Math's Eye View

# The real problem with rainbows

## Forget the colours: why is a rainbow that shape?

Posted Mar 02, 2011

In our part of the United Kingdom the winds often come from the west, and if a rainstorm passes overhead in the afternoon or evening it's always worth a look out of the front window towards the east, in the expectation of seeing a rainbow. Sometimes the spectacle is especially dramatic, with a double rainbow, the inner one brighter than the outer.

We all learned about rainbows at school, and what nearly all of use learned is that rainbows form because sunlight is composed of many different colours, and a drop of water splits white light into these components. Then we were shown how a prism does precisely that, and may even have done experiments to show that water refracts light.

It took about twenty years before I realised that this is not the most interesting question about rainbows. It may explain the colours, though I'm about to argue that it is inadequate even for that. But it definitely fails to explain --- does not even address --- the rainbow's shape. And in fact, the shape is crucial to any genuine explanation of the colours, as well.

I understand why we weren't told about the shape, because the explanation requires some sophisticated geometry and wouldn't really have been suitable for children of my age. But we should have been told that the explanation was only part of the story, and when we were older, we should have been told the rest. It would have made our mathematics and physics lessons much more interesting.

The problem with the usual explanation is straightforward, once someone points it out. Agreed, each droplet of water acts like a prism, producing a series of colours. But a rainbow involves millions of droplets, spread over a large volume of space. Why don't all those coloured rays get in each other's way, producing a muddy smeared-out pattern? Why do we see a concentrated band of light?

The answer lies in the geometry of light passing through a spherical droplet. Imagine a tight bunch of parallel light rays, from the Sun, encountering a single tiny droplet. Each ray is really a combination of rays of many distinct colours, so consider --- for the moment --- just one colour. That incoming light bounces round inside, and is reflected back by the droplet. What happens is surprisingly complex, but the part that creates the rainbow consists of rays that hit the front of the droplet, pass inside and are refracted (bent) by the water --- the prism analogy --- hit the back of the droplet and are reflected, and finally pass out again through the front, being further refracted.

The geometry of this process reveals that the light emitted by the droplet is mostly focused at a specific angle to the incoming rays. So most of that light can be thought of as forming a bright cone. Every droplet that receives sunlight produces the same cone. So when we look into the sky, most of the light that we see comes from those droplets whose cones happen to meet our eye. A bit more geometry shows that these droplets lie on a similar cone, with our eye at the tip, pointing in exactly the opposite direction. So what we observe is a bright circular arc. The other raindrops don't smear that out because hardly any of their light hits our eye.

Why incoming rays from many droplets form a cone

Different colours produce arcs of slightly different sizes, but with the same centre, the point on the far side of our eye in line with the Sun. This is why the rainbow is a coloured band.

What about the second rainbow that I sometime see? That is created by light that bounces round the droplet several times before escaping back towards the observer, and this also forms a cone, but the angle is different. The geometry also predicts that the region between the two rainbows should be darker than the rest of the sky, and once you know to look for this, it's usually obvious.

The mathematics involved in this explanation of the rainbow is known as singularity theory, and it has many other applications, ranging from capsizing ships to the formation of galactic clusters. So whenever I see a rainbow I am reminded of the power and generality of mathematics, and its inner beauty. Which, for me, enhances a different kind of beauty: the inspiring elegance of the rainbow itself.