The standard equation method in the dynamics of structurally inhomogeneous elastic media

Abstract

The development of the standard equation method is examined for studying harmonic wave propagation in stochastically inhomogeneous elastic media. The Helmholtz operator equation describing the propagation of a mean scalar field in a medium is investigated as the standard equation. For an arbitrary correlation function of the elastic coefficients of the medium, the roots of the dispersion equation are found by expanding them in a series in the dispersion parameter, and the eigenvectors of the operator are correspondingly determined approximately. For media of the exponential class, the roots and eigenfunctions of the standard problem are determined exactly. Results obtained in solving the standard problem, are used in investigating wave propagation in elastic madia; the roots and eigenvectors are found in the form of a series expansion in the dispersion. A relationship is set up between the spectra of the elastic operator and the operator of the standard problem. Formulas are obtained to find the mean elastic fields (including the eigenvectors) in terms of the mean standard functions in the form of scattering series. The elastic operator in an isotropic homogeneous body has eigenvectors in the form of longitudinal and transverse waves satisfying the Helmholtz equations. The eigenvalues and vectors of an elastic operator are a set of eigenvalues and vectors of the Helmholtz operator /1/. The elasticity equations do not split into Helmholtz equations or scalar equations in the general case in an inhomogeneous medium. This can be done for high frequencies /2/ and certain particular kinds of inhomogeneities /3/. The method of standard equations /4,5/ enables the form in which the eigenfunctions and eigenvalues of the problem under investigation must be sought, to be determined for stochastically inhomogeneous media, the eauations for the mean field are integro-differential: the medium possesses spatial dispersion. The dispersion equations are transcendental and the roots can only be found approximately. We take the Helmholtz operator equation /6/ as the standard equation, then the roots of the dispersion equation and the eigenfunctions are found exactly for the class of media characterized by an exponential correlation function. The results of solving the standard problem are used in solving the elastic problem, where the eigenvalues of the elastic operator are expressed in terms of the roots and eigenvectors of the standard problem. The standard solution describes qualitatively wave propagation in a structurally inhomogeneous medium, which enables us to speak of the similarity between corresponding dispersion laws and damping /7/. The dimensionless parameters in which the quantities in both problems are expanded are the dispersion and the product of the correlation radius by the wave number /7,8/.