Sympathetic vibration occurs when a vibrating object causes other objects to vibrate at the same frequency. In the case of a piano, when a key is struck, the corresponding strings vibrate. This vibration can cause other strings on the piano to vibrate, even if they are not being directly played. This is because the strings are all tuned to specific frequencies, and when one string vibrates, it can cause other strings that are tuned to the same frequency to vibrate as well.
Sympathetic vibration is an important part of the sound of a piano. It gives the instrument its rich, full sound and helps to create the characteristic resonance that is associated with the piano.