A general wave problem can be written as a set of coupled, linear, homogeneous,
first-order, partial-differential equations, which take the form

The lowest order WKB approximation is premised on the assumption that
depends so weakly on and that all of the
spatial and temporal dependence of the components of
is specified by a common factor
.
Thus,
Eq. (639) reduces to

In general, Eq. (640) has many solutions, corresponding to the many different types and polarizations of wave which can propagate through the plasma in question, all of which satisfy the dispersion relation

where . As is easily demonstrated (see Sect. 4.11), the WKB approximation is valid provided that the characteristic variation length-scale and variation time-scale of the plasma are much longer than the wave-length, , and the period, , respectively, of the wave in question.

Let us concentrate on one particular solution of Eq. (640) (*e.g.*,
on one particular type of plasma wave). For this solution, the dispersion
relation (643) yields

where (locally)

(646) |

The integral (645) averages to zero, except at a point of *stationary
phase*, where
(see Sect. 4.16). Here,
is the -space
gradient operator. It follows that the (instantaneous) trajectory of the pulse
matches that of a point of stationary phase: *i.e.*,

(647) |

(648) |

Let us now determine how the wave-vector, , and frequency, ,
of a pulse evolve as the pulse propagates through the
plasma. We start from the cross-differentiation rules
[see Eqs. (641)-(642)]:

Equations (644) and (649)-(650) yield (making use of the Einstein summation convention)

(651) |

(652) |

Partial differentiation of Eq. (644) with respect to gives

(653) |

(654) |

According to the above analysis, the evolution of a pulse
propagating though a spatially and temporally non-uniform
plasma can be determined by solving the
*ray equations*:

(655) | |||

(656) | |||

(657) |

The above equations are conveniently rewritten in terms of the dispersion relation (643):

Note, finally, that the variation in the amplitude of the pulse, as it propagates through though the plasma, can only be determined by expanding the WKB solutions to higher order (see Sect. 4.11).