Characterization of Surface Wave Scattering by Surface Breaking Cracks

Various analytical methods based on the real reciprocity relation are applied to the problem of Rayleigh wave scattering by a surface crack. In one formulation, the reflection coefficient observed at the transducer terminals is expressed in terms of an integral over the crack surface of the product of the perturbed and unperturbed fields. This integral is then converted to a volume integral and the Born Approximation is applied. In the other formulation a "Kirchoff" type approach is used such that the effect of the crack is expressed as an equivalent body force distribution. That force distribution is then approximated and normal mode techniques are used to find the scattered field amplitude. The two methods are compared with each other and with the results obtained from geometrical diffraction theory. An experimental procedure is also proposed for the inverse problem. INTRODUCTION New experimental results have been obtained on the surface scattering of Rayleigh ~1aves in the medium and short wavelength regimes during the past year, especially by Khuri-Yakub and co-workers at Stanford, by Tittman at Rockwell, and Quentin and co-workers at the University of Paris. The theoretical calculations we present here apply in the same regime and show agreement as far as the angular scattering from the surface crack is concerned. In the frequency response, the theory does not predict the resonances observed experimentally, but agrees well with the general behavior of the scattering. ....--+-T-SURFACE SF, We basically used two approaches to the problem. In the first one we used Auld's reciprocity relation 1 in volume integral form and applied the Born t.pproximation. In the second one, we replaced the crack by approximately equivalent body sources and found the scattered field using normal mode theory. We called this approach a "Kirchoff" method due to the fact that we calculate the scattering from a set of secondary sources that do not satisfy the boundary conditions everywhere. Born Approximation Consider the general scattering geometry shown in Fig. 1 . Starting from the piezoelectric reciprocity relation in a source free medium v·{~·T2-V2·~+~xH2-E2x~} 0 (1) It can be shown that 1 or21 = 1p {F {~ • T2V2 • "J.} n dS (2) where the field quantities with subscript 1 are those excited by incident power P at terminal 1 in the absence of the crack and those with subscript 2 are excited with an incident power P at terminal 2 in the presence of the crack. Here, ar21 is the change in the transmission coefficient from terminal 1 to terminal 2 due to the presence of the crack. We use primes with subscript 2 to emphasize the presence of the crack. 498 ENCLOSING VOLUME VF Jl....---1--t-SCATT ERER