Supernumerary Rainbows

Supernumerary rainbows are the faint rainbows that can sometimes be seen under the primary rainbow. They occur when the water droplets are small, from about 1.0 mm to 0.4 mm wide. The smaller the droplets, the wider the supernumary bands are spaced. Supernumary bands are most often seen near the top of the rainbow arc, as the water droplets are flattened slightly, and at the top the light passes through them at their narrowest. This is why in the above image, the supernumary bands fan outwards closer to the top of the rainbow

The wave nature of light is what causes supernumary bands. The small droplets allow us to see the effects of the waves, but with larger drops (1+ mm diameter) the wave nature is less pronounced.

Multiple Paths and their Angles

Its important to consider that light enters the droplets at different positions and exits the droplets at different angles. the most common angle is the one where the exit path has deviated the least from the entrance path; the other angles are less common thus less bright. Most of the other angles cluster around the ideal angle.

this diagram shows the ideal angle (note that 180 - 137.5 = 42.5)

one of many other angles, where the exit path is more deviated from the entrance path

this diagram shows multiple rays all together, including when they reflect and when they simply pass through the drop. These diagrams are from AtOpt

Also consider that the ideal angle is slightly different for each color because their wavelengths cause the angle or refraction to be different (red is 42.5 and blue is 40.7)

Notably, for each exit angle there are two entrance positions that produce it. These two paths will experience interference

Interference

This diagram shows contructive and destructive interference. When the peaks and valleys of two waves line up, they add together, increasing the amplitude of the wave (left). When a peak lines up with another waves valley, they cancel each other out, decreasing the amplitude of the resultant wave (right)

The two paths of each angle will interfere with each other. Since they will travel slightly different distances, the intererence pattern for each pair will differ. Some will result in constructive interference, some destructive, and some inbetween. This results in bands of light (the light bands being from constructive interference, and the dark bands being from destructive interference)

Overlapping Colors

Consider these two simulations. The left is a primary rainbow formed by larger water droplets, and the right is a primary rainbow with supernumary bands formed from 0.7 mm droplets. Both feature staggered colors, allowing a rainbow to form instead of just white light, but in the supernumary simulation, each wavelength of light is banded due to interference. Since the wavelengths are different, each color bands differently, allowing certain colors to show through where others are dark, thus rainbows form. Notably, further from the primary, the bands are more blended, creating duller rainbows further out

Even in the simple primary bow, the larger wavelengths overlap the smaller wavelengths before them, muting their colors slightly, except for the smallest red wavelength which has no colors before it. Thus the smaller the wavelength the more pure its color is.