Application of Fourier Elastodynamics to Direct and Inverse Problems for the Scattering of Elastic Waves from Flaws Near Surfaces

Abstract

In ultrasonic inspection one frequently encounters situations in which flaws lie too close to surfaces (or interfaces) for the applications of the conventional theory of scattering in an infinite host medium. One approach, pursued by Varadan, Pillai, and Varadan, is to rework scattering theory with a Green function suitably modified for the presence of surfaces. Our approach uses a Fourier elastodynamics formalism in which the scattering process is represented by the generalized transfer function for a slab of material containing the flaw. A sufficiently simple elastodynamic system can be built up of separately characterizable subsystems defined by intervals on the z-axis. Fortunately, it turns out that, in the case of propagating waves the generalized transfer function can be simply related to the scattering amplitude, the representation of scattering in the conventional theory. In the case where evanescent waves are present, the generalized transfer function can also be simply related to the scattering amplitude in which the incident and scattered direction $\mathop e\limits^{{ \to _1}}$ and $\mathop e\limits^{{ \to _S}}$ have been analytically continued from real to complex forms. In some cases it may be a good approximation to ignore completely the effects of evanescent waves. An integral equation has been derived whose solution gives the scattering from a system composed of a flaw and a nearby surface or interface. The computational results pertain to the scattering from spherical voids and inclusions in plastic at various distances from the plastic-water interface with the incident wave propagating through the water.