To understand how double rainbows work scientifically, you must first know several basic facts about single rainbows. Single rainbows are caused by refraction, dispersion and internal reflection. Refraction is the way sunlight strikes moisture, bends, and bounces back from the moisture. Dispersion is when the moisture acts as a prism. It then splits white light into seven colors. Internal reflection is when the light hitting the moisture passes straight through it and comes out on the other side. It reflects off the surface of the moisture and is sent in a different direction. This is how a single rainbow is created.
When refraction occurs to cause single rainbows, the light strikes and goes through moisture. Double rainbows are created in the same way except the sunlight bounces twice off the moisture before re-entering the sky. According to The Straight Dope, this second reflection inverts the colors when dispersion occurs and creates a 54-degree angle instead of a 50- to 53- degree angle, thus, creating a second rainbow.
Many people notice that second rainbows are fainter than single rainbows. This is because the light has been reflected twice and had two chances to be transmitted out the back of moisture in the air. Because the light is reflected toward the Earth, people see a fainter bow. According to The Straight Dope, many secondary rainbows can also only be seen at certain points along the arc.
Double rainbows also create what is called "Alexander's Band." According to Atmospheric Optics, Alexander's Band is the optical phenomenon caused by double rainbows and refers to the change in angles caused by them. If this occurs, a dark band in between the rainbows occurs, making the two rainbows appear brighter.