Pulse Propagation

The pulse structure is conveniently represented as

where is the electric field produced by the antenna, which is assumed to lie at . Suppose that the pulse is a signal of roughly constant (angular) frequency , which lasts a time , where is long compared to . It follows that possesses narrow maxima around . In other words, only those frequencies that lie very close to the central frequency, , play a significant role in the propagation of the pulse.

Each component frequency of the pulse yields a wave that propagates independently along the -axis, in a manner specified by the appropriate WKB solution [see Equations (6.17) and (6.18)]. Thus, if Equation (6.70) specifies the signal at the antenna (i.e., at ) then the signal at coordinate (where ) is given by

where

(6.72) |

Here, we have made use of the fact that .

Equation (6.71) can be regarded as a contour integral in
-space.
The quantity
is a relatively slowly varying function of
, whereas the phase,
, is a large and rapidly varying
function of
.
The rapid
oscillations of
over most of the path of
integration ensure that the integrand averages almost to zero. However,
this cancellation argument does not apply to places on the
integration path where the phase
is stationary: that is,
places where
has an extremum. The integral can, therefore, be
estimated by finding those points where
has a vanishing derivative,
evaluating (approximately) the integral in the neighborhood of each of
these points, and summing the contributions. This procedure is called
the *method of stationary phase* (Budden 1985).

Suppose that has a vanishing first derivative at . In the neighborhood of this point, can be expanded as a Taylor series,

(6.73) |

Here, the subscript is used to indicate or its second derivative evaluated at . Because is slowly varying, the contribution to the integral from this stationary phase point is approximately

(6.74) |

The previous expression can be written in the form

(6.75) |

where

(6.76) |

The integrals in the previous expression are known as

(6.77) |

It follows that

If there is more than one point of stationary phase in the range of integration then the integral is approximated as a sum of terms similar to that in the previous formula.

Integrals of the form (6.71) can be calculated exactly using the
*method of steepest descent* (Brillouin 1960; Budden 1985). The stationary
phase approximation (6.78) agrees with the leading term of the
method of steepest descent (which is far more difficult to implement
than the method of stationary phase) provided that
is
real (i.e., provided that
the stationary point lies on the real axis). If
is complex, however, then the stationary phase
method can yield erroneous results.

It follows, from the previous discussion, that the right-hand side of Equation (6.71) averages to a very small value, expect for those special values of and at which one of the points of stationary phase in -space coincides with one of the peaks of . The locus of these special values of and can obviously be regarded as the equation of motion of the pulse as it propagates along the -axis. Thus, the equation of motion is specified by

(6.79) |

which yields

Suppose that the -velocity of a pulse of central frequency at coordinate is given by . The differential equation of motion of the pulse is then . This can be integrated, using the boundary condition at , to give the full equation of motion:

A comparison of Equations (6.80) and (6.81) yields

The velocity is usually called the

The dispersion relation for an electromagnetic plasma wave propagating through an unmagnetized plasma is [see Equation (6.121)]

Here, we have assumed that equilibrium quantities are functions of only, and that the wave propagates along the -axis. The phase-velocity of waves of frequency propagating along the -axis is given by

According to Equations (6.82) and (6.83), the corresponding group-velocity is

It follows that

(6.86) |

Let us assume that , and for , which implies that the reflection point corresponds to . It is clear from Equations (6.84) and (6.85) that the phase-velocity of the wave is always greater than the velocity of light in vacuum, whereas the group-velocity is always less than this velocity. Furthermore, as the reflection point, , is approached from positive , the phase-velocity tends to infinity, whereas the group-velocity tends to zero.

Although we have only analyzed the motion of the pulse as it travels from the antenna to the reflection point, it is easily demonstrated that the speed of the reflected pulse at position is the same as that of the incident pulse. In other words, the group velocities of pulses traveling in opposite directions are of equal magnitude.