When a violin string is vibrating it is playing many notes at the same time:

The loudest note is the sine wave fitting the length of the string:

The next loudest is the sine wave of half the wavelength which fits the length of the string.

We have notes of wavelengths corresponding to each fraction.

Now let’s see the shape of the sound when we play all the notes together. The animation builds up the sound by adding each new harmonic.

If you have a guitar or violin you can hear these higher harmonic notes by performing the following experiment. Pluck the string. Lightly place you finger half way along the string. You should hear a note an octave higher. Try putting your finger a third of the way along the string. Here is the third harmonic.