Scattering of Plane Sh-Waves By A Semi-Parabolic Cylindrical Canyon In an Elastic Half-Space

Abstract

S U M M A R YThe 2-D scattering and diffraction of plane SH-waves by a semi-parabolic cylindrical canyon in an elastic half-space is analysed for the cases of SH-waves coming from the side of the half-space towards the front of the canyon. The exact analytic series solution of the problem is presented. It shows that the surface topography can have significant effects on the incident waves, which become prominent at high frequencies, or when the wavelengths of the incident waves are short compared with the focal length of the canyon. Higher frequencies will also lead to greater complexity of the pattern of surface displacement amplitudes involving abrupt changes of amplification or deamplification from one point to another.

Surface Displacement on and around a Semi-Parabolic Canyon Half-Space Induced by Plane SH-Waves Revisited: At (Five Times) Higher Frequencies

“…Therefore, the solution of resultant waves satisfying all boundary conditions defined in the model is given by: Lee [1990] computed the Weber Function numerically using the recursive relations in the reverse direction with respect to their orders, with algorithm given in Abramowitz and Stegan [1972]. With this numerical algorithm, Lee [1990] pointed out the convergence of the displacement in the infinite sum of equation became more difficult and gets harder and harder and less accurate as the angles of incidence increases.…”

“…This article is based on Lee [1990] and serves as an extension of the analytic work of Lee to further expands the results to higher frequency. Lee's [1990] model consists of a half-space in which a semi-infinite part of the surface is removed to form a parabolic canyon, as shown in Fig. 1.…”

“…This includes finite-difference, finite element, boundary element integral methods in both the analytic, numerical, and semi-analytical numerical methods [Chen et al, 2008[Chen et al, , 2011[Chen et al, , 2012. This article is based on Lee [1990] and serves as an extension of the analytic work of Lee to further expands the results to higher frequency. Lee's [1990] model consists of a half-space in which a semi-infinite part of the surface is removed to form a parabolic canyon, as shown in Fig.…”

“…We reinvestigated the surface displacement on a semi-parabolic canyon half-space on the basis of the classic elastic wave propagation study developed by Lee [1990]. In Lee, numerical computation for surface displacement could be only obtained at relative low frequency due to computation limitation on boundary condition determination and ability with numerical calculation.…”

Scattering and diffraction of plane SH-waves by periodically distributed canyons

“…The analytical method that provides a series of solutions for several simple geometric shapes in a homogenous half-space is valuable for revealing the nature of the topographic effect and for testing the accuracy of numerical methods. For example, analytical solutions for incident SH-waves include the semi-cylindrical canyon (Trifunac, 1973), semielliptical canyon (Wong, 1974), shallow circular-arc canyon (Lee, 1990;Todorovska and Lee, 1991;Yuan and Liao, 1994;Liang et al, 2000Liang et al, , 2003, V-shaped canyon (Tsaur and Chang, 2008;Tsaur et al, 2010;Gao et al, 2012), U-shaped canyon , truncated semicircular canyon (Tsaur and Chang, 2009) and circular sectorial canyon (Chang et al, 2013). Numerical methods are more fl exible than the analytical approach and modern high-speed computers make various numerical schemes more powerful to account for complex conditions and material nonlinear problems.…”