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    I don't really understand how the Fourier Transform works, I understand how to carry it out, but I don't see why the FT of the time domain is the frequency domain, or position is momentum. I was just wondering whether there is a physical interpretation of the FT.

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    well you can split up any time function into a fourier series, like cos(n*2pi t), etc, where n*2pi is the frequency - just from the definition of it

    and the fourier transform basically picks out each of these frequencies

    try taking the fourier transform of cos(4t) or something to get a feel for how it works
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    I really think Fourier analysis should be part of A Level Further Maths/Physics. I learnt it for my A2 Electronics course (it was needed to understand other concepts, not examinable directly) and found it very interesting. It's a shame people aren't introduced to it at this level.
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    I always think of the Fourier transform of something like a sound wave in the following way.

    If you imagine a vibrating string and look at the standing wave patterns that are possible, you get the familiar sequence of harmonics starting with the fundamental f then the next harmonic at 2f, next at 3f and so on. The string, when vibrating, vibrates in all these modes simultaneously, and the resulting sound wave is a complex shape made up of the sum of all these frequencies, each one being a sine wave. The higher frequencies at ever smaller amplitudes.

    Fourier analysis of this complex wave would be the reverse process. It takes the complex waveform, and calculates which harmonics are present in it, at what their relative amplitudes are. In other words, it takes the time variation of the original wave, and produces a frequency domain.
 
 
 
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