Perhaps I'm too dense to understand your answer. In the simplest example of N=2, what does the crystal look like & what are its modes? My thought was that the DFT's come from circulant matrices, so if you had a kind of necklace of masses bouncing against one another, this might work. But the other major problem is if you think of Fourier Transforms in terms of frequency & time, you now have discrete frequencies, which is ok, but discrete times (in a physical system) ? So perhaps we model the system with a set of finite difference equations ? At 01:31 PM 7/28/2013, Eugene Salamin wrote:

The vibrations of a finite crystal is what you want. The DFT is the expansion into normal modes.

-- Gene

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________________________________ From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Sent: Sunday, July 28, 2013 12:13 PM Subject: [math-fun] Really stupid physics question

We've all seen the classical analysis of a vibrating string, which leads to classical Fourier theory.

In the 20th century, we all became converts to DFT's (Discrete Fourier Transforms).

So....

I'd like to see the exact mechanical counterpart to the DFT.

Given N, the _length_ of the DFT, how do I arrange for N masses & (?) Hookean springs so that the dynamics give me exactly the DFT ?

In particular, for very small N (N<10, nowhere near the continuum case), what do such systems look like?