> How does one take a "Fourier analysis" of a bridge? Give the bridge an 'impulse. Then it's Fourier transform is has all frequencies. Then see what the brid

> How does one take a "Fourier analysis" of a bridge? Give the bridge an 'impulse. Then it's Fourier transform is has all frequencies. Then see what the bridge does: Get out the transfer function of the bridge. It might have a peak -- then don't march the troops at that frequency. Or the output motion of the bridge is the inverse Fourier transform of the Fourier transform of the input to the bridge from the soldiers times the transfer function of the bridge. If the signal from the soldiers puts too much power at a frequency that is high in the transfer function, then that is a "resonate frequency" and the bridge is threatened. For the violin, yes, the human ear does a lot of Fourier analysis with the little hairs inside that coiled up thing, whatever it is called, but for the violin I was considering the beats when bow two adjacent strings together. So, bow the A string at 440 Hz together with the E string at about (3/2)440 Hz, and listen to the third harmonic of the A string and the second harmonic of the E string -- they should be the same frequency. If their frequencies differ by x Hz, then the sound will have amplitude modulations, beats, of x per second. That's how a violinist tunes the instrument. The Fourier part? The sound from the A string is roughly periodic but not a sine wave. Similarly for the sound from the E string. Do Fourier analysis, that is, Fourier series on each of the two periodic signals. Take the third term from the Fourier series of the A string and the second term from the Fourier series for the E string, and listen to the beats. While the signals from each string are periodic, they are not sine waves, but the terms from the Fourier analysis are sine waves which is partly why the beats are so easy to hear. So, in effect, are using, say, the E string to find the Fourier coefficient of the third harmonic of the A string. The A and E strings are separated in frequency by an interval of a perfect fifth which is 7 semi-tones or, from a piano, about 2^(7/12) which is close to 3/2. Well, that's the story for a perfect fifth. But there is also a perfect fourth, 5 semi-tones, a major third, 4 semi-tones, and a minor third, 3 semi-tones, a perfect 6th, 9 semi-tones, and, of course, an octave, 12 semi-tones. In perfect tuning, each of these intervals has, for the two notes, overtones with the same frequency (for two small whole numbers, one of them times the frequency of the lower note is the same as the other times the frequency of the higher note) so that can tune the interval by listening to beats. Of course, the easiest one to hear is the octave. So, when using two strings this way, really are doing a Fourier analysis using one string to take the Hilbert space inner product of the two signals and using a sine wave from one of the overtones and its inner product with the other signal to scan that signal for beats and, thus, where the overtones are and, thus, really doing what Fourier analysis does, a projection via Hilbert space inner products onto orthogonal axes (sine waves at the overtone frequencies, that is, at frequencies that are whole number multiples, whole numbers again, of the frequency of the original periodic signal). So, adjust one of the two strings and in effect scan the sound from the other string for it's overtones -- that's essentially Fourier analysis. I will resist writing out all the math, but, really, a lot of Fourier theory is fairly intuitive stuff where a lot of the main ideas can be done with just pictures. Or, yes, it would be a little simpler to play just, say, the A string and have an audio oscillator that puts out a sine wave and, then, slowly sweep the frequency of the oscillator sine wave and listen for beats with the sound of the A string -- that's essentially just Fourier series analysis of the periodic signal of the A string. But if don't have an audio oscillator handy, and use another string, say, the D or E string, and its overtones, each of which is a sine wave, and adjust the frequency of the D or E string and, thus, sweep a selected (listen carefully!) overtone of the D or E string past selected overtones of the A string and do much the same as with the audio oscillator. Gee, I knew there was some Fourier theory in there somewhere!