Standing Waves: Nodes, Antinodes and Harmonics

When a traveling wave reflects back on itself in a medium of finite length, such as a guitar string fixed at both ends, the incident wave and the reflected wave constantly superpose. The result no longer moves along: it is a standing wave. Some points of the string remain motionless at all times; these are the nodes (N), while others oscillate with maximum amplitude; these are the antinodes (A).

For a string fixed at both ends, the ends are necessarily nodes, since they cannot move. This constraint means that only certain particular wavelengths are possible: these are the normal modes, or harmonics. For a string of length L, the harmonic of order n (n = 1, 2, 3...) corresponds to a wavelength λ_n = 2L / n, and therefore to a frequency f_n = n × v / (2L), where v is the speed of the wave on the string.

The first harmonic (n = 1), also called the fundamental, has only two nodes, at the ends, and a single antinode in the middle. The second harmonic (n = 2) adds an extra node at the center, forming two antinodes. In general, the harmonic of order n has n antinodes and (n + 1) nodes. It is this division into modes that gives each stringed instrument its characteristic timbre.

The same reasoning applies to a sounding pipe (as in a flute or an organ), with one important nuance: an open end behaves like a vibration antinode, whereas a closed end imposes a node. A pipe closed at one end therefore produces only the odd harmonics, which explains why its sound differs markedly from that of a pipe open at both ends.

Beware of the classic mistake of believing that an antinode is a point where matter moves the farthest without ever stopping: in reality, every point of the string oscillates, including at the antinode, with zero velocity at the extreme positions and maximum velocity when passing through equilibrium, exactly like a pendulum.

string fixed at both ends, length L

harmonic n=1 (fundamental) :     N-------A-------N
                                   (1 antinode, 2 nodes)

harmonic n=2 :                   N---A---N---A---N
                                   (2 antinodes, 3 nodes)

harmonic n=3 :                   N-A-N-A-N-A-N
                                   (3 antinodes, 4 nodes)

lambda_n = 2L / n        f_n = n x v / (2L)