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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.

To examine the effects of different earthquake sources, ground motion near the rupture for several dislocation models has been calculated. Quite different dislocation models of the rupture can give very similar displacements at stations located only one fault dimension from the fault in an infinite homogeneous elastic space. In particular: (1) models with fault motions and rupture geometry quite different from Haskell's propagating ramp can often have near-field motions very similar to those from a ramp model; and (2) quite similar near-field motions are found, for different ramp models in which the rupture velocity and rise time are varied together over rather large ranges (e.g. a factor of three in rise time). We infer that there is considerable ambiguity in interpreting near-field displacement data in terms of a model of fault motion. The ambiguity may be reduced by using acceleration data, although this entails a considerable increase in computer time for solving forward problems.

Because the computation of synthetic seismograms has become an invaluable tool for the study of the Earth, it is of major interest to know whether two different methods, the reflectivity method and the full wave theory, generate identical record sections when applied to the same body wave problem. The full wave theory employs approximations by assuming decoupling of P and SV potentials in arbitrarily thick radially inhomogeneous layers and by representing radial eigenfunctions by the first term in a uniformly asymptotic series. The reflectivity method employs approximations by transforming depth-dependent variables of a spherically symmetric Earth to those of a plane-layered medium and by replacing this medium by a stack of homogenous layers. We have compared record sections generated by the two methods using the same Earth model, source parameters, distance ranges and body waves. Specifically, we examined core phases, which should provide a stringent comparison because they contain a wide range of frequency-dependent phenomena including diffraction and the effects that arise from a cusp and a caustic. It is found that the synthetic record sections of the two techniques are identical when displacements are low-pass filtered. This is the case for many practical purposes, for displacement generally must be convolved with an instrument response and a source function of several seconds duration before comparison with observed seismograms. However, as the content of the higher frequencies becomes more significant, the waveforms of the reflectivity records become more contaminated by reverberations. These reverberations arise when the smooth velocity or density. profile is approximated by homogeneous layers whose thicknesses are too large for the wavelengths involved. The reflectivity method must choose between using coarser layering (at the expense of accuracy) or finer layering (and greater computation time). The full wave theory can also handle body waves with turning points at or near the centre of the Earth, a region where the earth-flattening approximation begins to break down.

An explicit surface integral expression is derived, for the divergence (div u) of elastic displacement in an inhomogeneous anisotropic medium. In isotropic media, simple ray theory methods permit approximate evaluation of the integral; and (to the same order) an approximate relation is found between div u and the irrotational component of u. The resulting formula for P-wave displacement is expected to find application in the study of seismic sources: it is used here to evaluate the frequencydependent interference properties of P-waves, near the shadow boundary of the Earth's core.