# General solutions of the wave equation

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 2.5a

Verify that and are solutions of the wave equation (2.5b).

### Background

When unbalanced stresses act upon a medium, the strains are propagated throughout the medium according to the general wave equation

**(**)

being a disturbance such as a compression or rotation. is propagated with velocity (see Sheriff and Geldart, 1995, Section 2.2). The disturbance is the result of unbalanced normal stresses, shearing stresses, or a combination of both. When normal stresses create the wave, the result is a volume change and is the dilitation [see equation (2.1e)], and we get the P-wave equation, becoming the P-wave velocity . Shearing stresses create rotation in the medium and is one of the components of the rotation given by equation (2.lg) ; the result is an S-wave traveling with velocity . Various expressions for and in isotropic media are given in Table 2.2a.

In one dimension the wave equation (2.5a) reduces to

**(**)

### Solution

We use subscripts to denote partial derivatives and primes to denote derivatives with respect to the argument of the function. Then, writing , we have

Substituting in equation (2.5b), we get the identity so is a solution. We get the same result when . A sum of solutions is also a solution, so is a solution.

## Problem 2.5b

Verify that is a solution of equation (2.5a), where are direction cosines.

### Solution

Let . We now must show that is a solution of equation (2.5a). Proceeding as before, we have

In the same way we get

But (see Sheriff and Geldart, 1995, problem 15.9a), so .

Following the same procedure we find that thus verifying that is a solution of equation (2.5a).

## Problem 2.5c

Show that

is a solution of the wave equation in spherical coordinates (see problem 2.6b) when the wave motion is independent of and :

**(**)

### Solution

The wave equation in spherical coordinates is given in problem 2.6b. When we drop the derivatives with respect to and , the equation reduces to equation (2.5c). Writing , we proceed as in part (a). Starting with the right-hand side, we ignore for the time being and obtain

Substitution in equation (2.5c) shows that is a solution. In the same way we can show that is also a solution, hence the sum is a solution.

## Continue reading

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Magnitudes of elastic constants | Wave equation in cylindrical and spherical coordinates |

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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
- 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
- Tube-wave relationships
- Relation between nepers and decibels
- Attenuation calculations
- Diffraction from a half-plane