Spatial autocorrelation functions for calculations of effective propagation constants in polycrystalline materials

Spatial autocorrelation functions effect the scattering processes and the effective propagation constants of elastodynamic waves in single-phase, polycrystalline media. The spatial autocorrelation function W(r1,r2) is the probability that the two points r1 and r2 are in the same grain. A ‘‘grain’’ is a discrete, homogeneous domain within an inhomogeneous medium which is composed completely of such domains. One way to calculate W(r1,r2) is empirically. It employs cord lengths measured on a micrograph, and thus provides a means of determining the autocorrelation function for real materials. A second way to calculate W(r1,r2) is analytically, and employs expressions for the probability density of the cord lengths. Analytical calculation allows the use of assumptions about the geometrical nature of the grains in a material to be incorporated into theoretical expressions for effective propagation constants, and gives measures of effective ‘‘grain size.’’ Two specific assumptions have been important for such calculations, and so serve as examples in the present paper: that the cord lengths have Poisson statistics, and that the grains are spherical. Analytical calculation is also useful for the prediction of errors caused by empirically calculating W(r1,r2) from a finite data set of measured cord lengths.