The 14 th World Conference on Earthquake Engineering October 12-17, 2008, Beijing, China Free-Field (Elastic or Poroelastic) Half-Space Zero-Stress or Related Boundary Conditions Vincent W Lee 1 and Jianwen Liang 2 1 Professor, Civil & Environmental Engineering, Univ. of Southern California, Los Angeles, CA 90089 USA, vlee@usc.edu 2 Professor, School of Civil Engineering, Tianjin University, Tianjin 300072, China, liang@tju.edu.cn ABSTRACT : The boundary-valued problem for solving for waves scattered and diffracted from surface and sub-surface topographies have attracted much attention to earthquake and structural engineers and strong-motion seismologists since the last century. It is of importance in the design, construction and analysis of earthquake resistant surface and sub-surface structures in seismic active areas that are vulnerable to near field or far field strong-motion earthquakes. The half-space medium can be elastic, or poroelastic and fluid saturated, the later case has attracted much new research in recent years. The presence of a surface or sub-surface topography, like the case of a surface canyon, valley, canal or structural foundations, or an underground cavity, tunnel or pipe, will result in scattered and diffracted waves being generated. Combined with the free-field input waves, they will together satisfy the appropriate stress and/or displacement boundary conditions at the surface of the topography present in the model. For those problems where analytical solutions are preferred in the studies, this often involves a topography that is either: circular, elliptic, spherical or parabolic in shape. This is because in those coordinate systems, the scattered waves are expressible in terms of orthogonal wave functions, and the surface of the topography often allows the orthogonal boundary conditions to be applied, so that the wave coefficients can be analytically defined. However, the presence of the half-space boundary makes the problem much more complicated. The scattered wave functions are no longer orthogonal on the flat half-space surface, and the zero-stress or related boundary conditions are no longer simple nor straight forward to apply. This paper will examine the available numerical and approximate methods that have been attempted or proposed, and the direction all future research is taking us to solve this part of the problem. KEYWORDS: Diffraction, Free-Field, Half-Space, Elastic, Poroelastic 1. INTRODUCTION Large topographies at half-space surfaces, like, canyons, valleys, canals and structural foundations, large, underground sub-surfaced structures like cavities, pipes, subways or tunnels, are always present in metropolitan areas associated with infrastructure developments in cites. As a result of excitation by incoming seismic waves, their presence will often generate additional waves from diffraction and scattering, resulting amplifications and de-amplification of the input waves, which will affect the deformations, distributions and concentrations of stresses on nearby ground surfaces and the structures on them. It is thus important to fully understand the theoretical and engineering aspects of the effects of such diffraction. This calls for a need for accurate, analytical method to be developed to study the in-plane responses of buried topographies to input waves. In general, for any wave propagation boundary valued problems, it is necessary to select the right coordinate systems that will allow the vector wave equations of the elastic waves to be separable into scalar wave equations for the P- and SV- wave potentials. However, because of the complexity of the problem involved, simple analytic solutions to diffraction problems by these underground topographies, are limited in both quantities and qualities. For the case of finite topographies in a finite region, like that of