SINGLE SCATTERING AND DIFFUSIVE LIMITS OF THE ULTRASONIC RADIATIVE TRANSFER EQUATION Joseph A. Turner and Richard L. Weaver Department of Theoretical and Applied Mechanics University of Illinois at Urbana-Champaign Urbana, Illinois 61801 INTRODUCTION Microstructural characterization of polycrystalline metals is often performed using ultrasonic backscatter techniques [1,2,3]. The backscattered diffuse or incoherent signals, also called grain noise, contain microstructural information about grain size, orientation, and composition which is useful for materials characterization. The grain noise can also interfere with flaw detection. Understanding the scattering mechanism is thus important. When the time and/or length scales of a backscatter experiment are long compared with the time and length scales of the random scattering events occurring within the medium, multiple scattering effects become important. The multiple scattering problem has two limits. In the limit of early times or weakly scattering materials, and for experiments involving focussed transducers, a single scattering approximation has been successful for modeling grain noise [1,2,3]. This assumption implies that the incident wave strikes, on average, a single scatterer before being detected. In the opposite limit, at late times after the energy has scattered many times, the behavior is governed by a diffusion equation [4,5]. The intermediate multiple scattering regime has not, however, been fully utilized for microstructural characterization possibly because of the lack of an adequate theory with which to describe corresponding experiments. A method was recently proposed to model the multiple scattering of diffuse ultrasound in polycrystalline materials [6,7]. It has its foundations in optical radiative transfer theory which was developed to quantify the diffuse scattering of light from planetary and stellar atmospheres. The ultrasonic radiative transfer equation (URTE) is derived for a polycrystalline medium through an examination of ensemble averaged responses of the elastic wave equation by use of the Bethe-Salpeter equation [6,7]. The URTE is expected to be valid within the limit of its primary assumption that the material heterogeneity is weak. Many materials of interest satisfy this requirement and thus are expected to be modeled appropriately by the URTE. Both the single scattering and diffusive limits of the URTE are discussed in this paper. The URTE is shown to have the appropriate behavior in each limit which provides confidence in the URTE as a descriptor for multiply scattered diffuse ultrasound. The range of validity of each limit is also examined. It is shown that intuition regarding the time scales of the problem may not be entirely correct for predicting the valid range for each limit of the multiple scattering problem. ULTRASONIC RADIATIVE TRANSFER EQUATION Consider a polycrystalline specimen with Voigt average longitudinal and transverse wave speeds, C L and Cn immersed in a water bath with wave speed cf" The fourth rank elastic moduli tensor is assumed to be of the form Review of Progress in Quantitative Nondestructive Evaluation. Vol. 14 Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New Yark, 1995 75