Strange Bows & Supernumeraries

Brian Turner pictured this sight over London.

A discontinuous main primary bow is on the right. Broad supernumerary fringes fan out from it to the left, their separation and width increasing towards the top.

©Brian Turner, shown with permission

Supernumeraries can be 'explained' in several ways. One way invokes a (dubious?) mixture of classical rays and of wave optics.

For a given angle inside the main primary bow there are always two classical ray paths contributing light.

Enter waves: Imagine them strung along the two ray paths. The paths have different lengths and the two sets of waves may, or may not, be in phase when they leave the drop. In-phase waves give light, out-of-phase darkness. The path difference depends on the angle and there are thus a series of bright and dark fringes - supernumeraries.

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This bow is the product of small droplets that probably did not even reach the ground as rain.

The split primary bow could be due to two sheets or layers of different size droplets each producing a bow of different size. As drop size decreases diffraction effects become increasingly significant. The rainbow broadens and shrinks.  

Supernumeraries are also diffraction effects. They were first called ‘supernumeraries’ because, in ray optics, they were not supposed to exist.   Their separation and width increases as droplets get smaller. Here the drops are evidently smaller towards the top of the picture.