The pulse structure is conveniently represented as
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
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,
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
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 dispersion relation for an electromagnetic plasma wave propagating through an unmagnetized plasma is [see Equation (6.121)]
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.