Nodes
The crests and troughs of a standing wave do not travel, or propagate, down the string.
Instead, a standing wave has certain points, called nodes, that remain fixed at the
equilibrium position. These are points where the original wave undergoes complete
destructive interference with its reflection. In between the nodes, the points that oscillate
with the greatest amplitude—where the interference is completely constructive—are
called antinodes. The distance between successive nodes or antinodes is one-half of the
wavelength,.
Resonance and Harmonic Series
The strings on musical instruments vibrate as standing waves. A string is tied down at
both ends, so it can only support standing waves that have nodes at both ends, and thus
can only vibrate at certain given frequencies. The longest such wave, called the
fundamental, or resonance, has two nodes at the ends and one antinode at the center.
Since the two nodes are separated by the length of the string, L, we see that the
fundamental wavelength is. The string can also support standing waves with
one, two, three, or any integral number of nodes in between the two ends. This series of
standing waves is called the harmonic series for the string, and the wavelengths in the
series satisfy the equation , or:
In the figure above, the fundamental is at the bottom, the first member of the harmonic
series, with n = 1. Each successive member has one more node and a correspondingly
shorter wavelength.
EXAMPLE