  One is led to study the asymptotic domain in which the parameter \(\varepsilon\) in Schema R approaches 0. In the example above, this is the short wavelength limit. Michael Berry (1980; 1990; 1994a; 1994b) has done much research on this and other asymptotic domains. He has found that in the asymptotic borderlands between such theories there emerge phenomena whose explanation requires in some sense appeal to a third intermediate theory. This is a claim (Batterman 2002) that when taken literally, has raised a number of hackles in the literature. However, understood in terms of the mathematics of characteristics and wavefronts, as was originally intended, the current author believes some of the debates are misdirected. The emergent structures (the rainbow itself is one of them) are not fully explainable either in terms of the finer wave theory or in terms of the ray theory alone. Instead, aspects of both theories (through asymptotic investigation of the wave equations) are required for a full understanding of these emergent phenomena. 