Effects of cross correlations between inhomogeneities of the parameters of an isotropic medium on the spectrum and damping of elastic waves

Abstract

The dispersion and damping laws have been investigated for elastic waves in an isotropic medium with one- and three-dimensional inhomogeneities of the density p(x) of the material and the elastic force constants μ(x) and λ(x) with allowance for the cross correlations between these inhomogeneities. It has been demonstrated that the positive cross correlations between μ(x) and λ(x), as well as the negative cross correlations between p(x) and μ(x) or p(x) and λ(x), lead to an enhancement of the modification of the dispersion law and an increase in the damping of waves. The positive cross correlations between p(x) and μ(x) or p(x) and λ(x), as well as the negative cross correlations between μ(x) and λ(x), result in the opposite effects: a weakening of the modification of the dispersion law and a decrease in the damping. An analysis of the results obtained in this paper and in our recent work [15] has made it possible to formulate the general regularity of the effects of cross correlations, irrespective of the physical nature of the waves: the effects of cross correlations between inhomogeneities of any two parameters of the material on the wave spectrum depend on whether both parameters related by the cross correlations belong to the same part of the Hamiltonian (i.e., if they both belong to either the kinetic part or the potential part of the Hamiltonian) or they belong to different parts of the Hamiltonian. The positive cross correlations lead to a greater modification of the dispersion law and to an increase in the damping of waves in the former case and to a decrease in these characteristics in the latter case. Correspondingly, the negative cross correlations in each of these cases result in the opposite effects. This regularity has been explained qualitatively.