Determination of Grain-Size Distributions from Ultrasonic Attenuation

The overall research goal of this project is the nondestructive measurement of grain-size distribution parameters using ultrasonic attenuation. Ultrasonic attenuation α is dependent upon the sound wavelength (λ), the size of the grains (D), and in many cases elastic constants and sound velocities of the material. Assuming that multiple scattering can be ignored, the expression for the wavelength dependence of the attenuation is 1$$ \alpha (\lambda ) = \int\limits_0^\infty {N(D)\,\alpha (\lambda, D)\,} \,dD $$ where N(D) is the grain-size distribution. In previous work [1] the sizes were assumed to be distributed following a power-law with exponent γ: 2$$ N(D) = K{D^{ - \gamma }},0 < D < \infty $$ Different justifications for this assumption have been provided [1]. After substituting the power-law expression for the grain-size distribution, the attenuation was shown to be a power-law: 3$$ \alpha (\lambda ) = K'\,{\lambda^{ - \gamma }} $$