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
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(2.7a)
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(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,
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Also in this chapter
External links
find literature about Sum of waves of different frequencies and group velocity
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