[Musicforawhile (OP)]

So I am given to understand that the sunlight refracts through the water droplets like a light ray travelling through a prism and being split into the colours of the rainbows; but if this happens to each raindrop then why aren't there thousands of tiny rainbows in the sky rather than one (or two) big one(s)?

Are these decent books on light and rainbows?

 Feynman's Six Easy Pieces
Catching the Light: The Entwined History of Light and Mind by Arthur Zajonc published in 1995.

[chiralSPO]

I think each droplet of water produces a very faint but large rainbow. The apparent size of the rainbow is determined by the angles that the light is deflected and how far away the observer is from the droplet, so if all of the droplets are causing the same angles of deflection (there is some variation based on droplet size, but they are usually pretty close), and all the droplets are essentially the same distance from the observer (also a pretty safe bet if they are all way up in the sky), then all of the faint but roughly equally sized rainbows will add up to one bright rainbow of the same apparent size.

[RD]

... but if this happens to each raindrop then why aren't there thousands of tiny rainbows in the sky rather than one (or two) big one(s)?

Each person sees their own "personal" rainbow of the great many which are possible ... http://www.atoptics.co.uk/rainbows/primcone.htm

[PmbPhy]

So I am given to understand that the sunlight refracts through the water droplets like a light ray travelling through a prism and being split into the colours of the rainbows; but if this happens to each raindrop then why aren't there thousands of tiny rainbows in the sky rather than one (or two) big one(s)?

Each drop reflects light by an angle amount determined by the wavelength of the light impinging on it. Therefore the colors you see in the rainbow coming from the direction you see them are a direct result of those angles. See:
http://www.canon.com/technology/s_labo/light/001/02.html

[yor_on]

Had such a lovely book on light by a female author, that I think (?) either was a professor, or university teacher, in chemistry. Living and teaching in the UK but with a Italian (?) name, if I now remember correctly? Discussing both the history and physics. One of those books you don't want too lose, which you then naturally as a direct effect of that wish will do, sooner than later. If I ever find it again I will post it Or if anyone else recognize it?

This one is also about rainbows. http://eo.ucar.edu/rainbows/

[evan_au]

I would say that there are millions of rainbows, one for every raindrop in the sky.

But all of these diffracted photons exit the raindrop in different directions, and the vast majority of them go in directions that do not include your eyeballs, so you never see them.

If some photons from one raindrop do reach your eye, they will be of a narrow wavelength range (ie a single color) - because any photons of different wavelength would have been diffracted in a different direction, so you don't see them.

All of the rainbows from all of the raindrops seem to form a uniform whole, with colors determined by the angle between the raindrop, your eye and the Sun.

Of course as individual raindrops fall, they will diffract different colors into your eyes at different times, because the angle between the raindrop, the Sun and your eye changes over time.

[CliffordK]

This one is also about rainbows. http://eo.ucar.edu/rainbows/

That has a good description in the middle of the article.  Which is almost what I figured out before reading it.  So, I'll go ahead with the description I was working out, but refer you to that article for details.

 [ Invalid Attachment ]

So, say you consider a single droplet.  Light enters on one side, refracts, reflects off the back, and refracts again, which then separates the light to the color spectrum.

Now, consider two observers, Bob & Jane. 

If Bob & Jane are at the right distance apart, one will see the droplet as blue, and the other will see the same droplet as red.  Other droplets in other directions may not be seen at all, or at least the path of refracting through the surfaces and reflecting of of the back of the drop would not be visible (in the visible light spectrum)

It turns out this pathway of refraction/reflection projected onto tiny spheres would create a circle around the droplet of the point of maximum reflection off the back. When observed from a distance, each droplet then reflects in a single color back to the observer if it is in the right position to refract/reflect the color.

So, your rainbow is like the French Impressionist Paintings, where each line or color in the rainbow is made up of millions of tiny dots, each dot from a single raindrop that is in turn being visualized by the observer as a single color.