  We propose a renormalization process of a two phase WKB solution, which is
based on an appropriate surgery of local uniform asymptotic approximations of
the Wigner transform of the WKB solution. We explain in details how this
process provides the correct spatial variation and frequency scales of the wave
field on the caustics where WKB method fails. The analysis has been thoroughly
presented in the case of a fundamental problem, that is the semiclassical Airy
equation, which arises from the model problem of acoustic propagation in a
layer with linear variation of the sound speed.
