# Sum of waves of different frequencies and group velocity

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 2.7a

A pulse composed of two frequencies, , can be represented by factors involving the sum and difference of the two frequencies. If the two components have the same amplitudes, we can write

where ,

Show that the composite wave is given approximately by the expression

where .

### Background

When different frequency components in a pulse have different *phase velocities* (the velocity with which a given frequency travels), the pulse changes shape as it moves along. The *group velocity* is the velocity with which the envelope of the pulse travels.

The envelope of a pulse comprises two mirror-image curves that are tangent to the waveform at the peaks and troughs, and therefore define the general shape of the pulse.

### Solution

Adding the two components and using the identity

we get for the composite wave

## Problem 2.7b

Why do we regard as the amplitude? Show that the envelope of the pulse is the graph of plus its reflection in the -axis.

### Solution

The solution in (a) can be written . We regard as the amplitude for two reasons: (1) repeats every time that increases by or each time that increase by . But , , so must repeat more slowly than . (2) Each time that attains its limiting values, , has the value and therefore never exceeds ; thus the curves of and pass through the maxima and minima of and therefore constitute the envelope.

## Problem 2.7c

Show that the envelope moves with the group velocity where

**(**)

**(see Figure 2.7a).**

### Solution

If we consider the quantity as a wave superimposed on the primary wavelet , comparison with the basic wave type of problem 2.5a shows that takes the place of , i.e., is the velocity with which the envelope travels. In the limit, is given by

where is the frequency. The result is

We introduce the phase velocity by noting that , so

Thus,

To replace the derivative with , we find the relation between and ; to do this we let be constant so that

hence,

## Continue reading

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Wave equation in cylindrical and spherical coordinates | Magnitudes of seismic wave parameters |

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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
- 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