PROPAGATING MODES OF A HETEROGENEOUS, MACRO-HOMOGENEOUS CONTINUUM

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Abstract
A method is developed to calculate the dispersion of propagating elastic modes in heterogeneous, macro-homogeneous media. A time-periodic solution of the wave equation is defined inside a volume V by imposing a plane-wave character on the boundary Σ completely surrounding V. The equation is transformed into an integral equation with use of a Green's function for a " model-medium ", with constant elastic coefficients and density. It is brought in such a form, that the limit V → ∞ can be taken. A uniform" effective medium " is defined, in terms of the solution of this integral equation, by taking the space average of the wave equation. This gives an implicit equation for the dispersion relation. An approximation analogous to the CPA is obtained. The integral equation is approximated in terms of an imbedding method, through which the average wave amplitude is obtained as an ensemble average of solutions for small neighbourhoods embedded in the model medium. By imposing the condition that the model-medium and the effective medium are identical, the errors of imbedding are minimized. This yields a self-consistent iterative procedure. The case of long wavelength or low density contrast leads to simplified equations analogous to those for the static limit. This case can be approximated with use of correlation functions, as shown elsewhere

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