# Boundary conditions in terms of potential functions

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 |

## Contents

## Problem 2.11

Using the definitions of stress and strain in problem 2.1 and the potential functions in equation (2.9b), show that the boundary conditions at the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {xy}}**
-plane separating two semi-infinite solids require that, for a wave traveling in the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {xz}}**
-plane, the following functions must be continuous:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left({\phi}_{z} -{\chi}_{x} \right),\qquad \left({\phi}_{x} +{\chi}_{z} \right), \end{align} }****(**)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\mathbf{\lambda \nabla ^{2} \phi}} +{\mathbf{2\mu}}\left({\phi}_{zz} -{\chi}_{xz} \right),\qquad {\mu} \left({\mathbf{2\phi}}_{xz} +{\chi}_{zz} -{\chi}_{xx} \right), \end{align} }****(**)

where subscripts denote partial derivatives. These terms are, respectively, the normal and tangential stressses and the normal and tangential displacements.

### Background

As stated in problem 2.10, all stresses and displacements must be continuous at an interface between two different media.

### Solution

From equations (2.9d) and (2.9e) we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u=\mathrm{\phi}_{x} +\mathrm{\chi}_{z}, \; w=\mathrm{\phi} -\mathrm{\chi}_{x}. \end{align} }**

The normal displacement is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w}**
while the tangential displacement is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u}**
, so these two functions must be continuous.

The normal stress is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\sigma}_{zz}}**
and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{\sigma}_{zz} =\mathrm{\lambda}\Delta +2\mathrm{\mu} \mathrm{\varepsilon}_{zz} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} w_{z} =\mathrm{\lambda} \nabla ^{2} \mathrm{\phi} +2\mathrm{\mu} \left(\mathrm{\phi}_{z} -\mathrm{\chi}_{xz} \right), \end{align} }**

the last step being obtained by differentiating the above expression for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle w}**
.

The tangential stress is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{\sigma}_{xz}}**
and equations (2.1c), (2.1i), (2.9d), and (2.9e) give

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{\sigma}_{xz} =\mathrm{\mu \varepsilon}_{xz} =\mathrm{\mu} \left(u_{z} +w_{x} \right)=\mathrm{\mu} \left(2\mathrm{\phi}_{xz} +\mathrm{\chi}_{zz} -\mathrm{\chi}_{xx} \right). \end{align} }**

Since the normal and tangential stresses must be continuous, these two functions must also be continuous.

## Continue reading

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---|---|

Boundary conditions at different types of interfaces | Disturbance produced by a point source |

Previous chapter | Next chapter |

Introduction | Partitioning at an interface |

## 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
- 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