From AMS Glossary
(Or supernumerary bows.) A series of fine weakly colored bows that can frequently be seen just inside the primary rainbow.
When formed in rain showers, where there is a broad distribution of drop sizes, these bows are mainly seen near the top of the rainbow arch, but fade toward the vertical portions of the primary bow. They owe their name (beyond the prescribed number) to the fact that an explanation of rainbows based upon a treatment of light as a series of rays is incapable of accounting for them. However, when light is treated as a wave, the supernumerary bows become higher-order interference maxima, for which the primary bow is but the first maximum. In this sense, the supernumerary bows are as much a part of the primary bow as are, say, its colors.