Weighted Residual Method for Diffraction of Plane P-Waves in a 2D Elastic Half-Space Revisited: On an Almost Circular Arbitrary-Shaped Canyon

Scattering and diffraction of elastic in-plane P- and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P- and SV-scattered-waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattering and diffraction of these in-plane waves on an on an almost-circular surface canyon that is arbitrary in shape.