# Boundary conditions at different types of interfaces

Series | Geophysical References Series |
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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 2.10a

Justify on physical grounds the boundary conditions for solid-fluid media in contact.

### Background

The basic principles underlying the boundary conditions are (i) all stresses must be continuous (that is, no changes in values) at the interface; this must be true to avoid unbalanced forces which would produce accelerations; and (ii) strains must be continuous at the interface; this must be true for normal strains to avoid interpenetration of one medium into the other, or creation of a vacuum between them, or sliding of solid on solid for unequal tangential strains.

When a wave is incident on a boundary, one to four boundary conditions must be satisfied, depending on the types of media. The angles of reflection and refraction are fixed by the laws of reflection and refraction [see equation (3.1a)], so the only parameters that can be adjusted to satisfy these conditions are the relative amplitudes of reflected and/or refracted P- and S-waves generated by the incident wave. An S-wave generated by an incident P-wave, or a P-wave generated by an incident S-wave, is called a *converted wave*.

### Solution

Normal stress and displacement (strains) are continuous, stress to avoid normal acceleration and displacement to avoid interpenetration or a vacuum. Tangential stress is zero everywhere in the fluid and so the tangential stress in the solid must be zero at the interface. There is no restriction on tangential displacement.

## Problem 2.10b

Justify on physical grounds the boundary conditions for solid-vacuum media in contact.

### Solution

Stresses are zero in a vacuum, so normal and tangential stresses in the solid vanish at the interface.

## Problem 2.10c

Justify on physical grounds the boundary conditions for fluid-fluid media in contact.

### Solution

Normal stresses and displacements are continuous at the interface for the same reasons as in (a). Tangential stresses cannot exist in a fluid, and tangential displacements are zero.

## Problem 2.10d

Justify on physical grounds the boundary conditions for fluid-vacuum media in contact.

### Solution

Stresses are zero in a vacuum, so the normal stress in the fluid vanishes at the interface.

## Continue reading

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Potential functions used to solve wave equations | Boundary conditions in terms of potential functions |

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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 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