17146 JUNE 1982 BODY WAVE EXCITATION OF EMBEDDED HEMISPHERE By Vincent W. Lee ,' and Mihailo D. Trifunac2 ABSTRACT : Solutions of three-dimensional vibrations of an embedded hemi- spherical foundation are presented for excitation by incident plane P, SV, and SH waves with arbitrary angles of incidence. The effects of embedment are illustrated by comparing the motion of the embedded hemispherical foundation with those of a rigid disc bonded to a half space of equal radius and excited by the same waves. The impedence matrix and the foundation motion are analyzed and compared with similar and related approximate solutions. In the context of Fourier synthesis, the results can be viewed as representing the transfer functions for the forces and motions of the foundation. For arbitrary transient excitation which can be represented by plane body waves, the results can be used to calculate the transient motion of the foundation. INTRODUCTION EM3 Dynamic response of a rigid foundation embedded in an elastic half-space plays an important role in the calculation of response of structures to seismic excitation. At present, many soil structure interaction studies are restricted to rigid surface foundation, and the development of methods for considering the effects of embedment is still in progress. In simple two-dimensional models, when embedment is considered (e.g., Luco, et al. (8) and Wong and Trifunac (13)), the limitation of two dimensions exclude the possibility of analyzing con- tributions to the response from, e.g., torsion. Common assumptions that seismic waves arrive vertically may diminish further the validity of the models and the methods of analysis, since it appears that strong shaking contains significant contributions from surface waves (10). In general, for most practical cases of excitation to strong earthquake shaking, it must be assumed that all waves con- tribute to excitation and, thus, it is necessary to develop techniques which are capable of considering all components of motion in three dimensions. The aim of this paper is to present such a solution for vibration of a rigid hemisphere embedded into an elastic half-space and excited by incident plane P, SV, and SH waves, with arbitrary angles of incidence. The effects of embed- 'Research Asst. Prof., Dept. of Civ. Engrg., Univ. of Southern California, University Park, Los Angeles, Calif. 90007. 'Prof., Dept. of Civ. Engrg., Univ. of Southern California, University Park, Los An- geles, Calif. 90007. Note.-Discussion open until November 1, 1982. To extend the closing date one month, a written request must be filed with the Manager of Technical and Professional Publi- cations, ASCE. Manuscript was submitted for review for possible publication on Septem- ber 3, 1980. This paper is part of the Journal of the Engineering Mechanics Division, Proceedings of the American Society of Civil Engineers, ©ASCE, Vol. 108, No. EM3, June, 1982. ISSN 0044-7951/82/0003-0546/$01.00. 546