Standing waves. What is a harmonic?

When two interfering waves of equal amplitude, wavelength and frequency travel in opposite directions along a string, the resultant wave is called a standing wave.

A standing wave pattern always consists of an alternating pattern of nodes, that never move, and antinodes that undergo simple harmonic motion of maximum amplitude 2A. Both nodes and antinodes are always located at the same position.

Moreover, if both ends of the string of length L are fixed (for example in a guitar or in a violin) standing waves will only be produced at specific frequencies of the interfering waves. That is so because the displacement of both ends of the strings is always zero and therefore they are nodes.

The position of the nodes is given by:

And imposing the condition that the end of the string of coordinate L is a node, we get:

And using the definition of wave speed:

where n is a positive integer.

The previous expression gives us the allowed frequencies of the standing waves propagating along a string of length L and fixed ends. These frequencies are called natural frequencies or resonant frequencies.

On the other hand, the wave speed is given by:

where T is the tension in the string and μ its linear mass density. Therefore, the allowed frequencies will depend on both the tension and the physical properties of the string.

Different values of n correspond to modes of oscillation called harmonics.

In the figure above the shape of the standing wave for the first three values of n is shown.

The lowest harmonic, with n = 1 , is called the fundamental harmonic, given by:

All the frequencies of the harmonics are whole-number multiples of this fundamental frequency. Higher values of n give the second harmonic, third harmonic, and so on.

Timbre and tone

Each note that comes out of an instrument is a mixture of different harmonics. Mathematically, we can describe the resultant wave as a linear combination of standing waves of different frequencies:

Even playing the same note (that is, the same fundamental frequency or tone), each instrument has a distinct sound, called timbre. Timbre is the result of a superposition of harmonics, which have different relative amplitudes A_n for each instrument. Whe can use Fourier analysis to see what harmonics are involved in a certain waveform.

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