Analytic approximation to the scattering of antiplane shear waves by free surfaces of arbitrary shape via superposition of incident, reflected and diffracted rays

The scattering induced by surface topographies of arbitrary shapes, submitted to horizontally polarized shear waves (SH) is studied analytically. In particular, we propose an analysis technique based on a representation of the scattered field like the superposition of incident, reflected and diffracted rays. The diffraction contribution is the result of the interaction of the incident and reflected waves, with the geometric singularities present in the surface topography. This splitting of the solution into different terms, makes the difference between our method and alternative numerical/analytical approaches, where the complete field is described by a single term. The contribution from the incident and reflected fields is considered using standard techniques, while the diffracted field is obtained using the idea of a ray as was introduced by the geometrical theory of diffraction. Our final solution however, is an approximation in the sense that, surface-diffracted rays are neglected while we retain the contribution from corner-diffracted rays and its further diffraction. These surface rays are only present when the problem has smooth boundaries combined with shadow zones, which is far from being the typical scenario in far-field earthquake engineering. The proposed technique was tested in the study of a combined hill-canyon topography and the results were compared with those of a boundary element algorithm. After considering only secondary sources of diffraction, a difference of 0.09 per cent (with respect to the incident field amplitude) was observed. The proposed analysis technique can be used in the interpretation of numerical and experimental results and in the preliminary prediction of the response in complex topographies.