Pulse Propagation
Consider a onedimensional wave pulse,

(9.1) 
made up of a linear superposition of cosine waves, with a range of different wavenumbers, all
traveling in the positive direction. The angular frequency, ,
of each of these waves is related to its wavenumber, , via the socalled
dispersion relation, which can be written schematically as

(9.2) 
In general, this relation is derivable from the wave disturbance's equation of motion.
Up to now, we have only considered sinusoidal waves that have linear dispersion
relations of the form

(9.3) 
where is a constant. The previous expression immediately implies that such waves have the same phase velocity,

(9.4) 
irrespective of their frequencies.
Substituting Equation (9.3) into Equation (9.1),
we obtain

(9.5) 
which is the equation of a wave pulse that propagates in the positive
direction, at the fixed speed , without changing shape. (See Chapter 8.) The previous analysis
seems to suggest that arbitrarilyshaped wave pulses generally propagate at the same speed as sinusoidal
waves, and do so without dispersing or, otherwise, changing shape. In fact,
these statements are only true of pulses made up of superpositions of sinusoidal waves with linear dispersion
relations. There are, however, many types of sinusoidal wave whose dispersion relations
are nonlinear. For instance, the dispersion relation of sinusoidal
electromagnetic waves propagating through an unmagnetized plasma is (see Section 9.3)

(9.6) 
where is the speed of light in vacuum, and
is a
constant, known as the (electron) plasma frequency, that depends on the properties of
the plasma. [See Equation (9.28).] Moreover,
the dispersion relation of sinusoidal surface waves in deep
water is

(9.7) 
where is the acceleration due to gravity, the surface tension of water, and
the mass density. Sinusoidal waves that satisfy nonlinear dispersion relations, such as (9.6) or (9.7), are known
as dispersive waves, as opposed to waves that satisfy linear
dispersion relations, such as (9.3), which are called nondispersive waves. As we saw previously, a wave pulse made up of a linear superposition of nondispersive sinusoidal waves,
all traveling in the same direction, propagates at the common phase velocity
of these waves, without changing shape. How does a wave pulse
made up of a linear superposition of dispersive sinusoidal waves evolve in time?
Suppose that

(9.8) 
In other words, the function in Equation (9.1) is a Gaussian, of characteristic width , centered on wavenumber . It follows, from the properties of the
Gaussian function, that is negligible for
.
Thus, the only significant contributions to the wave
integral

(9.9) 
come from a small region of space centered on . Let us
Taylor expand the dispersion relation,
, about .
Neglecting secondorder terms in the expansion, we obtain

(9.10) 
It follows that

(9.11) 
where
, and

(9.12) 
is a constant with the dimensions of velocity.
If is sufficiently small then the neglect of secondorder terms in the expansion (9.11) is a good approximation, and Equation (9.9) becomes
where use has been made of a standard trigonometric identity. (See Appendix B.)
The integral involving
is zero, by symmetry. Moreover,
an examination of Equations (8.15)–(8.18) reveals that

(9.14) 
where
. Hence, by analogy with this expression, Equation (9.13) reduces to

(9.15) 
This is the equation of a wave pulse, of wavenumber , and angular frequency ,
with a Gaussian envelope, of characteristic width , whose peak (which is located by setting the argument of the exponential to zero) has the
equation of motion

(9.16) 
In other words, the pulse peak—and, hence, the pulse itself—propagates at
the velocity , which is known as the group velocity. In the case of nondispersive waves, the group
velocity is the same as the phase velocity (because, if
then
). However, for the case of dispersive waves, the two velocities are, in general,
different.
Equation (9.15) indicates that, as the wave pulse propagates, its envelope
remains the same shape. Actually, this result is misleading, and is only obtained because
of the neglect of secondorder terms in the expansion (9.11). If we keep more terms in this expansion then we can show that the wave pulse
does actually change shape as it propagates. However, this demonstration is most readily effected by means of the
following simple argument. The pulse extends in Fourier space from
to
, where
. Thus, part of the pulse
propagates at the velocity
, and part at the
velocity
. Consequently, the pulse spreads out as it propagates, because some parts of it move faster than others.
Roughly speaking, the spatial extent of the pulse in real space grows as

(9.17) 
where
is the extent of the pulse at .
Hence, from Equation (9.12),

(9.18) 
We, thus, conclude that the spatial extent of the pulse grows linearly in time, at a rate
proportional to the second derivative of the dispersion relation with respect to (evaluated at the pulse's central wavenumber). This effect is
known as pulse dispersion. Incidentally, there is no pulse dispersion in a nondispersive
medium, because, by definition,
in such a medium.
In summary, a wave pulse made up of a linear superposition
of dispersive sinusoidal waves, with a range of different wavenumbers, propagates at the group velocity, and also gradually disperses as time progresses.