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   Supernumerary Bow Formation

      Two rays

Each have the same deflection.

Classical theory predicts their light always combines to make the bow brighter in that direction.

This does not happen.

      Two waves

One wave has further to travel through the drop than the other.

When they leave the wave crests are no longer always in phase.

When completely out of phase the waves cancel and there is darkness. When in phase there is light.

The result, dark and light bands inside the primary bow and a broadening of it

   Left: A pair of classical rays with the same deviation through a droplet.

For each deviation angle there are always two** rays at different distances from the drop centre that have the same overall deviation.

Geometric optics incorrectly *** predicts that their light combines to produce extra brightness in that direction.

It is quite unable to explain the bands of light and dark, supernumeraries, inside the humble rainbow.

  Right: The same ray path pair replaced by a representation* of light waves. Wave crests of opposite amplitude are shaded green and blue.

The rays have different path lengths. This has the result that after they have left the drop their wave crests are not necessarily aligned .

For some pairs of equal deviation rays the crests will be aligned (in phase). Those of other pairs and ray directions will be completely misaligned (out of phase) in the sense that crests of opposite amplitude will correspond instead. The diagram shows an intermediate condition.

When crests having the same amplitude direction are aligned the two waves reinforce each other and there is bright light at that particular deviation angle. When crests of opposite amplitude are aligned the waves cancel and there is darkness. This is interference.****

Imagine now the ray pairs for a range of deviation angles. As the angle changes the wave pairs alternately reinforce or cancel each other to produce light and dark fringes. Each bright fringe is a supernumerary bow.
The waves are diagrammatic, do not take them over literally. The only accurate portrayal of them is mathematical.
**  At the angle of minimum deviation, the rainbow angle, the two rays coincide.
***  Geometric optics gives an incorrect prediction because it has been used outside its limits of applicability.
****  The intensity is give by the vector sum of the two waves rather than the scalar sum predicted by classical theory.