View source for Disturbance produced by a point source

{{publication
 | name    = Problems in Exploration Seismology and their Solutions
 | image   = [[File:897.Def.Lz-cropped.png|frameless|180px|link=Problems in Exploration Seismology and their Solutions]]
 | width   = 120px
 | series  = Geophysical References Series
 | title   = [[Problems in Exploration Seismology and their Solutions]]
 | part    = 
 | chapter = 2
 | pages  = 7 - 46
 | author  = [[Lloyd P. Geldart]] and [[Robert E. Sheriff]]
 | doi     = 10.1190/1.9781560801733
 | pdf     = 
 | store   = https://shop.seg.org/shop/products/detail/897
 | isbn    = ISBN 9781560801153
}}
__TOC__
== Problem 2.12 ==
A source of seismic waves produces on a spherical cavity of radius <math>{\mathbf{\mathit{r_{0}}}}</math> enclosing the source a step displacement of the form

<math>
\begin{align}
\mathrm{step}_0 (t)&=k,\qquad t\ge 0,\\
&=0,\qquad t\le 0.
\end{align}
</math>
	
Starting with equation (2.12a) below, show that the displacement at distance <math>{r}</math> is given by

<math>
\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}
</math>

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

the displacement <math>u\left(r,\; t\right)</math> is given by

{{NumBlk|:|<math>
\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}
</math>|{{EquationRef|2.12a}}}}

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

<math>
\begin{align}
\mathrm{step}_0 (t)&=k,\quad t\ge 0, \\
&=0,\quad t\le 0,
\end{align}
</math>

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

<math>
\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}
</math>

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

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

== Continue reading ==
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| align="center" | [[Boundary conditions in terms of potential functions]]
| align="center" | [[Far- and near-field effects for a point source]]
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| align="center" | [[:Category:Introduction to Problems_in_Exploration_Seismology_and_their_Solutions|Introduction]]
| align="center" | [[:Category: Partitioning at an interface| Partitioning at an interface]]
|-
| align="center" colspan="2" |
[[Problems_in_Exploration_Seismology_and_their_Solutions|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]]

== External links ==
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[[Category:Problems and Solutions]]
[[Category:Publications]]
[[Category: Theory of seismic waves]]