# Disturbance produced by a point source

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 2.12

A source of seismic waves produces on a spherical cavity of radius enclosing the source a step displacement of the form

Starting with equation (2.12a) below, show that the displacement at distance is given by

Is the motion oscillatory? What is the final (permanent) displacement?

### Background

When a source, such as an explosion, creates very high stresses, the wave equation does not apply near the the source because the medium does not obey Hooke’s law in this region. For a symmetrical point source, this situation can be handled mathematically by enclosing the source with a spherical surface centered at the source and specifying the displacement at all points on the spherical surface at . If the source generates a wave such that the displacement at each point on the surface of radius is

the displacement is given by

**(**)

where [see Sheriff and Geldart, 1995, Section 2.4.5, equations (2.76)
and (2.77)].
The step function, step (*t*) , is defined in Sheriff and Geldart, 1995, Section 15.2.5.

### Solution

Equation (2.12a) gives when . If we let , in the limit when , the displacement of the spherical surface becomes

which is the given type of source. Setting in equation (2.12a) we find that

If the motion is oscillatory, must change sign at least once, that is, the value of the expression in the square brackets must pass through zero. But , so and the exponential term is always positive, therefore oscillation is not possible.

At , which is the permanent displacement.

## Continue reading

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Boundary conditions in terms of potential functions | Far- and near-field effects for a point source |

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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 at different types of interfaces
- Boundary conditions in terms of potential functions
- 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