Magnitude of disturbance from a seismic 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 2.3a

Firing an air gun in water creates a pressure transient a small distance away from the air gun with peak pressure ${\mathcal {P}}$ of 5 atmospheres ($5.0\times 10^{5}$ Pa). If the compressibility of water is $4.5\times 10^{-10}$/Pa, what is the peak energy density?

Background

Air guns (see problem 7.7) suddenly inject a bubble of high‐pressure air into the water to generate a seismic wave.

Stresses acting upon a medium cause energy to be stored as strain energy, because the stresses are present while the medium is being displaced. Strain energy density (energy/unit volume) $E$ is equal to [see Sheriff and Geldart, 1995, equation (2.22)]

${\begin{aligned}{E={\frac {1}{2}}\sum \limits _{i}\sum \limits _{j}{\mathrm {\sigma } }_{ij}{\mathrm {\varepsilon } }_{ij}={\frac {1}{2}}{\mathrm {\lambda } }{\mathrm {\Delta } }^{2}+{\mathrm {\mu } }\left({\mathrm {\varepsilon } }_{xx}^{2}+{\mathrm {\varepsilon } }_{yy}^{2}+{\mathrm {\varepsilon } }_{zz}^{2}\right)}\\{{\qquad }{\qquad }{\qquad }{\qquad }+{\frac {1}{2}}{\mathrm {\mu } }\left({\mathrm {\varepsilon } }_{xy}^{2}+{\mathrm {\varepsilon } }_{yz}^{2}+{\mathrm {\varepsilon } }_{zx}^{2}\right).}\end{aligned}}$

(2.3a)

Solution

From problem 2.1c, we see that $k=1/\left(4.5\times 10^{-10}\right)=2.2\times 10^{9}$ Pa. Also, ${\mathrm {\mu } }=0$ for water, so $E={\frac {1}{2}}{\mathrm {\lambda } }{\mathrm {\Delta } }^{2}$. From equation (7,5) of Table 2.2a we find that ${\mathrm {\lambda } }=k$ when ${\mathrm {\mu } }=0$. Also, $k=-{\mathcal {P}}/{\mathrm {\Delta } }$ (see equation (2.1f)), so

${\begin{aligned}\Delta =\left|{\cal {P}}/k\right|=5.0\times 10^{5}/2.2\times 10^{9}=2.3\times 10^{-4}.\end{aligned}}$

Using equation (2.3a) we find that

${\begin{aligned}E={\frac {1}{2}}\left(2.2\times 10^{9}\times 2.3^{2}\times 10^{-8}\right)=58\,\,\,\mathrm {J/m} ^{3}.\end{aligned}}$

[The dimensions of $E$ are the same as those of stress, since strains are dimensionless. Thus, stress units are ${\rm {N/m}}^{2}={\rm {Nm/m}}^{3}={\rm {J/m}}^{3}$.]

Problem 2.3b

If the same wave is generated in rock with ${\mathrm {\lambda } }={\mathrm {\mu } }=3.0\times 10^{10}$ Pa, what is the peak energy density? Assume a symmetrical $P$-wave with ${\mathrm {\varepsilon } }_{xx}={\mathrm {\varepsilon } }_{yy}={\mathrm {\varepsilon } }_{zz},{\mathrm {\varepsilon } }_{ij}=0$ for $i\neq j$.

Solution

We have ${\mathrm {\Delta } }={\mathrm {\varepsilon } }_{xx}+{\mathrm {\varepsilon } }_{yy}+{\mathrm {\varepsilon } }_{zz}=3{\mathrm {\varepsilon } }_{xx}$, ${\mathrm {\varepsilon } }_{ij}=0,i\neq j$, so equation (2.3a) becomes

${\begin{aligned}E={\frac {1}{2}}{\mathrm {\lambda } }{\mathrm {\Delta } }^{2}+3{\mathrm {\mu } }{\mathrm {\varepsilon } }_{xx}^{2}={\mathrm {\lambda } }{\mathrm {\Delta } }^{2}\left({\frac {1}{2}}+{\frac {1}{3}}\right)={\frac {5}{6}}{\mathrm {\lambda } }({\mathcal {P}}/k)^{2}.\end{aligned}}$

Equation (9,3) in Table 2.2a gives $k={\mathrm {\lambda } }+{\frac {2}{3}}{\mathrm {\mu } }={\frac {5}{3}}{\mathrm {\lambda } }$ so that

${\begin{aligned}E={\frac {5}{6}}{\mathrm {\lambda } }{\mathcal {P}}^{2}\left/\right.\left({\frac {5}{3}}{\mathrm {\lambda } }\right)^{2}=0.3\times (5.0\times 10^{5})^{2}/3\times 10^{10}=2.5\ \mathrm {J/m} ^{3}.\end{aligned}}$

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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
• 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
• Disturbance produced by a point source
• 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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