Disturbance produced by a point source

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	2
Pages	7 - 46
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 2.12

A source of seismic waves produces on a spherical cavity of radius ${\mathbf {\mathit {r_{0}}} }$ enclosing the source a step displacement of the form

${\begin{aligned}\mathrm {step} _{0}(t)&=k,\qquad t\geq 0,\\&=0,\qquad t\leq 0.\end{aligned}}$

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

${\begin{aligned}u={\frac {r_{0}^{2}k}{r}}\left[\left({\frac {1}{r_{0}}}-{\frac {1}{r}}\right)e^{-V\zeta /r_{0}}+{\frac {1}{r}}\right].\end{aligned}}$

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 $t=0$. If the source generates a wave such that the displacement at each point on the surface of radius $r_{0}$ is

${\begin{aligned}u_{0}\;\left(r_{0},\;t\right)&=ke^{-at},\quad k>0,\;t\geq 0,\;a>0,\\&=0,\qquad \quad t\leq 0,\end{aligned}}$

the displacement $u\left(r,\;t\right)$ is given by

${\begin{aligned}u(r,t)={\frac {r_{0}k}{r\left(V/r_{0}-a\right)}}\left({\frac {V}{r_{0}}}e^{-V\zeta /r_{0}}-ae^{-a\zeta }-{\frac {V}{r}}e^{-V\zeta /r_{0}}+{\frac {V}{r}}e^{-a\zeta }\right),\end{aligned}}$

(2.12a)

where $\zeta =t-\left(r-r_{0}\right)/V$ [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 $u\left(r,\;t\right)$ when $u_{0}\left(r_{0},\;t\right)=ke^{-at}$. If we let $a\to 0$, in the limit when $a=0$, the displacement of the spherical surface becomes

${\begin{aligned}\mathrm {step} _{0}(t)&=k,\quad t\geq 0,\\&=0,\quad t\leq 0,\end{aligned}}$

which is the given type of source. Setting $a=0$ in equation (2.12a) we find that

${\begin{aligned}u\left(r,\;t\right)=\left(r_{0}^{2}k/r\right)\left[\left(1/r_{0}-1/r\right)e^{-V\zeta /r_{0}}+1/r\right].\end{aligned}}$

If the motion is oscillatory, $u\left(r,\;t\right)$ must change sign at least once, that is, the value of the expression in the square brackets must pass through zero. But $r>r_{0}$, so $\left(r_{0}^{2}k/r\right)(1/r_{0}-1/r)>0$ and the exponential term is always positive, therefore oscillation is not possible.

At $t=+\infty ,\zeta =+\infty ,u\left(r,\;\infty \right)=k(r_{0}/r)^{2}$, which is the permanent displacement.

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Boundary conditions in terms of potential functions	Far- and near-field effects for a point source
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Introduction	Partitioning at an interface
Table of Contents (book)

Also in this chapter

• The basic elastic constants
• Interrelationships among elastic constants
• Magnitude of disturbance from a seismic source
• Magnitudes of elastic constants
• General solutions of the wave equation
• Wave equation in cylindrical and spherical coordinates
• Sum of waves of different frequencies and group velocity
• Magnitudes of seismic wave parameters
• Potential functions used to solve wave equations
• Boundary conditions at different types of interfaces
• Boundary conditions in terms of potential functions
• Far- and near-field effects for a point source
• Rayleigh-wave relationships
• Directional geophone responses to different waves
• Tube-wave relationships
• Relation between nepers and decibels
• Attenuation calculations
• Diffraction from a half-plane

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