ON THE ANALYSIS OF WAVE MOTIONS IN A MULTI-LAYERED SOLID by B. B. GUZINA (Department of Civil Engineering, University of Minnesota, Minneapolis, Minnesota 55455-0220, USA) and R. Y.S. PAK (Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, Colorado 80309-0428, USA) [Received 15 December 1998. Revises 28 October 1999 and 8 March 2000] Summary A rigorous treatment of the singular visco-elastodynamic solutions for a semi-infinite multi- layered solid is presented. It is shown explicitly via an asymptotic analysis of the propagator matrices that the singular components of the dynamic Green’s functions, which are critical to the theoretical foundation of boundary integral equation methods, correspond fully to the static point-load solutions for an appropriate bi-material full-space. With the aid of the analytical expressions for the bi-material response, a computational formulation for the multi- layered Green’s functions is also developed where the integral representation of the solution is decomposed into a closed-form singular part and a residual component which is amenable to numerical contour integration. With the foregoing treatment, the multi-layered fundamental solutions can be accurately and efficiently evaluated for a wide range of material and geometric configurations, including the special cases of elastic strata and the source points at the interface between two layers. As an illustration, the performance of the method in simulating the exact solution for an elastic half-space with a linear wave velocity profile is demonstrated. 1. Introduction Dynamic fundamental solutions for a semi-infinite multi-layered solid have long been of interest in seismology for their usefulness in representation of surface waves from a fault. Owing to the nature of the latter physical problem, most of the earlier treatments (for example, (1 to 6)) were focused primarily on the far-field response. Prompted by the developments in boundary integral formulations and their applications in modelling of semi-infinite domain problems arising in earthquake engineering, soil dynamics, and non-destructive testing, several authors (7 to 10) have developed point-load solutions for layered media which are more suitable for simulating various effects of proximate sources. Recognition of the need of further improvement can be found in (11 to 13) which suggests an evaluation of the improper integrals involved by means of these adhoc asymptotic wave expansion schemes. While these studies represent substantial advances in the subject, they have not addressed with generality a number of important issues such as the near-field mechanics at material interfaces and the character of load-induced singularities in an elastic or viscoelastic multi-layered half-space at an arbitrary source location. As these problems are central to the theoretical foundation of boundary integral equation methods (14 to 16) which require an explicit fundamental repre- sentation of the Green’s function’s singularity at an arbitrary interfacial or boundary point, a clear Q. Jl Mech. Appl. Math. (2001) 54 (1), 13–37 c  Oxford University Press 2001