# Disturbance produced by a point source

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

## Problem 2.12

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

{\displaystyle {\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 ${\displaystyle {r}}$ is given by

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

{\displaystyle {\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 ${\displaystyle u\left(r,\;t\right)}$ is given by

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

{\displaystyle {\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 ${\displaystyle a=0}$ in equation (2.12a) we find that

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

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