Supernumerary bow formation

Supernumerary Bow Formation: A Phenomenon of Light and Dark Bands

When we look up at the sky after a rain shower, we often marvel at the beauty of a rainbow. But did you know that within the primary bow of a rainbow, there are intricate patterns of light and dark bands known as supernumerary bows? These fascinating phenomena have puzzled scientists for centuries and continue to captivate our imagination. In this article, we will delve into the science behind supernumerary bow formation and explore the factors that contribute to their creation.

The Intriguing Behavior of Light Waves

To understand supernumerary bow formation, we must first grasp the behavior of light waves as they interact with water droplets in the atmosphere. When sunlight enters a raindrop, it undergoes a process called refraction, bending as it passes from air into water and then back into air. This bending causes the different colors of light to spread out, creating a beautiful spectrum.

Within each raindrop, two rays of light are deflected at the same angle but travel different distances before exiting the droplet. As a result, the wave crests of these rays are no longer always in phase when they leave the droplet. In some instances, the wave crests align perfectly, leading to reinforcement and the appearance of bright light. In other cases, the wave crests are misaligned, resulting in cancellation and darkness.

Exploring Interference and Wave Crest Alignment

The phenomenon of interference plays a crucial role in the formation of supernumerary bows. When wave crests of the same amplitude align, they reinforce each other, creating bright fringes or bands of light. Conversely, when wave crests of opposite amplitudes align, they cancel each other out, leading to dark fringes or bands.

As the angle of deviation changes, the pairs of rays alternately reinforce or cancel each other, producing a series of light and dark fringes. Each bright fringe corresponds to a supernumerary bow. These intricate patterns can be observed within the primary bow of a rainbow, adding a captivating complexity to an already stunning natural phenomenon.

Challenging Classical Theory

Classical theory, based on geometric optics, predicts that the light from the two rays will always combine to make the bow brighter in the direction of the deflection. However, this prediction does not align with the observed bands of light and dark within supernumerary bows. Geometric optics fails to explain the intricate patterns that arise, leaving scientists searching for a more accurate explanation.

Embracing Mathematical Representations

To fully comprehend supernumerary bow formation, it is essential to embrace mathematical representations of light waves. While diagrams can provide a visual aid, they should not be taken literally. The accurate portrayal of these wave interactions lies in mathematical models that allow us to understand the underlying physics.

The Rainbow Angle and Coinciding Rays

At the angle of minimum deviation, known as the rainbow angle, the two rays coincide. This alignment contributes to the vivid colors and distinct shape of a rainbow. Understanding this angle and its relationship to supernumerary bows provides further insight into the complexities of atmospheric optics.

The Limitations of Geometric Optics

Geometric optics, while a valuable tool for understanding many optical phenomena, falls short when attempting to explain supernumerary bow formation. This theory, which assumes light behaves as rays, fails to account for the interference patterns and intricate fringes observed within supernumerary bows. Recognizing the limitations of geometric optics allows scientists to explore alternative explanations and delve deeper into the mysteries of atmospheric optics.

Vector Sum versus Scalar Sum

In contrast to classical theory, which predicts a scalar sum of light intensity, the intensity within supernumerary bows is determined by the vector sum of the two waves. This distinction highlights the importance of considering the complex interactions and interferences that occur between light waves within raindrops.

Unveiling the Wonders of Supernumerary Bows

Supernumerary bows add an extra layer of enchantment to the already awe-inspiring phenomenon of rainbows. Their intricate patterns of light and dark bands within the primary bow challenge our understanding of light behavior and showcase the remarkable complexities of atmospheric optics. By unraveling the mysteries behind supernumerary bow formation, scientists continue to deepen our knowledge of the natural world and ignite our curiosity about the wonders that surround us.


Supernumerary bow formation remains a captivating subject within atmospheric optics. By exploring the behavior of light waves, the concept of interference, and the limitations of geometric optics, we gain a deeper understanding of these mesmerizing phenomena. As we marvel at the beauty of rainbows, let us not forget to appreciate the intricate patterns of light and dark bands within supernumerary bows, reminding us of the boundless wonders that await our exploration.

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.

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Reference Atmospheric Optics

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