Pulse Propagation

The pulse structure is conveniently represented as

Each component frequency of the pulse yields a wave which
propagates *independently* along the -axis, in a manner specified by the
appropriate WKB solution [see Eqs. (569)-(570)]. Thus, if Eq. (622)
specifies the signal at the antenna (*i.e.*, at
), then the signal at coordinate (where
)
is given by

(624) |

Equation (623) 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*: *i.e.*,
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 neighbourhood of each of
these points, and summing the contributions. This procedure is called
the *method of stationary phase*.

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

(625) |

(626) |

(627) |

(628) |

(629) |

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 like the above.

Integrals of the form (623) can be calculated exactly using the
*method of steepest decent*.^{} The stationary
phase approximation (630) agrees with the leading term of the
method of steepest decent (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, the stationary phase
method can yield erroneous results.

It follows, from the above discussion,
that the right-hand side of Eq. (623) 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

(631) |

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:

The velocity is usually called the

The dispersion relation for an electromagnetic plasma wave propagating
through an unmagnetized plasma is

(636) |

(637) |

(638) |

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.