Since a fraction of light always undergoes additional internal reflection instead of re-entering the air, a single drop of water is theoretically capable of giving an infinite number of rainbows. But the higher the order, the fainter the rainbow. Now, rainbows up to k=13 (i.e. with light reflecting 13 times inside the drop before re-entering the air) have been observed from a single drop of water suspended from the end of a wire in the lab. And rainbows up to k=19 have been observed in thin, falling streams of water, again in the lab.
But the rainbows with k=3 and beyond are observed only in lab. To my knowledge they have never been observed outdoors. Here's why:
You'd see the primary and secondary rainbow only if the Sun is behind your. But the third (k=3) and fourth (k=4) rainbows are formed behind your back, the fourth just above the third. You'll never see them, as they are much fainter than the first two, and you'd have to look in the direction of the Sun, where a glare from the sunlight will make an observation impossible.
http://www.jal.cc.il.us/~mikolajsawicki/ex_optics.html
Originally Posted by KT.
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