Imagine an idealized stretched string with fixed ends vibrating the first 4 modes of the standing waves. This can be expressed as the relationship between wavelength, speed and frequency, a basic formula where the wavelength is inversely proportional to the frequency when speed is a constant (k) since it is the same string:
Let's work out the relationships among the
frequencies of these modes. For a wave, the frequency is the ratio of the speed
of the wave to the length of the wave: f = k/wavelength. Compared to the string
length L, you can see that these waves have lengths 2L, L, 2L/3, L/2. We could
write this as 2L/n, where n is the number of the harmonic.
The fundamental or first mode has frequency f1
= k/wavelength = k/2L,
The second harmonic has frequency f2 = k/wavelength = 2k/2L = 2f1
The third harmonic has frequency f3 = k/wavelength = 3k/2L = 3f1,
The fourth harmonic has frequency f4 = v/wavelength = 4k/2L = 4f1,
and, to generalize,
All waves in a string travel with the same speed, so these waves with different wavelengths have different frequencies as shown. The mode with the lowest frequency (f1) is called the fundamental. Note that the nth mode has frequency n times that of the fundamental. All of the modes (and the sounds they produce) are called the harmonics of the string. The frequencies f, 2f, 3f, 4f etc are called the harmonic series.
The diagram below displays the harmonics in a span of 5 octaves, where the fundamental is C with the frequency of 32 Hz. As the octaves progress the numberes of new harmonics increase with the factor of 2.