proper numerical models for understanding the physics of elastic wave propagation and scattering in the more general Novel formulations for numerical modelling of elastic waves in block media are developed in this paper. A single differential-differblock media.
ence equation, which can be discretized to give explicit finite differ-There are several approaches for studying elastic block ence models of wave propagation in elastic block media, is obtained media numerically. The traditional one is to let the wave after incorporating continuity conditions of stresses into the equafield in the interior of each individual block be computed tion of motion. Further decompositions of the differential-difference from the discretized homogeneous and isotropic elastic equation also lead to a parallel algorithm for computing the wave field. แฎ 1997 Academic Press wave equation of the block. The wave field along any interface is then computed from the imposition of the continuity of displacements and stresses, see [9] for example.
1. Introduction
A more general approach treats elastic block media as inhomogeneous media, see Boore [10], Temple [11,12], Elastic wave diffraction at multimedia interfaces is an and Cunha [13], though special care has to be taken at interesting problem encountered in many different applicaplaces where material parameters are discontinuous but tions, including geophysics, seismology, and ultrasonic stresses have to be continuous. Typically, the discontinunon-destructive evaluation (NDE). Mathematical models ities of the material parameters have to be smoothed over in geophysics and seismology often treat large scale regions some narrow artificial transition zones. The actual smoothof the earth as multi-layered elastic bodies with wavy intering is usually done by a simple averaging or blending profaces between the layers, see Kelly et al. [1]. In ultrasonic cess. Additional computational difficulties may arise from NDE, small scale defects or inhomogeneities are usually such a treatment. The width of a transition zone is small, of interest, which include voids, inclusions, and cracks.
normally one or two grid spacings only. Thus material Reflections, transmissions, and diffractions due to ultraparameters as functions of spatial variables may not be sound interaction with defects in the test material carry accurately discretized, which can affect the overall accucrucial information about the properties of the material racy of the numerical model. To overcome this problem, and provide the basis for ultrasonic NDE, see [2][3][4][5].
Cunha [13] has tried ''long'' and ''short'' operators on Any complicated elastic space can be approximated by displacements and material parameters separately to minidiscrete polygonal elastic sub-spaces (blocks). This might mize the effect of discontinuities in material parameters. include many problems of practical interest such as elastic
In their recent paper, Zahradnฤฑ ยดk and Priolo [14] have used wave scattering from regularly shaped inhomogeneities the Dirac delta function and Heaviside's step function to and propagation in stratified media. Curved interfaces can explicitly describe the wave motion and continuity condialso be approximated as a series of rectilinear steps. A tions in elastic block media in one single differential equamodelling approach which is based on discretization of the tion. This new differential equation is then discretized globcomputational space into blocks renders itself amenable ally for numerical computations. to numerical solutions using a method such as finite differ-
In this study, we consider elastic media composed of ences, see Harker [6].
rectangular blocks also in a global manner. A series of The interface between two homogeneous solid halfgeneralized equations of motion will be developed with spaces provides the simplest example of an interface probcontinuity conditions across interfaces automatically incorlem in elastic block media. Analytical solutions often exist porated. We give a detailed derivation and analysis of the for such cases, see [7,8]. The problem of elastic wave generalized equation of motion along horizontal interfaces propagation in more general block media, however, is and list the analogies for vertical and vertex interfaces. much more difficult to solve. There are no complete analytical solutions available. It is therefore necessary to have Neighbouring blocks do not have to be all distinct. Thus, 299