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

## TL;DR

This paper introduces a renormalization method for two-phase WKB solutions using local asymptotic approximations of the Wigner transform, effectively capturing wave behavior near caustics where traditional WKB fails.

## Contribution

It presents a novel renormalization process for two-phase WKB solutions based on Wigner transform approximations, improving accuracy near caustics.

## Key findings

- Provides a detailed analysis for the semiclassical Airy equation
- Demonstrates the method's effectiveness in modeling wave behavior at caustics
- Enhances understanding of wave propagation in layered media

## Abstract

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.