# Tube-wave relationships

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 2.16a

A tube wave has a velocity of 1050 m/s. The fluid in the borehole has a bulk modulus of Pa and a density of . The wall rock has and density . Calculate and for the wall rock.

### Background

Several types of tube waves exist (Sheriff and Geldart, 1995, Section 2.5.5). The classical type consists of a P-wave traveling in a fluid within a tubular cavity (such as a borehole) in a solid medium, the wall of the tube expanding and contracting as the pressure wave passes. Because the wall material interacts with the fluid, the tube-wave velocity depends upon the properties of both the wall material and the fluid. The formula for the tube-wave velocity is [see Sheriff and Geldart, 1995, equation (2.97)]

**(**)

being the density and the bulk modulus of the fluid while is the rigidity modulus of the wall material.

### Solution

Assuming the wall material to be rock, we write , , , , and for the rock, and for the fluid. We solve equation (2.16a) for but first we note that and are in units of , so we express as . Then,

Using equation (5,5) of Table 2.2a, we have

To get , we have

## Problem 2.16b

Repeat for and 1300 m/s. What do you conclude about the accuracy of this method for determining ?

### Solution

When ,

When ,

Summarizing the results for versus , we get the following:

1.05 | 3.44 | — | — |

1.20 | 6.80 | +14 | +156 |

1.30 | 35.7 | +24 | +938 |

Since the relative change in is very much larger than the relative change in , the method is very sensitive to changes in , hence the accuracy is very poor.

## Continue reading

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Directional geophone responses to different waves | Relation between nepers and decibels |

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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
- Disturbance produced by a point source
- Far- and near-field effects for a point source
- Rayleigh-wave relationships
- Directional geophone responses to different waves
- Relation between nepers and decibels
- Attenuation calculations
- Diffraction from a half-plane