It's been raining a lot lately in the bay area of California (thankfully! keep it coming rain gods!) which puts me and my dad in a contemplative mood. The rain has given us something to contemplate: rainbows. Every time we see a rainbow my dad is amazed by their beauty. He asked the titled question: Why are rainbows always in an arch shape? We know that there can be rainbows that have a different shape when, for example, they form in oily water or when they are in bubbles. However, we want to know about the traditional rainbows that you see in the sky.
I came up with a partial explanation to why I think this happens. I do not expect this to be right because I didn't do any research or any deep thinking about rainbows. I don't even know how rainbows are formed! Additionally, I stress that this is only a partial explanation. With that in mind, here it is.
When you see a rainbow in the sky, what you are seeing is different wavelengths of light from the sun enter rain droplets and exit in an organized fashion. A little more precisely, different wavelengths of light are reflected (or refracted? IDK!) in distinct directions when they exit the rain droplet. Each rain droplet has all of the wavelengths of light exiting it, but only one wavelength is reflected in the direction of my eyes. The specific direction is dictated by the position of the rain droplet relative to the position of my eye. When you're in a car and you're driving toward a rainbow, your position is changing, so the droplets you see a moment before the next are no longer reflecting the same wavelength of light. But you normally still see the same rainbow. It must then be the case that that rainbow was already there, waiting to be seen by some ponderer.
Of course, none of this explains the arch shape. I suspect it can be explained mathematically. But I don't know any of the formulas. That isn't to say that you shouldn't put a good formula in your explanation if it helps. I am knowledgeable in mathematics up to linear algebra. My dad is not on the other hand (I don't think he can even do algebra 1). Soooo yeah...
Anyways, enough of my mumbling. Why is it the shape that it is?
