The disturbance due to a line source in a semi-infinite elastic medium

Abstract

When a cylindrical pulse is emitted from a line source buried in a semi-infinite homogeneous elastic medium, the subsequent disturbance at any point near the surface is much more complex than for an incident plane pulse. The curvature of the wave-fronts produces diffraction effects, of which the Rayleigh-pulse is the most important. In this paper the exact formal solution is given in terms of double integrals. These are evaluated approximately for the case when the depths of source and point of reception are small compared with their distance apart, allowing a description of the sequence of pulses which arrive at the point of reception. When that point is at the surface and distant from the epicentre, the disturbance there can be regarded as made up of the following pulses, in order of arrival: ( a ) for initial P -pulse at source: P -pulse, surface. S -pulse and Rayleigh-pulse; ( b ) for initial S -pulse: surface P -pulse, S - pulse and Rayleigh-pulse. If the initial pulse has the form of a jerk in displacement, the P - and S - pulses arrive as similar jerks, whereas the Rayleigh-pulse is blunted, having no definite beginning or end. The surface P-pulse takes a minimum-time path and arrives with a jerk in velocity. The surface S -pulse, on the other hand, is confined to the neighbourhood of the surface and arrives as a blunted pulse. Moreover, part of the S -pulse arrives before the time at which it would be expected on geometrical theory. Although derived on very restricting hypotheses, these results may throw some light on seismological problems. In particular, it is shown that when the sharp S -pulse of ray theory is converted by the presence of the surface S -pulse and the spreading of S into a blunted pulse, the duration of this composite pulse is of the same order of magnitude as the observed scatter of readings of Sg and other distortional pulses from near earthquakes.