Ray Tracing

A general wave problem can be written as a set of coupled, linear, homogeneous, first-order, partial-differential equations, which take the form (Hazeltine and Waelbroeck 2004)

The vector-field has components (e.g., might consist of , , , and ) characterizing some small disturbance, and is an matrix characterizing the undisturbed plasma.

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, Equation (6.87) reduces to

where

In general, Equation (6.88) has many solutions, corresponding to the many different types and polarizations of waves that can propagate through the plasma in question, all of which satisfy the dispersion relation

where . As is easily demonstrated (see Section 6.2), the WKB approximation is valid provided that the characteristic variation lengthscale and variation timescale of the plasma are much longer than the wavelength, , and the period, , respectively, of the wave in question.

Let us concentrate on one particular solution of Equation (6.88) (e.g., on one particular type of plasma wave). For this solution, the dispersion relation (6.91) yields

that is, the dispersion relation yields a unique frequency for a wave of a given wave-vector, , located at a given point, , in space and time. There is also a unique associated with this frequency, which is obtained from Equation (6.88). To lowest order, we can neglect the variation of with and . A general pulse solution is written

where (locally)

(6.94) |

and is a function that specifies the initial structure of the pulse in -space.

The integral (6.93) averages to zero, except at a point of stationary phase, where . (See Section 6.7.) Here, is the -space gradient operator. It follows that the (instantaneous) trajectory of the pulse matches that of a point of stationary phase:

(6.95) |

where

(6.96) |

is the group-velocity. Thus, the instantaneous velocity of a pulse is always equal to the local group-velocity.

Let us now determine how the wavevector, , and the angular frequency, , of a pulse evolve as the pulse propagates through the plasma. We start from the cross-differentiation rules [see Equations (6.89) and (6.90)]:

Equations (6.92), (6.97), and (6.98) yield [making use of the Einstein summation convention (Riley 1974)]

(6.99) |

or

(6.100) |

In other words, the variation of , as seen in a frame co-moving with the pulse, is determined by the spatial gradients in .

Partial differentiation of Equation (6.92) with respect to gives

(6.101) |

which can be written

(6.102) |

In other words, the variation of , as seen in a frame co-moving with the pulse, is determined by the time variation of .

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

(6.103) | ||

(6.104) | ||

(6.105) |

The previous equations are conveniently rewritten in terms of the dispersion relation (6.91) (Hazeltine and Waelbroeck 2004):

Incidentally, the variation in the amplitude of the pulse, as it propagates through the plasma, can only be determined by expanding the WKB solutions to higher order. (See Exercises 3 and 4.)