# 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

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 -plane separating two semi-infinite solids require that, for a wave traveling in the -plane, the following functions must be continuous:

**(**)

**(**)

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

The normal displacement is while the tangential displacement is , so these two functions must be continuous.

The normal stress is and equations (2.1b), (2.1h), (2.9e), and (2.9f) show that

the last step being obtained by differentiating the above expression for .

The tangential stress is and equations (2.1c), (2.1i), (2.9d), and (2.9e) give

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