Determination of grain-size distribution from ultrasonic attenuation: Transformation and inversion

Although researchers often describe a particular medium with only a single grain size or an average grain size, distributions of grain sizes can have significant effects on the physical characteristics of the material. Ultrasonic attenuation due to energy scattered at the grain boundaries depends on the ultrasonic wavelength, the grain size, and certain material properties such as elastic constants. This paper examines the relationship between the wavelength dependency of ultrasonic attenuation and grain-size distributions. A transformation is developed which solves a general class of Fredholm equations of the first kind. The equation relating attenuation to grain-size distributions belongs to this class of equations. The transformation allows attenuation to be calculated by discrete convolution, and provides a method for estimating the grain-size distribution directly from attenuation measurements.