Riemannian geometrical optics: Surface waves in diffractive scattering

The geometrical diffraction theory, in the sense of Keller, is here reconsi dered as an obstacle problem in the Riemannian geometry. The first result i s the proof of the existence and the analysis of the main properties of the "diffracted rays", which follow from the non-uniqueness of the Cauchy prob lem for geodesics in a Riemannian manifold with boundary. Then, the axial c austic is here regarded as a conjugate locus, in the sense of the Riemannia n geometry, and the results of the Morse theory can be applied. The methods of the algebraic topology allow us to introduce the homotopy classes of di ffracted rays. These geometrical results are related to the asymptotic appr oximations of a solution of a boundary value problem for the reduced wave e quation. In particular, we connect the results of the Morse theory to the M aslov construction, which is used to obtain the uniformization of the asymp totic approximations. Then, the border of the diffracting body is the envel ope of the diffracted rays and, instead of the standard saddle point method , use is made of the procedure of Chester, Friedman and Ursell to derive th e damping factors associated with the rays which propagate along the bounda ry. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated.