  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.
