Propagation of polarization in transmission problems

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 Hörmander. 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.