Soil Dynamics and Earthquake Engineering 11 (1992) 445-456 Diffraction of SV waves by underground, circular, cylindrical cavities V.W. Lee & J. Karl Civil Engineering Department, University of Southern California, Los Angeles, California 90089, USA Communicated by M.D. Trifunac (Received 10 December 1991; revised version received 12 August 1992; accepted 21 September 1992) The scattering and diffraction of plane SV waves by underground, circular, cylindrical cavities at various depths in an elastic half space is studied in this paper. The cavities, studied here, are at depths of two to five cavity radii, measured from the surface to the center of the cavity. Fourier-Bessel series are used to satisfy the wave equation and the boundary conditions. When the angle of incidence of the plane SV wave exceeds the critical angle, surface waves are generated, which are expanded in terms of Fourier series, which also involve Bessel functions. The surface displacement amplitudes and phases that are presented show that the results depend on the following parameters: (1) The angle of incidence, 0;~; (2) the ratio of cavity depth to the cavity radius, h/a; (3) the dimensionless frequency of the incident SV wave, r/; and (4) Poisson's ratio, u. The presence of the cavity in the half space results in significant deviation of both the displacement amplitudes and phases on the nearby half space surface from that of a uniform half space. 1 INTRODUCTION The earliest research on incident elastic waves and cylindrical cavities involved solving the incident SH wave problem since this could be solved with an imaging method. The case of SH wave scattering by an underground cylindrical tunnel was solved using an imaging method. 1 The case of SH wave scattering by an underground cylindrical tunnel with a concentric lining around the cylindrical wall, with a different elastic material than the half space, was also solved using an imaging method. 2 The SV wave cylindrical cavity problem, however, could not be solved by an imaging method, and is solved here by a different method. Scheidl & Zieglar 3 proposed the use of a large radius circular surface to approximate the half space surface in the vicinity of the obstacle. The expansion of surface waves that they used were not convergent, as pointed out by Lee & Cao. 4 The numerical results they presented thus involved calculations using the non-convergent series. Lee and Cao 4 replaced the non-convergent series expansion by one that is convergent, using finite Fourier Soil Dynamics and Earthquake Engineering 0267-7261/93/$06.00 © 1993 Elsevier Science Publishers Ltd. series. Their method was successfully applied later to diffraction problems involving incident plane elastic P, SV, SH and Rayleigh surface waves on cylindrical canyons and valleys. 4-1° There has been other earlier work on the diffraction of waves by an underground cavity in an elastic half space. Gregory ll'12 proposed a solution for such problems by the use of orthogonal wave functions, but he presented no numerical results. Numerical implementation of his theory seems to be very complicated, as was presented by Datta) 3 The general method, used here for SV waves on cylindrical cavity cases, is simple in theory and easy to apply numerically. It was tested 7 by first applying it to the case of incident SH waves and then comparing it with that from the imaging solution for the incident SH wave and cylindrical cavity I with good agreement. 445 2 MODEL The model for the incident SV wave and the cylindrical cavity is seen in Fig. 1. The halfspace is elastic, isotropic, and uniform except for the circular, cylindrical section that is removed to make a cylindrical cavity. The cylindrical cavity is at a depth h measured from the