Diffusivity of ultrasound in polycrystals

The diffusivity of ultrasound in an untextured aggregate of cubic crystallites is studied theoretically with a view towards nondestructive characterization of microstructures. Multiple scattering formalisms for the mean Green's Dyadic and for the covariance of the Green's Dyadic (and therefore for the energy density) based upon the method of smoothing are presented. The first-order smoothing approximation used is accurate to leading order in the anisotropy of the constituent crystallites. A further, Born, approximation is invoked which limits the validity of the calculation to frequencies below the geometrical optics regime. Known results for the mean field attenuations are recovered. The covariance is found to obey an equation of radiative transfer for which a diffusion limit is taken. The resulting diffusivity is found to vary inversely with the fourth power of frequency in the Rayleigh, long wavelength, regime, and inversely with the logarithm of frequency on the short wavelength, stochastic, asymptote. The results are found to fit the experimental data.