What causes sympathetic vibrations in a string?
You pluck the A string of a guitar, then damp it. You notice that the (high) E string is now vibrating, although you didn’t touch it. Or you play an A on the G string of your violin, and the open A string starts to vibrate. What is going on? When you pluck the low A, the vibration is not a simple sine wave with one frequency. Rather, it is a complex vibration whose spectrum has quite a lot of frequency components, most of which are nearly harmonic. So, as well as a fundamental at the pitch A2, it also has a second harmonic at A3, a third harmonic at E4 and so forth. So the string vibrates the bridge and soundboard of the guitar at all of these frequencies. Of course the top E string is tuned to E4, and the vibration of the bridge is exciting it at its own natural frequency, so it absorbs energy from the bridge. Then, when you damp the A string, no more energy comes into the E string, but the energy it has stored is now given back to the bridge and soundboard, as its sound fades away. Y
