3-D Surface Motions of Long Semi-Circular Longitudinal Canyons I: Incident Plane P Waves Vincent W. Lee*. Mohammed A. Sabban** and Tarun K. Ghosh*** SUMMARY — The methodology previously used for solving the two-dimensional scattering and diffraction of elastic waves by a semi-circular canyon (Lee and tao, 1989; (‘tao and Lee, 1989, 1990) has been extended here to diffraction problems by three- dimensional semi-circular canyons (canals) in a half space (Fig. I). The half-space is assumed to be elastic, homogeneous. and isotropic. Longitudinal (P) incident plane waves are considered here. In order to satisfy the boundary conditions the incident and reflected waves are expanded in terms of Bessel-Fourier series and the surface of the half space is approximated as the outer surface of a circular cylinder of infinite length (Fig. 2). The surface displacements and phases are calculated and are compared to the available results of the two-dimensional analyt ic solutions. The orientation of the incident waves is represented now by two angles, 8j and 2 (Fig. 3). As expected the incidence angles 6,, 2. and the frequency of the incident waves affect the amplitude and phase of the displacement at the surface and haif space nearby. KEYWORDS: P waves, Scattering, Canyons, Hankel functions, Free Stress. 1. Introduction As more earthquakes of greater amplitudes are re corded, the concern of earthquake engineers grows deeper. The amplifications of wave amplitudes due to various geological and topographical inclusions have been one of the main concerns of seismologists and earthquake engineers. Depending on the type and direc tion of the incident waves, an obstacle such as a can yon or canal acting as a scatterer could shield an adja * Assoc. Prof Civil Engrg Dept.. Univ. of Southern California, Los Angeles. CA 90089-1114. Assistant Prof.. Univ. of ljmm, AiQura. Saudi Arabia. Dynamics Engineer. Rocketdyne Div.. Rockwell International Corp. Canoga Park. CA 91309-7922. Received december 1994. Revised february 1995. cent area from a destructive amount of strong motions. At the same time the barrier might help to amplify or deamplify the displacement amplitudes. Over the last seventy years simple models have been studied to try to understand this important phenomena. In general, for any wave propagation study it is es sential to choose the right coordinate systems that render the wave equation separable into scalar wave equations. Coupling in the governing equations for elastic wave propagation and boundary conditions that are difficult to satisfy are among the challenges to face further studies of the scattering of elastic waves. When considering two-dimensional anti-plane strain problems, only the SH-wave is analyzed using the imaging meth od, since the P and SV waves do not contribute to the displacement in the direction considered. Therefore, the scattering of the two-dimensional SH-wave is consid ered the simplest in wave propagation studies. The cas es of P and SV incident waves are more difficult to study. The coupling between P and SV waves prevents the use of the imaging technique. The wave functions’ solutions must then satisfy both half-space and curve boundary conditions. The objective of the current study is to investigate the three-dimensional motion of the half-space near a semi circular longitudinal canyon. Incident plane P-waves are considered. The waves are directed arbitrarily in the three-dimensional space. The method of analysis used in this investigation has been used by Scheidi and Zieglar (1978), Lee and Cao (1989), Cao and Lee (1989, 1990) and Lee and Karl (1992, 1993) in the two-dimensional study of scattering of plane elastic waves. Transforma tion of Bessel-Fourier series between two coordinate systems is required to satisfy the boundary conditions of the problem. This transformation has been used by Lee (1977), Lee and Trifunac (1979), Lee and Cao (1989), and Cao and Lee (1989, 1990) to study the diffraction near a cylindrical canyon, an underground circular cav ity and a circular tunnel. The current study provides an 12 EUROPEAN EARTHOUAE ENGINEERING 3, 1995