Ray tracing via Snellius

A wave travelling between two media denoted by different wave propagation velocities is subject to refraction at the interface between the media. The refraction is regulated by the Snellius law, where the interface is assumed infinitesimally thin. The jump in propagation velocity at the interface results in a discontinuous propagation direction for the wave. We consider a continuously changing medium, where the wave propagation velocity is assumed to be a continuous field. We reduce the Snellius law to its linear expansion at the interface between two regions of the medium with infinitesimally different propagation velocities. The linearised Snellius law connects the curvilinear coordinates associated with the propagation process from a point source and the spatial distribution of the propagation velocity. The coordinates map the rays evolving from the source and the wavefronts, orthogonal to the rays. Curved rays determine local osculating planes, spanned by the tangent to the ray and the gradient of the propagation velocity. The wavefront curvature is determined parallel to the tracing of each ray. Intersections of the wavefront are considered, with the osculating plane and with the longitudinal plane of the ray. For curved rays, the determined wavefront curvatures are different for the different planes. A numerical implementation of the model is used to approach an exemplary test case, regarding sound radiation in a stratified medium.