(8.114) |

Equation (8.8) can be generalized to give

Moreover, it follows from Equation (8.71) that

(8.116) |

In other words, there is no perturbed magnetic field associated with an electrostatic wave. Equation (8.69) yields

Here, and , whereas the Cartesian components of and are written and , respectively. The equilibrium magnetic field takes the form . Equations (8.115) and (8.117) can be combined to give

After some tedious analysis, the previous expression reduces to the so-called

For Maxwellian distribution functions of the form (8.79), we can explicitly perform the velocity-space integrals in the Harris dispersion relation to give

where , , , and . Here, the are modified Bessel functions (Abramowitz and Stegun 1965c), whereas is a plasma dispersion function. (See Section 8.4.) In deriving the previous expression, use has been made of the identity (Watson 1995)

Consider electrostatic waves propagating parallel to the equilibrium magnetic field. In this case, and , so the dispersion relation (8.120) reduces to

(8.122) |

with the eigenvector . (Recall that for an electrostatic wave.) It can be seen that this expression is identical to the dispersion relation (8.97) for longitudinal plasma waves. Consider electrostatic waves propagating perpendicular to the equilibrium magnetic field. In this case, and , so the dispersion relation (8.120) reduces to

with the eigenvector . Making use of the identity (8.121), as well as the fact that (Abramowitz and Stegun 1965c), the previous expression can be rearranged to give

It can be seen that this expression is identical to the dispersion relation (8.111) for Bernstein waves. Thus, we can now appreciate that plasma waves and Bernstein waves are merely different aspects of a more general type of electrostatic wave. This wave takes the form of a plasma wave when propagating parallel to the equilibrium magnetic field, of a Bernstein wave when propagating perpendicular to the magnetic field, and takes an intermediate form when propagating obliquely to the magnetic field.