Disturbance produced by a point source

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Problem 2.12

A source of seismic waves produces on a spherical cavity of radius enclosing the source a step displacement of the form

Starting with equation (2.12a) below, show that the displacement at distance is given by

Is the motion oscillatory? What is the final (permanent) displacement?

Background

When a source, such as an explosion, creates very high stresses, the wave equation does not apply near the the source because the medium does not obey Hooke’s law in this region. For a symmetrical point source, this situation can be handled mathematically by enclosing the source with a spherical surface centered at the source and specifying the displacement at all points on the spherical surface at . If the source generates a wave such that the displacement at each point on the surface of radius is

the displacement is given by


(2.12a)

where [see Sheriff and Geldart, 1995, Section 2.4.5, equations (2.76) and (2.77)]. The step function, step (t) , is defined in Sheriff and Geldart, 1995, Section 15.2.5.

Solution

Equation (2.12a) gives when . If we let , in the limit when , the displacement of the spherical surface becomes

which is the given type of source. Setting in equation (2.12a) we find that

If the motion is oscillatory, must change sign at least once, that is, the value of the expression in the square brackets must pass through zero. But , so and the exponential term is always positive, therefore oscillation is not possible.

At , which is the permanent displacement.

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