Raindrops' target practice

Every raindrop splits the incoming sunlight into light of different wavelengths (ie colours), each colour being refracted (bent) by a slightly different angle; the droplet then reflects the colours back at various angles. From 0 (horizontal) through steeper downward angles until about 40 the reflected colours are disorganised and jumbled together to produce what we call "white light". Then at 40 many violet rays are concentrated into a bright beam; at 41 there is a green beam; and so on as far down as a red beam at 42. At any angle greater than 42 no light is reflected at all.

So, starting from the top, each drop reflects:
  • a broad band of white light (W), then
  • an inverted rainbow (violet, ... , green, ... , red) (VGR), then
  • nothing (X)

Let's suppose the sun is low in the sky behind us, so its rays are almost parallel to the ground, while we are looking in the direction of the falling rain. Now, instead of looking at the rainbow from our side, let's look at things from the individual raindrops' point of view. Imagine they are all trying to get seen by shooting reflected light rays at the same target: our eye. In each case we'll quote the droplet's report and then give a few words of explanation:

  1. A droplet fairly high in the sky: "Target at 45: it's in my X region, so I can't hit it. Oh, wait a minute, Target No 2's at 42—I've just got him with a red."
    [We can't see any reflected light at all from this droplet; but someone else's eye—"Target No 2"—looking out through an upstairs window and observing the droplet at a shallower angle, sees it shining red.]

  2. A droplet lower than the first: "Target at 42: in R region. Shot it with a red!"
    [This droplet appears red.]

  3. Another droplet even lower down: "Target at 40: in my V region. Got it with a blast of violet! Target No 2 now in W region."
    [This droplet appears violet to us, but just "bright"—already below the rainbow—to the upstairs observer.]

  4. A droplet quite low down; "Target at 35, in my W region. Gave it a burst of white!"
    [Just looks "bright".]
If you multiply each of these typical droplets by several million, and remember that there is a whole spectrum (range) of colours between red and violet, you can now understand why the "target" eye sees a spectrum of colours in the sky at an angle of about 40 - 42 to the sun's rays. For further explanation and diagrams showing why we see a bow rather than, say, a horizontal band, see Beverly Lynds' excellent website. Hint: the ends of the rainbow must be on the horizon at about 40 to your right and left.

The important thing to understand is that, although each droplet reflects an entire spectrum of colours, we will see only one of the colours from any given droplet. Which colour we see depends on the angle at which we are viewing that droplet—in other words, its position in the sky.

The explanation may seem to make good sense, except for one fact that you've had to take on trust. Just why does each droplet reflect the sunlight in the form of an inverted rainbow, with no light being reflected at a greater angle than 42? To understand this we need to look in detail at the geometry (and trigonometry) of refraction and reflection within a raindrop: and a convenient way of doing this is to use an interactive Excel program.

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