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For the explanation of the supernumerary arcs, the principals of classical optics fail. The occurance of minima and maxima in the light distribution can be explained by interfering rays which experience a difference in the length of their paths while passing through the water droplet. Those rays which differ by half of their wavelengths interfer destructively and lead to the minima of light between the supernumerary arcs. However, also effects of polarization of deflected rays, sizes and oscillations of the water droplets have to be taken into account to properly model the supernumerary arcs.
More detailed information about rainbows can be found here.
M.G.J. Minnaert: "Light and Color in the Outdoors", Springer-Verlag
Nussenzveig H. M.: "The Theory of the Rainbow" in: Light from the Sky, Scientific American, 1977