Asymptotic approximation of the winger function in two-phase geometric optics

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.