Potential functions used to solve wave equations

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 2.9a

Show that equation (2.9a) relating the potential functions ${\phi }$ and ${\chi }$ to the vector displacement ${\zeta }$ requires that ${\Delta }$ and ${\theta _{z}}$ [see equations (2.1e) and (2.1g)] be solutions of the P- and S-wave equations, that is, of equation (2.5a) with ${\psi }$ replaced by ${\Delta }$ and ${\theta _{z}}$, respectively.

${\mathbf {\zeta =\nabla \left(\mathrm {\phi } +{\frac {\partial \chi }{\partial z}}\right)-\nabla ^{2}\mathrm {\chi } k} },$

(2.9a)

${\phi }$ and ${\chi }$ being solutions of the P- and S-wave equations, respectively.

Background

The dilatation $\mathrm {\Delta }$ and component of rotation $\mathrm {\theta } _{x}$ are defined in equations (2.1e,g).

While solutions of the wave equation (see problem 2.5) furnish values of $\mathrm {\Delta }$ or a component of rotation $\mathrm {\theta } _{i}$, we often need to know the displacements $\left(u,\;v,\;w\right)$ (defined in problem 2.1) which are not easily found given $\mathrm {\Delta }$ or $\mathrm {\theta } _{i}$. This difficulty can be avoided by using potential functions that are solutions of the wave equations and from which $\left(u,\;v,\;w\right)$, hence $\mathrm {\theta } _{i}$ also, can be found by differentiation.

Note that derivatives of a solution of a differential equation are also solutions.

The vector operator $\nabla$ (called “del”) and its properties are discussed in Sheriff and Geldart, 1995, Section 15.1.2c.

Solution

From equation (2.1e) and the definition of $\nabla$, we get for the dilatation $\Delta$

${\begin{aligned}\Delta &={\nabla }{\cdot }{\zeta }=\nabla ^{2}\left(\mathrm {\phi } +{\frac {\partial \mathrm {\chi } }{\partial z}}\right)-\left[{\nabla }{\cdot }{k}\right]\left(\nabla ^{2}\mathrm {\chi } \right)\\&=\nabla ^{2}\mathrm {\phi } +\nabla ^{2}\left({\frac {\partial \mathrm {\chi } }{\partial z}}\right)-\left({\frac {\partial }{\partial z}}\right)\left(\nabla ^{2}\mathrm {\chi } \right)\\&=\nabla ^{2}\mathrm {\phi } ,\end{aligned}}$

(2.9b)

since $\nabla ^{2}\left({\frac {\partial \mathrm {\chi } }{\partial z}}\right)={\frac {\partial }{\partial z}}\left(\nabla ^{2}\mathrm {\chi } \right)$ and $\nabla \cdot k=\partial /\partial z$. Because $\mathrm {\phi }$ is a solution of the P-wave equation, $\mathrm {\Delta }$ must also be a solution.

We have

${\begin{aligned}{\theta }&={\frac {1}{2}}\nabla \times {\zeta }={\frac {1}{2}}{\nabla }\times {\nabla }\left(\mathrm {\phi } +{\frac {\partial \mathrm {\chi } }{\partial z}}\right)-{\frac {1}{2}}{\nabla }\times \left[\left(\nabla ^{2}\mathrm {\chi } \right)k\right]\\&=0-{\frac {1}{2}}{\nabla }\times \left[\left(\nabla ^{2}\mathrm {\chi } \right)k\right]={\frac {1}{2}}\nabla ^{2}\left(\chi _{x}-\chi _{y}\right)\left(\chi _{y}i-\chi _{x}j\right)\end{aligned}}$

[see Sheriff and Geldart, 1995, equations (15.13) and (15.9)]. Since $\mathrm {\chi }$ is a solution of the S-wave equation, ${\theta }$ is also a solution.

Problem 2.9b

In two dimensions, the potential function $\zeta$ can be defined as

${\begin{aligned}{\zeta ={\mathbf {\nabla \phi } }+{\nabla }\times {\chi },\quad {\chi }=-\left|{\chi }\right|j}.\end{aligned}}$

(2.9c)

Show how to obtain the displacements ${\mathbf {\left(u,\;v,\;w\right)} }$, the dilatation ${\Delta }$, and rotation ${\theta }$ from this equation (see Sheriff and Geldart, 1995, Section 15.1.2c and problem 15.5c).

Solution

From equation (2.1d) we see that $u$ is the $x$-component of ${\zeta }$, that is, of ${\nabla }\mathrm {\phi }$ and ${\nabla }\times {\chi }$, so $u={\frac {\partial \mathrm {\phi } }{\partial x}}+x$-component of ${\nabla }\times {\chi }$. From Sheriff and Geldart, 1995, equation (15.13) we have

${\begin{aligned}{\nabla }\times {\chi }=\left|{\begin{array}{ccc}i&j&k\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\{0}&{-\mathrm {\chi } }&{0}\end{array}}\right|={\frac {\partial \mathrm {\chi } }{\partial z}}i-{\frac {\partial \mathrm {\chi } }{\partial x}}k=\mathrm {\chi } _{z}i-\mathrm {\chi } _{x}k.\end{aligned}}$

Thus,

${\begin{aligned}u=\mathrm {\phi } _{x}+\mathrm {\chi } _{z}\end{aligned}}$

(2.9d)

and

${\begin{aligned}w=\mathrm {\phi } _{z}-\mathrm {\chi } _{x}.\end{aligned}}$

(2.9e)

To get the dilatation $\Delta$, we use equation (2.1e) and Sheriff and Geldart, 1995, problem 15.5c and obtain

${\begin{aligned}\Delta ={\nabla }{\cdot }{\zeta }=\nabla ^{2}\mathrm {\phi } .\end{aligned}}$

(2.9f)

The rotation ${\theta }$ can be obtained by taking the curl of equation (2.9c) but an easier method is to substitute equations (2.9d) and (2.9e) in equation (2.1g). This gives

${\begin{aligned}{\theta }={\frac {1}{2}}\left|{\begin{array}{ccc}i&j&k\\{\frac {\partial }{\partial x}}&0&{\frac {\partial }{\partial z}}\\u&0&w\end{array}}\right|={\frac {1}{2}}\left({\frac {\partial u}{\partial z}}-{\frac {\partial w}{\partial x}}\right)j\end{aligned}}$

Thus ${\theta }$ has only a $y$-component given by

${\begin{aligned}\mathrm {\theta } _{y}={\frac {1}{2}}\left[{\frac {\partial }{\partial z}}\left(\mathrm {\phi } _{x}+\mathrm {\chi } _{z}\right)-{\frac {\partial }{\partial x}}\left(\mathrm {\phi } _{z}-\mathrm {\chi } _{x}\right)\right]={\frac {1}{2}}\nabla ^{2}\mathrm {\chi } .\end{aligned}}$

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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
• 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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