In ultrasonic nondestructive evaluation (NDE) testing, sound waves are beamed into a material. Any flaw that may be present in the material will interrupt the incident sound beam and give rise to scattered waves. One important objective of ultrasonic NDE testing is to use this scattered response to determine the nature of the flaw present. In this work, elastodynamic ray theory is employed to obtain both zeroth and first order asymptotic solutions for the far-field leading edge response of a volumetric scatterer caused by incident longitudinal and transverse waves. The explicit results given here for voids have some important implications for the validity of equivalent flaw sizing schemes, such as the inverse Born approximation, and, as shown, can form the basis for new equivalent flaw sizing methods. These results also can be used to develop a more exact flaw sizing algorithm that does not involve any a priori assumption regarding the flaw shape. Other details contained in this work include the asymptotic expansions of integrals defined on arbitrary curved surfaces and a unified treatment of the curvature tensors of the reflected and refracted wavefronts; these results are of fundamental importance for the solution of scattering problems via ray theory.