> 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!