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

Summary 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/d δ) 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'score, 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.