Green's function for the vector wave equation in a mildly heterogeneous continuum

Abstract

In this work, a fundamental solution is derived for the case of time-harmonic elastic waves originating from a point source and propagating in a three-dimensional, unbounded heterogeneous medium with a Poisson's ratio of 0.25. The first step in the solution procedure is to transform the displacement vector in the Navier equations of dynamic equilibrium through scaling by the square root of the position-dependent shear modulus. Following imposition of certain constraints that are subsequently used to derive the depth profile of the elastic moduli and of the density, it becomes possible to employ Helmholtz's vector decomposition so as to generate two scalar wave equations for the dilational and rotational components of the wave motion, a process which again generates additional constraints. The corresponding Green's function is then synthesized in the conventional way followed for homogeneous media. Consideration of all intermediate constraints shows that the elastic moduli and the density all have quadratic variation with respect to the depth coordinate z, while the pressure and shear wavespeed profiles are constant and correspond to reference values at z = 0. The present methodology is based on earlier algebraic transformation techniques applied for the case of scalar wave propagation. The methodology is finally illustrated through a number of examples involving a commonly encountered geological medium.