Sum of waves of different frequencies and group velocity

From SEG Wiki
Jump to: navigation, search

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


(2.7a)

(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


Figure 2-7a  Illustrating group and phase velocity.

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

Previous section Next section
Wave equation in cylindrical and spherical coordinates Magnitudes of seismic wave parameters
Previous chapter Next chapter
Introduction Partitioning at an interface

Table of Contents (book)

Also in this chapter

External links

find literature about
Sum of waves of different frequencies and group velocity
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png