At Øredev I had a discussion with some people about how the double rainbow comes about so in the spirit of sharing knowledge I thought I would share how it works.
The first rainbow comes from light being reflected once in the raindrop and similar to a prism the refraction of the light in the drop splits the white light into different colors.
Compare with the prism on Pink Floyd's Dark Side of the Moon.
The secondary rainbow on the other hand comes from the light being reflected twice inside the drop.
This is the reason the colors of a real secondary rainbow are inverted.