# Potential functions used to solve wave equations

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 2.9a

Show that equation (2.9a) relating the potential functions and to the vector displacement requires that and [see equations (2.1e) and (2.1g)] be solutions of the P- and S-wave equations, that is, of equation (2.5a) with replaced by and , respectively.

**(**)

and being solutions of the P- and S-wave equations, respectively.

### Background

The dilatation and component of rotation are defined in equations (2.1e,g).

While solutions of the wave equation (see problem 2.5) furnish values of or a component of rotation , we often need to know the displacements (defined in problem 2.1) which are not easily found given or . This difficulty can be avoided by using potential functions that are solutions of the wave equations and from which , hence also, can be found by differentiation.

Note that derivatives of a solution of a differential equation are also solutions.

The vector operator (called “del”) and its properties are discussed in Sheriff and Geldart, 1995, Section 15.1.2c.

### Solution

From equation (2.1e) and the definition of , we get for the dilatation

**(**)

since and . Because is a solution of the P-wave equation, must also be a solution.

We have

[see Sheriff and Geldart, 1995, equations (15.13) and (15.9)]. Since is a solution of the S-wave equation, is also a solution.

## Problem 2.9b

In two dimensions, the potential function can be defined as

**(**)

**Show how to obtain the displacements , the dilatation , and rotation from this equation (see Sheriff and Geldart, 1995, Section 15.1.2c and problem 15.5c).**

### Solution

From equation (2.1d) we see that is the -component of , that is, of and , so -component of . From Sheriff and Geldart, 1995, equation (15.13) we have

Thus,

**(**)

and

**(**)

To get the dilatation , we use equation (2.1e) and Sheriff and Geldart, 1995, problem 15.5c and obtain

**(**)

The rotation can be obtained by taking the curl of equation (2.9c) but an easier method is to substitute equations (2.9d) and (2.9e) in equation (2.1g). This gives

Thus has only a -component given by

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Magnitudes of seismic wave parameters | Boundary conditions at different types of interfaces |

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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
- 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