On the decoupling of P and S waves in inhomogeneous elastic media

Abstract

Summary. A representation derived by Richards (1974) for P-SV wave displacements in spherically symmetric elastic media is extended to general displacements in a general inhomogeneous isotropic elastic medium, with suitably differentiable elastic constants. This representation in terms of appropriate potentials gives rise to a partial decoupling of the P and S (weighted) displacements, which satisfy simple second order wave equations with lower order coupling terms. The highest order P and S components satisfy homogeneous wave equations that depend only on the P and S velocities a and respectively and are unaffected by the density other than at the source and observer positions. In the study of small high-frequency particle displacements in homogeneous elastic media the motion may be completely separated into P and S components. Each component has a characteristic wave speed and satisfies a simpler equation than the general elastic wave equation. In inhomogeneous media, this separation of P and S waves may still be observed, for example as distinct phases on seismograms, but there is now lower order coupling between the phases. Also a modified form of the Helmholtz representation in terms of a gradient and curl is required for other than the highest order components. In this paper we examine the partial first order decoupling of the P and S components in a general isotropic inhomogeneous medium with differentiable material constants. We begin by deriving for the harmonic displacement u exp (-- iwt) a representation in terms of potentials P and S, u = V(P/p”2) t g,(P/p”Z) + v x (S/p’/Z) t g, x (S/p”2) + C(u)