The problem of effective parameters of a mixture of two isotropic dielectrics distributed in space-time and the conservation law for wave impedance in one-dimensional wave propagation

Abstract

The problem of effective parameters of a composite medium assembled from the original constituents distributed in space–time is formulated in covariant tensor form. I introduce the fourth rank tensor of material constants using Maxwel's system for a moving dielectric medium as a model example. For one–dimensional wave propagation, if a mixture is composed from two dielectrics with the same ratio ϵ/μ of permittivity ϵ to permeability μ, then the ratio E / M of an effective permittivity to an effective permeability of the mixture will preserve the value ϵ/μ. This statement may be rephrased as the conservation law for the relevant wave impedances √μ/ϵ this is similar to the law known for two–dimensional polycrystals in an analogous elliptic situation. The tensor concept developed for a dielectric medium is based on the idea of a relativistic invariance of Maxwel's system. The idea of relativistic invariance is fundamental for an adequate description of the effective parameters of any material assemblage in space–time regardless of its physical embodiment. Problems related to structural vibrations or acoustics could be completely understood on the basis of the relevant relativistic equations.