Non-homogeneous elastic waves in soils: Notes on the vector decomposition technique

Geological media are invariably non-homogeneous, which complicates considerably the analysis of seismically induced wave propagation phenomena. Thus, closed-form solutions in the form of Green's functions are difficult to construct, but are quite valuable in their own right and often play the role of kernels in boundary integral equation formulations that are used for the solution of complex boundary-value problems of engineering importance. In this work, we examine in some detail the types of wave-like equations that result from vector decomposition of the equations of motion for the infinitely extending non-homogeneous continuum, which would be a first step for evaluating Green's functions. Specifically, an eigenvalue analysis is first performed, followed by computations using the finite difference method for a specific example involving a soil layer with quadratically varying material parameters. The aforementioned wave-like equations, defined in terms of dilatational and rotational strains, are originally coupled. Their uncoupling involves use of algebraic transformations, which are in turn valid for certain restricted categories of non-homogeneous materials. Numerical solution of these equations clearly shows attenuation patterns and phase changes that are manifested as the incoming wave disturbance is continuously scattered by non-constant material stiffness values encountered along the propagation path.