Parabolic equation and Pearcey-type integral for a transmitted wavefield

High-frequence wavefield of a point source situated near an interface of two homogeneous media and placed in the faster medium is considered. We are interested in description of a vicinity the critical ray in the slower medium. The description is given via the parabolic equation method. The result involves a Pearcey-type integral. The approach is not based on the Fourier method and can be generalised to inhomogeneous media with curved interfaces.