Riemannian Geometrical Optics: Surface Waves in Diffractive Scattering

TL;DR

This paper reinterprets geometrical diffraction as a Riemannian obstacle problem, analyzing diffracted rays, caustics, and wave asymptotics using advanced geometric and topological methods, and explicitly computes diffraction amplitudes for a sphere.

Contribution

It introduces a Riemannian geometric framework for diffraction theory, linking Morse theory, homotopy classes, and asymptotic wave approximations, providing new insights into surface wave behavior.

Findings

Proved existence and properties of diffracted rays in Riemannian geometry.

Connected Morse theory and Maslov construction for wave asymptotics.

Explicitly calculated diffraction amplitudes for a sphere.

Abstract

The geometrical diffraction theory, in the sense of Keller,is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is 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 problem for geodesics in a Riemannian manifold with boundary. Then,the axial caustic is here regarded as a conjugate locus, in the sense of the Riemannian 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 diffracted rays. These geometrical results are related to the asymptotic approximations of a solution of a boundary value problem for the reduced wave equation. In particular, we connect the results of the Morse theory to the Maslov construction, which is used to obtain the uniformization of the asymptotic approximations. Then, the border of the diffracting body is the envelope 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 the damping factors associated with the rays which propagate along the boundary. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated.