analyzer for light.  It spreads out light of different frequencies, allowing us to analyze how much of each frequency is present in a given source.
