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The scattering and reflection of SH waves by a slope on an elastic wedged space is investigated. A series solution is obtained by using the wave function expansion method. The slope on a wedged space is divided into two subregions by an artificial, auxiliary circular arc. The wave fields with unknown complex coefficients within each sub-region are derived. Applying Graf addition theorem, the scattered waves in the sub-regions are expressed in a global coordinate system. Fourier transform is adopted to derive a consistent form of standing waves in the inner region using the orthogonality of the cosine functions. The boundary-valued problem is solved by stress and displacement continuity along the artificial, auxiliary arc to obtain the unknown complex coefficients. Parametric studies are next performed to investigate how the topography from the slope on the wedged space will affect the scattering and diffraction, and hence the amplification and de-amplification of the SH waves. Numerical results show that the surface motions on the slope of the wedged space is influenced greatly by the topography. Amplification of the surface motions near the slope vertex is significant. The corresponding phases along the wedged space surfaces are consistent with the direction that the SH waves are propagating. [ABSTRACT FROM AUTHOR]

The analytical solution of a two-dimensional moon-shaped alluvial valley embedded in an elastic half-space is analyzed for incident plane SH waves, using the wave function expansion and the Discrete Cosine Transform (DCT). A series of solutions with different depth-to-radius ratios have been computed, analyzed, and discussed. It is shown that amplification of incident motions along the thinning valley segment can be significant. The phenomena of combined action of the waves resulting from (a) turning (reversing the direction of propagation), (b) focusing, and (c) diffraction from the half space into the valley have been examined with an emphasis on the significance for surface-motion amplification and the power to damage man-made structures. [ABSTRACT FROM AUTHOR]

A closed-form wave function analytic solution of two-dimensional scattering and diffraction of incident plane SH-waves by a shallow semi-elliptical hill with a concentric elliptical tunnel on an elastic half-space is presented. This research extends the previous solution by authors in 2013 which addressed only the semi-elliptical hill, to a hill-tunnel system. Ground surface displacement near the hill are studied and exhibited. It was observed from the results that the aspect ratio of hill and tunnel profiles, frequency of incident waves, and incident angles are the primary factors that may affect the surface ground motion amplitudes near the hill-tunnel system. [ABSTRACT FROM PUBLISHER]

The surface displacement in the vicinity of a semi-parabolic canyon was solved over 20 years ago. Previous study presented results only for a range of low frequencies, with computational difficulty coming from convergence of the Weber wave functions. We revisit this problem and employ new algorithm computing results at much higher frequencies for surface spectral displacements on the parabolic canyon by incident SH-waves. A transformation with the Wronskian allows computing displacement up to dimensionless frequency ofthen tonow! In terms of wavelength, the displacements were computed for waves with wavelengths five times smaller, or frequencies five times higher, a great improvement! [ABSTRACT FROM PUBLISHER]

Abstract: A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly. [Copyright &y& Elsevier]

Abstract: The paper presents an exact, analytical solution to the boundary-valued problem of the two-dimensional scattering of anti-plane (SH) waves by a shallow, semi-elliptical hill on an elastic half-space, based on the method of wave-function expansion in elliptical coordinate and elliptical cosine half-range expansion. It is expressed in terms of an infinite system of simultaneous linear equations that is later truncated for numerical computation. Some numerical results when semi-elliptical tends to semi-circular are compared with the existing results presented by Lee et al. (2006) [12]. Complicated effects on ground motion due to the existence of an elliptical hill at various aspect ratios and angles of wave incidence are illustrated. [Copyright &y& Elsevier]

Abstract: The ground motion and wave propagation resulting from the scattering of anti-plane (SH) waves by a deep semi-elliptical hill on an elastic half-space are investigated. The solution is the closed-form one based on the method of wave-function expansion, and the expansion of elliptical sine and cosine over the half-range [−π/2, π/2]. The analysis shows that the interference of ground motion depends on several factors such as frequency of the incident waves, incident angle, and the aspect ratio of elliptical hills. Similar to part I: Shallow Hill, numerical results when semi-elliptical tends to semi-circular are compared with the work done by Lee et al. [17]. [Copyright &y& Elsevier]

Abstract: The zero-stress boundary conditions at the surface of the half-space in the presence of surface and sub-surface cavities for in-plane, incident cylindrical P- and SV-waves have always posed challenging problems. The outgoing cylindrical P- and SV-waves can be represented by Hankel functions of radial distance coupled with the sine and cosine functions of angle. Together, at the half-space surface the P- and SV-wave functions are not orthogonal over the semi-infinite radial distance from 0 to infinity. Thus, to simultaneously satisfy the zero in-plane, normal, and shear stresses, an approximation of the geometry is often made. This paper presents an analytical formulation of the boundary-valued problem, where the Hankel wave functions are expressed in integral form, changing the representation from cylindrical to rectangular coordinates, so that the zero-stress boundary conditions at the half-space surface can be applied in a more straightforward way. It is sometimes desirable to use alternate, simpler, or approximate solutions to the boundary-valued problems, without going to great lengths to have all of the boundary conditions satisfied. The free-stress boundary conditions at the half-space surface to be solved here are among the most complicated boundary conditions to be satisfied in the present class of problems. It is thus of interest to illustrate what the solution would be like if the half-space boundary conditions are not imposed—in other words, if they are “relaxed.” A section of this paper is devoted to this approach, and the results of the solutions with the half-spaced, stress-free boundary conditions “imposed” and “relaxed” are presented and compared. The method introduced here may also be applied to more complicated wave-propagation problems, like the diffraction problems involving surface and sub-surface inhomogeneities in a poroelastic half-space medium. [Copyright &y& Elsevier]

Diffraction of a two-dimensional (2D) semi-circular cavity in a half-space under incident SH-waves is studied using the classic wave function expansion method with a new de-coupling technique. This so-called 'improved cosine halfrange expansion' algorithm exhibits an excellent performance in reducing displacement residual errors at two rim points of concern. The governing equations are developed in a manner that minimizes the residues of the boundary conditions. Detailed derivation and analysis procedures as well as truncation of infinite linear governing equations are presented. The semi-circular cavity model presented in this paper, due to its simple profile, is expected to be used in seismic wave propagation studies as a benchmark for examining the accuracies of various analytical or numerical methods for mixed-boundary wave propagation problems. [ABSTRACT FROM AUTHOR]