Journal of Earthquakes

Volume 2015 (2015), Article ID 543128, 21 pages

http://dx.doi.org/10.1155/2015/543128

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

Astani Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089-2531, USA

Received 2 May 2015; Accepted 12 July 2015

Academic Editor: Antonio Morales-Esteban

Copyright © 2015 Vincent W. Lee and Heather P. Brandow. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

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. It presents a solution for any arbitrary-shaped canyons where the depth of the canyon is approximately half the width of the canyon, such as a semicircle, ellipse, or trapezoid.

Researchers continue to study the effects of scattering and diffraction of waves on two-dimensional canyons in an elastic, isotropic, and homogeneous medium. These studies, which assist researchers to understand earthquake ground motions in and around topographic features, initially, addressed incident SH-waves [1–3]. In solving SH-waves, the method of images, which assumes equal and opposite scattered waves upon reflection, has been used [4]. Because of the mode conversion, an incident P-wave, which produces both a reflected P- and SV-wave, is more complex. Thus, the P- and SV-waves diffraction problems cannot be solved using the method of images. Much of the latest and most recent work on diffraction and soil-structure interaction for topographies involving analytic methods is so far thus limited mostly to SH-wave problems [5].

Recently 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 half-space using an analytic solution to satisfy the zero-stress boundary conditions [6]. In past approaches, numerical approximations of geometry and/or wave functions were made to satisfy the half-space boundary condition using wave functions that are a function of both sine and cosine [3, 7]. Lee and Liu’s new method redefines the cylindrical-wave functions for both the longitudinal P- and shear SV-wave so that they are now a function of sine or cosine, but not a combination of both. Using the Fourier half-range expansion—and because each function itself is orthogonal in the half-space—the functions satisfy the zero-stress boundary condition along the half-space. The normal- and shear-stress boundary condition along the half-space in Figure 1 will be zero since the stress function contains , which equals zero when , .