Scattering of plane SH-waves by cylindrical canals of arbitrary shape NASSER MOEEN-VAZIRI and M. D. TRIFUNAC Ovil Engineering Department, University of Southern California, University Park, Los Angeles, CA 90089, USA A boundary method is used to solve numerically the problem of scattering and diffraction of SH-waves by cylindrical canals of arbitrary shape in a homogeneous, linear, elastic, and isotropic half-space. A least-square technique has been employed to solve this problem. Results are obtained using a multiple expansion in terms of Hankel's functions. Displacements and stresses near and in the canal wall have been investigated and a comparison with known exact and approximate solutions for SH-wave motion has been studied. Numerical results for displacement and stress amplitudes for different geometries are presented. The stress and displacement amplitudes in the canal wall and on nearby ground surface change rapidly from one point to another. The higher excitation frequencies lead to greater complexity of the computed motions. For grazing and nearly grazing incidences, a shadow zone is developed behind the canal. INTRODUCTION In this paper, the problem of scattering and diffraction of plane SH-waves by cylindrical canals of arbitrary shape has been investigated. This type of problem is of practical interest, for example, in analysis and design of reinforced concrete canals. Related to this problem there have been previous studies on: (1) vibrations of a semi-circular canal excited by plane SH-waves;~ (2) scattering of SH-waves by two-dimensional canyons of arbitrary shape, using the boundary method 2 and (3) ground motion in alluvial valleys under incident plane SH-waves. 3 MODEL The cross-section of a two-dimensional model studied in this paper is shown in Fig. 1. It represents an unbounded region'E' enclosing a bounded region'D'. The outer boundary of D, the common boundary between E and D, is S~, and li'-= 20 2b " ' ;';;);'// // ////// Figure I. Cylindrical canal and the surrounding half-space Accepted April 1984. Discussion closes March 1985. 0261-7277/85/010018-0652.00 © 1985 CML Publications 18 Soil Dynamics and Earthquake Engineering, 1985, Vol. 4, No. 1 the inner boundary of D is $2. The unbounded region E is assumed to be a linear, elastic, homogeneous and isotropic medium of density Pl, rigidity /al, and with velocity of shear waves /~1. The corresponding quantities for the bounded region D, the canal wall, are P2,/a2, and/~2. The bounded region D has characteristic horizontal linear dimensions 2b and 2a (Fig. 1). Two coordinate systems are introduced: the rectangular coordinate system with positive x pointing to the right and positive y pointing down. The cylindrical coordinate system, consisting of the radial distance r and the angle 0, measured from the positive x coordinate, has a common origin with the rectangular system. EXCITATION AND SOLUTION OF THE PROBLEM Excitation i The excitation of the half-space, Uz, is assumed to con- sist of an infinite train of plane SH-waves with frequency and particle motion in the z-direction as follows: u / = expiw ---- (1) cx For an incident angle 7 the phase velocities along the x-axis, Cx, and y-axis, Cy, are given by c= =--; cy - (2) cos 7 sin 3' In the absence of the scattering region D, i.e. the canal, the incident motion would reflect from the plane free surface (y = 0), and incident waves u / and reflected waves r would interfere to give the resulting motion of the Uz half-space: i + r =2exp iw -- cos (3) Uz Uz \ Cy !