Abstract
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.
Highlights
This paper studies the subject on the diffraction of in-plane Plane longitudinal- (P-)waves in an elastic half-space by arbitrary-shaped canyons using the weighted residual method
On the flat half-space surface, the stress-free boundary conditions can be satisfied for the incident P and the reflected Pand SV-wave by utilizing the boundary conditions: σy = τyxy=0 = 0, σθ = τθrθ=0,π = 0. This results in the following set of equations for the reflected P-wave and SV-wave potentials defined in the x-y plane as Reflected Plane P-Waves φr = K1 exp ikα (x cos θα + y sin θα), (6)
The results for the case of the semicircular canyon can be compared theoretically by matching the boundary condition equations derived from the weighted residual method with those derived from Lee and Liu’s exact closed-form solution [6]
Summary
This paper studies the subject on the diffraction of in-plane P-waves in an elastic half-space by arbitrary-shaped canyons using the weighted residual method. In 2014, Lee and Liu analyzed the harmonic motion induced by an incident P-wave for a two-dimensional diffraction around a semicircular canyon in an elastic halfspace using an analytic solution to satisfy the zero-stress boundary conditions [6]. The new method uses only one set of scattered P- and SV-waves, using the method of Lee and Liu [6] to automatically satisfy the freestress boundary conditions at the half-space surface Using this improved weighted residual method, the results for Lee and Liu’s semicircle were verified and new results for an ellipse, trapezoid, and rectangle are presented.
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