Calculation of Body Waves, for Caustics and Tunnelling in Core Phases†

Abstract

The known failure of classical ray theory at caustics has led to a reconsideration of displacement (in the frequency domain), expressed as an integral over ray parameter p. The integrand contains saddle-points on the real p-axis which correspond to rays for the associated physical problem, and it is shown here that direct computation of the complex integral is still straightforward, even when two saddle-points (rays) have coalesced to form a caustic. WKBJ theory is still usable for the vertical wave-functions, but one may avoid both the Taylor series expansion for the phase, and the steepest-descents approximation. Attention is first directed towards the PKKP caustic near 119°, to calculations of both amplitude and the phase slowness (dT/dΔ) as a function of frequency, and to a criticism of some uses of plane wave reflection coefficients across the core-mantle boundary. It is then shown that short-period P-wave energy is efficiently tunnelled into and out of the Earth's core, from body waves having their turning point just above the core-mantle boundary. This provides an explanation for observations of multiply reflected core phases, PmKP with m > 2, which are found usually at distances beyond the cutoff one would expect from requiring real angles of incidence (≤ 90°) from mantle to core. To obtain body wave pulse shapes in the time domain, a method is described which appears to offer some strong advantages over Cagniard-de Hoop inversion.