While experimenting with piano acoustics I came across this weird phenomenon. Having an open B string and then hitting the C above will cause the B to resonate! This also works with open C and playing B below or B above! I cannot explain this with overtones since there isn't any close harmonic relationship between m2 or M7. Also how can any note produce frequencies below the fundamental?
The answer lies in a mixture of engineering and "pure" physics. And a lot of Fourier Transforms :-) .
As we all know, the strongest resonances happen when the second string's pitch is an integral multiple of the first string. To see why, let's start with just the sound wave in air reaching the second string. The first crest excites the string, and just as the string's natural resonance pulls it back, the first valley helps "pull" the string in that direction as well. Thus the second string absorbs lots of energy from the sound wave. Next, look at an emitted frequency of F and a string resonating at 2F. Here the energy transfer matches the resonance only half the time (roughly speaking).
Now, if the second string's resonance is something like 12/11 *F, the sound wave is almost aligned with the resonant frequency, so the string will absorb energy for a few cycles -- and then it will dump energy as it goes out of phase! This leads to the "beat" phenomenon we all hear when two instruments are close but not exactly in tune.
Now to the engineering part: a physical piano does not have perfect acoustic reflection of the first string's travelling wave at the two ends of the string. Some energy, both acoustic and mechanical, gets into the sound board and the piano body. This energy starts dispersing (i.e. into random frequencies) and feeding back into the posts of all the other strings, thus exciting every string to some extent. How much energy makes it into a given note depends on the resonant profile of the entire system, not just the acoustic wave generated directly from the first string.