Kirchhoff Approximation Revisited—Some New Results for Scattering in Isotropic and Anisotropic Elastic Solids

Through a series of numerical studies that compare the Kirchhoff approximation to more exact scattering theories, it is demonstrated that the Kirchhoff approximation can accurately predict the pulse–echo peak-to-peak responses of spherical pores and circular cracks in isotropic media over a very wide range of cases that extend well beyond the limits normally associated with this approximation. The reason for this good agreement is shown to lie in the ability of the Kirchhoff approximation to model accurately the very early time response of the flaw. It is also shown that in the Kirchhoff approximation the pulse–echo response of an arbitrary traction-free scatterer in an isotropic elastic solid is identical to the same response obtained using a scalar (fluid) scattering model. This leads to simple analytical expressions for the pulse–echo far-field scattering amplitude of some canonical geometries (circular cracks, spherical voids, cylindrical holes) and to simplified numerical expressions for more general scatterers. For general anisotropic volumetric flaws in a anisotropic elastic solid, it is shown that a high-frequency asymptotic evaluation of the Kirchhoff approximation yields an explicit analytical expression for the pulse–echo leading-edge response of the flaw. Explicit expressions are also given for the pitch–catch response of an elliptical-shaped flat crack in a general anisotropic solid.