This travel time of an Alfvén wave from one point to another on the same field line is

(2) |

In the following calculations we will adopt the model of a hydrogen plasma with a density distribution that varies as

(3) |

(4) |

Consider an Alfvén wave that travels from the spacecraft location
following the field line to the ionosphere and travels back through
the same path. The latitude at the foot of the field line on the
ionosphere is
or
depending on the ionosphere of concern. Therefore the travel time is

(5) |

It should be noted that the above calculation can only be considered
as a rough estimate of the Alfvénic travel time. First of all,
the magnetospheric field at larger *L* shells deviates from the
dipole model. Another reason, which might be even more serious,
is that the power law density model as used in the calculation
is a crude estimation. The *m* value in the density model can
range from 0 to 6 [cf. *Cummings et al.,* 1969].
For *m* = 3, (A4) or (A5) can also be written as

(6) |

We are indebted to C. A. Cattell at University of Minnesota, Minneapolis, for providing the ISEE 1 electric field data. One of us (P. J. C.) is grateful to M. G. Kivelson, R. J. Strangeway, and G. Le at IGPP, UCLA for their helpful comments. This work is supported by the National Aeronautics and Space Administration under grants NAGW-3974 and NAG 5-3171, and by the National Science Foundation under grant ATM 96-23163.

The Editor thanks W. J. Hughes and K. Takahashi for their assistance in evaluating this paper.