Effect of Depth of a Point Source on the Motion of the Surface of an Elastic Solid Sphere

An exact solution, obtained previously for the displacement of the surface of a uniform elastic solid sphere due to an impulsive compressional pulse from a point-source situated at depth d below the surface of a sphere of radius a is applied here to the problem of propagation from a source at d = a/32 and a/6400 and compared with results for d = a/8. Theoretical seismograms, radial and angular component, are given at distances 0 < θ < π from the source. Rayleigh waves are found due to all sources. In the high frequency limit they have the velocity of Rayleigh waves in a homogeneous half space. There is dispersion towards higher velocities. The Rayleigh waves from the shallow source have amplitudes more than ten times the amplitude of body waves. In the seismograms from the medium and deep sources their amplitude is smaller, they have less high-frequency components and show more dispersion. The amplitude of Rayleigh waves has a minimum at 90° and 270°. At 180° and 360° the radial component has a maximum of ten, seventeen and forty times the radial component at 90° for the deep, medium and shallow sources respectively. The ratio of angular to radial amplitude is largest at 90° and 270°, decreasing to zero at 180° and 360°. The Rayleigh waves show phase shift, almost elliptical retrograde particle motion, change in form of ellipses with depth of source and deviations of the direction of the major axis from the vertical direction similar to results in observed seismograms. Higher modes of surface waves are found and their connection with groups of reflected pulses of mixed types given. As in several observed seismograms the second mode is especially predominant in the angular component. Whereas diffracted pulses connected with μ times reflected Pμ are encountered mainly in deep-focus events, diffraction phenomena connected with the transformed phases PμSv are found here at all depths and have in several cases amplitudes comparable to, or larger than, the reflected pulses.