A Contribution to the Theory of High Frequency Elastic Waves, with Applications to the Shadow Boundary of the Earth's Core

The diffraction of P and S waves by various obstacles is studied theoretically, in order to evaluate frequency dependent corrections to ray theory for elastic waves which travel nearly along the Earth's core shadow boundary. Most of the properties of this scattering process are conveniently illustrated by a simple Earth model, which gives rise to a problem in plane strain. This model is an infinite homogeneous elastic solid in which a steady state plane body wave (of the type P, SV, or SH) is incident on a circular cylindrical cavity. A Poisson summation is used for the scattered elastic potentials, and contributions from waves diffracted at least once around the cylinder are neglected. Simple approximation formulae are developed to examine the behavior of P, SV, and SH waves on and near their geometrical shadow boundary behind the fluid. Computed numerical results are believed to be valid for frequencies above 0.03 Hz. The solution method, which may be regarded as a corrected Fresnel theory, is taken through four successive stages of generalization to study increasingly realistic Earth models: (i) diffraction of cylindrical waves from a line source. For this problem our solution is in excellent agreement with the results of an ultrasonic model experiment conducted by Teng and Wu (1968). (ii) Diffraction by a fluid cylinder of cylindrical waves from a line source. (iii) Diffraction by a spherical fluid of spherical waves from a point source. Here we find good agreement between numerical results from our approximate method, and computation of the exact Poisson line integral. The final stage of generalization, to study (iv) diffraction by a spherical fluid/solid discontinuity in a realistic radially heterogeneous Earth, is obtained by methods similar to (iii), but after an extensive revision of Hook's (1961) discussion of elastic potentials in general media. In our approach, we recognize that the designation of P and S displacements is somewhat arbitrary in heterogeneous elastic media, but becomes precise in the high frequency limit of ray theory (in which P and two S components are decoupled). These facts are used for radially heterogeneous isotropic Earth models to establish three potentials (P,S,T) with the properties (a) that T(_~r,t) is decoupled from P and S, and is a potential for SH motion, (b) the coupling of P and SV waves is reflected in a system of coupled scalar equations for P(_~r,t) and S(_~r,t), and (c) in the high frequency limit we have P(_~r,t) and S(_~r,t) satisfying canonical uncoupled wave equations with the respective velocities (λ+2µ/ρ)^(1/2),( μ/ρ)^1/2. Many possibilities are suggested by the coupled equations for P(_~r,t) and S(_~r,t), apart from their use in the solution of (iv) above. They lead to a statement of conditions on the Earth model under which P and SV waves can propagate independently (at any frequency). We also use them to obtain approximate reflection coefficients for upper mantle transition regions which generate observed precursors to the phase PKPPKP, finding that the "extent of velocity gradient anomaly in such regions must be less than about 4 km, in order to observe short period (1 sec) reflections. Our numerical study of core diffraction provides an explanation for the observed polarization towards SH of diffracted S waves, and also shows that there is a slight dispersion effect dT/dΔ data, obtained for P in the range beyond 90°, which can and must be allowed for in accurate Herglotz-Wiechert inversion studies. The numerical methods developed for discussion of (iv) are expected to have wider applications in seismological studies of the Earth's core, mantle, and crust.