Propagation of Polarization in Transmission Problems

TL;DR

This paper develops a framework for understanding how polarization states propagate in transmission problems, including elastodynamics and surface waves, by deriving a transport equation and establishing reflection and refraction laws.

Contribution

It introduces a transport equation for polarizations in geometric systems of real principal type and applies it to elastodynamics, extending beyond isotropic elasticity.

Findings

Derived a transport equation for polarization propagation.

Established reflection and refraction laws at boundaries and interfaces.

Applied spectral factorizations to analyze wave behavior.

Abstract

For geometric systems of real principal type, we define a subprincipal symbol and derive a transport equation for polarizations which, in the scalar case, is a well-known equation of Duistermaat and H\"ormander. We apply the transport equation to propagation of polarization in transmission problems of elastodynamics, to interior bulk waves as well as to free (Rayleigh) surface waves. Using spectral factorizations of matrix polynomials having real spectrum, we establish reflection and refraction laws of polarizations at the boundary and at interior interfaces. The results are not limited to isotropic elasticity.