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
Show that the composite wave is given approximately by the expression
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.
Adding the two components and using the identity
we get for the composite wave
Why do we regard as the amplitude? Show that the envelope of the pulse is the graph of plus its reflection in the -axis.
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.
Show that the envelope moves with the group velocity where
(see Figure 2.7a).
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
Illustrating group and phase velocity.
where is the frequency. The result is
We introduce the phase velocity by noting that , so
To replace the derivative with , we find the relation between and ; to do this we let be constant so that
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Sum of waves of different frequencies and group velocity