Propagation of a P-pulse in a solid sphere

Abstract

abstract An exact solution is obtained for the displacement of the surface of a uniform elastic solid sphere of radius a due to an impulsive compressional pulse from a point-source situated at a distance b from the center. The duration of the source is δa/c where c denotes the shear-wave velocity, and its time-variation is such that the surface-displacement stays finite when the time tends to infinity. The solution is applied to a source at a distance of one-eighth of the radius below the surface, approximating a deep-focus earthquake. Theoretical seismograms, radial and angular component, are given at distances 0 < ϑ < π for a source of duration 0.03a/c. Rayleigh waves are clearly seen at ϑ ≧ 45 °. Groups of reflected waves, especially predominant in the angular component, have the velocity of the lowest Airy phase in the group-velocity dispersion-curves. Diffracted waves, discussed in a previous paper, are found here again and in certain cases have an amplitude seven times larger than the amplitude of the direct pulse and also larger than any of the reflected pulses at the same distance. The transformed phases PSn, P2Sn have in general larger amplitude than the reflected Pn. Arrival times, initial amplitudes, reflection and convergence coefficients of pulses are obtained by steepest descents analysis and compared with the complete results.