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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathbf{\mathit{r_{0}}}} enclosing the source a step displacement of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{step}_0 (t)&=k,\qquad t\ge 0,\\ &=0,\qquad t\le 0. \end{align}

Starting with equation (2.12a) below, show that the displacement at distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {r} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=0 . If the source generates a wave such that the displacement at each point on the surface of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r_{0} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} u_{0} \; \left(r_{0}, \; t\right)&=ke^{-at},\quad k>0,\; t\ge 0,\; a>0,\\ &=0,\qquad\quad t\le 0, \end{align}

the displacement Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u\left(r,\; t\right) is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

(2.12a)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u\left(r,\; t\right) when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): u_{0} \left(r_{0}, \; t\right)=ke^{-at} . If we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a\to 0 , in the limit when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=0 , the displacement of the spherical surface becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathrm{step}_0 (t)&=k,\quad t\ge 0, \\ &=0,\quad t\le 0, \end{align}

which is the given type of source. Setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a=0 in equation (2.12a) we find that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 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{align}

If the motion is oscillatory, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): r>r_{0} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t=+\infty, \zeta =+\infty, u\left(r,\; \infty \right)=k(r_{0} /r)^{2} , which is the permanent displacement.

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