The general dispersion equation for whistler-mode propagation, instability or damping in a non-relativistic plasma with the electron distribution function in the form (1.90) has already been derived in Section 3.1 (see equation (3.20)). Assuming, as in Sections 1.2 and 3.2, that whistler-mode growth or damping does not influence wave propagation we can simplify equation (3.20) to:

where N, ω and Y hereafter in this chapter are assumed to be real, the argument of the Z function is ξ1 = ξ = (1 Y)/Nῶ∥, ῶ∥ = w∥/c (cf. similar assumptions in Section 3.2),

(cf. the definition of the Z function by equation (1.21)), and Ae = (j + 1) w⊥2/w∥2 (when deriving (4.1) we have generalized equation (3.20) for arbitrary integer j).

As follows from the analysis of Chapter 3, the non-relativistic approximation and, in particular, equation (4.1) is valid in a relatively dense plasma when ν ≫ 1 and N2 ≫ 1, in general. Hence, the second term ‘1’ in equation (4.1) will either be neglected altogether or taken into account when calculating the perturbation of N2 due to non-zero ν− 1 (cf. equation (3.34)).

Although equation (4.1) is much simpler than the corresponding weakly relativistic dispersion equation (cf. equation (3.10)), it still has no analytical solution in general.