What is meant by 'harmonics' in a musical note, and how are harmonics explained in physics? How is this relevant to how the same note played on different musical instruments sound different?
Harmonics are additional higher-frequency vibrations in a standing wave, such as occurs in the string or air column of a musical instrument.
If a guitar string is plucked or a flute is blown, a standing wave of a particular frequency will be created. This is the 'fundamental', and is the particular note being played - maybe an A.
The A just below middle-C on a piano has a frequency of 440 Hz.
An A played on a flute sounds quite different from an A played on a guitar, though. This is because there will also be a blend of harmonics playing, at a range of volumes. This blend is called the 'timbre' of the sound.
Harmonics are often multiples of the frequency of the fundamental. A tone one octave higher - the first A above middle-C for example - has a frequency twice as high: 880 Hz for that A. Another octave higher, the frequency doubles again, and so on. There are also other multiples, like 3/2.