Multivalued solutions to the eikonal equation in stratified media

In the present paper we study the geometric properties of the multivalued s olutions to the eikonal equation and we give the appropriate classification theorems. Our motivation stems from geometrical optics for approximating h igh frequency waves in stratified media. We consider the case of a fixed Ha miltonian imposed by the medium, and mie present the geometric framework th at describes the geometric solutions, using the notion of Legendrian immers ions with an initial point source or an initial smooth front. Then, we stud y the singularities of the solutions in the case of a smooth or piecewise H amiltonian in a boundaryless stratified medium. Finally we study the singul arities of the solutions in a domain with a boundary that describes the pro pagating field in a waveguide.