Plucking or striking a chord of a musical instrument you will cause a motion that, if there were no friction, would be harmonic. In addition each point, swinging, induces its neighbors to do the same, creating a wave that propagates along the string to one of the two ends fixed, where it is reflected, generating a second wave that propagates in the opposite direction. The motion of a point of the string is then the result of the overlap of two waves: a progressive and a regressive one.
The effect is that each point of the string vibrates in harmonic motion, with amplitude variable depending on the location. Note from figures 1 and 2 that in a wave single (no matter whether progressive or regressive) there is phase shift in the motion of the individual points, in the sense that while a point is in its highest position another is in the intermediate, another in the lower one, and so on. On the contrary, when a point in a standing wave is at the maximum, the others or do the same or are in the minimum. It is said that the points that rise and fall together vibrate in phase, although with different speed and then reaching different heights; on the contrary, those that have opposite movements (if one goes up, the other goes down) are in opposed phase. Since all points of the string are either in phase or in opposed phase,
the wave does not propagate more: then it is called standing wave.
As you can see from Figure 3, in addition, there are some points (called nodes) absolutely motionless, and others (called anti-nodes) that vibrate with maximum amplitude. Since the string is fixed at its ends, which thus can not vibrate, there must be at least two nodes; also every two consecutive nodes there is an anti-node in their mid-point.
Ultimately, the string can vibrate in one of the following ways:
- two nodes and one anti-node;
- three nodes and two anti-nodes;
- four nodes and three anti-nodes;
and so on.
