# Far- and near-field effects for 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 |

## Problem 2.13a

Show that for harmonic waves of the form

**(**)

**the displacement is**

**(**)

### Background

If we set in equation (2.9a), we obtain the result , where is a solution of the P-wave equation. Furthermore if is independent of latitude and longitude, the wave equation reduces to equation (2.5c), and the solution of problem 2.5c shows that equation (2.13a) is a P-wave solution of equation (2.5c).

### Solution

Since in equation (2.13a) is a solution of equation (2.5c), it represents a spherically symmetrical P-wave and therefore the only displacement is along . If we take the *x*-axis along , equation (2.9d) shows that

## Problem 2.13b

Show that the two terms in equation (2.13b), which decay at different rates, are of equal importance at distance .

### Solution

The two terms are of equal importance when the two amplitudes are equal, that is, when or .

## Continue reading

Previous section | Next section |
---|---|

Disturbance produced by a point source | Rayleigh-wave relationships |

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
- Rayleigh-wave relationships
- Directional geophone responses to different waves
- Tube-wave relationships
- Relation between nepers and decibels
- Attenuation calculations
- Diffraction from a half-plane