A result in equivalent medium theory

Francis Muir

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ABSTRACT

A point Green's function for a heterogeneous, one-dimensional elastic material is expanded in a Taylor series and examined term by term. As expected, the first-order term gives the kinematic result predicted by Schoenberg/Muir and like theories. However, the second-order term of this expansion (subject to a minor stationarity constraint on the heterogeneity) is unexpected in that it depends on first-order averages alone. A consequence is that ensemble average responses for materials with the same statistical properties show no second-order transmission loss. Energy transferred from the direct to the scattered field must involve higher order terms in the Taylor series expansion. It follows that there is no simple `central limit' asymptotic solution, and no equivalent homogeneous visco-elastic medium.