New solution to Airy's equation for modeling beams near turning points

TL;DR

This paper introduces a generalized wave solution near turning points, transitioning from Airy functions to hyperbolic umbilic functions, to better model wave focusing effects, with potential applications in fusion research.

Contribution

It presents a novel solution that extends classical Airy function models to include hyperbolic umbilic functions for wave behavior near turning points.

Findings

Solution transitions from Airy to hyperbolic umbilic functions with asymptotic matching.

Accounts for wave (de-)focusing due to gradient-index lensing.

Applicable to modeling waves in fusion plasma environments.

Abstract

Here I present a summarised description of the general behavior of waves near turning points. I show that when asymptotic matching to a prescribed incident wavefield is performed, the solution changes from being the Airy function to being more generally the hyperbolic umbilic function. This solution accounts for (de-)focusing of the incident wavefield due to gradient-index lensing in addition to the familiar Airy physics, which importantly do not superimpose. Possible applications to fusion research are discussed.