  A wave near an isolated turning point is typically assumed to have an Airy
function profile with respect to the separation distance. This description is
incomplete, however, and is insufficient to describe the behavior of more
realistic wavefields that are not simple plane waves. Asymptotic matching to a
prescribed incoming wavefield generically introduces a phasefront curvature
term that changes the characteristic wave behavior from the Airy function to
that of the hyperbolic umbilic function. This function, which is one of the
seven classic 'elementary' functions from catastrophe theory along with the
Airy function, can be understood intuitively as the solution for a linearly
focused Gaussian beam propagating in a linearly varying density profile, as we
show. The morphology of the caustic lines that govern the intensity maxima of
the diffraction pattern as one alters the density lengthscale of the plasma,
the focal length of the incident beam, and also the injection angle of the
incident beam are presented in detail. This morphology includes a
Goos-H\"anchen shift and focal shift at oblique incidence that do not appear in
a reduced ray-based description of the caustic. The enhancement of the
intensity swelling factor for a focused wave compared to the typical Airy
solution is highlighted, and the impact of finite lens aperture is discussed.
Collisional damping and finite beam waist are included in the model and appear
as complex components to the arguments of the hyperbolic umbilic function. The
observations presented here on the behavior of waves near turning points should
aid the development of improved reduced wave models to be used, for example, in
designing modern nuclear fusion experiments.
